Atomic scale simulation of oxide and metal film growth

Atomic scale simulation of oxide and metal film growth

Ivan Lazić

Atomic scale simulation of oxide and metal film growth

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties in het openbaar te verdedigen op maandag 14 december 2009 om 12:30 uur door Ivan LAZIĆ Diplomirani Inženjer Elektrotehnike – Elektrotehnički Fakultet Univerziteta u Beogradu (Servië) geboren te Knjaževac, Servië

Dit proefschrift is goedgekeurd door de promotor: Prof. dr. B. J. Thijsse

Samenstelling promotiecommissie: Rector Magnificus, voorzitter Prof. dr. B. J. Thijsse, Prof. dr. ir. C. R. Kleijn, Prof. dr. G. C. A. M. Janssen, Prof. dr. J. Th. M. de Hosson, Prof. dr. A. Bogaerts, Dr. ir. W. G. Sloof, Dr. ir. M. H. F. Sluiter,

Technische Universiteit Delft, promotor Technische Universiteit Delft Technische Universiteit Delft Universiteit van Groningen Universiteit van Antwerpen Technische Universiteit Delft Technische Universiteit Delft

This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organization for Scientific Research (NWO). This research was carried under the project number 02EMM31 in the framework of the Research Program of the Materials innovation institute (M2i) (www. M2i.nl).

ISBN 978-90-77172-53-7 Copyright © 2009 by Ivan Lazić, All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means without the prior written permission of the copyright owner. Printed by CPI, Wöhrmann Print Service, Zutphen, The Netherlands

Mojim roditeljima (To my parents)

Content

1 Introduction…………………………………………………………..…………………………...1 1.1 Aim of the work………………………………………………………………………………..1 1.2 Al-O potential construction for self-repairing oxide growth…………………………………...2 1.3 Large scale MD simulation of the Cu film growth on Ta (100) bcc substrate…………………5 2 Theory and methods………………………………………………………………………………9 2.1 Molecular dynamics…………………………………………………………………………...9 2.2 Al/O simulations: The CTIP+MEAM potential……………………………………………...11 2.2.1 General formulation……………………………………………………….…..............12 2.2.2 Separation into electrostatic and non-electristatic parts……………………………....14 2.3 Electrostatic part: The CTIP potential…………………………………………………..……15 2.3.1 Particle-Particle-Particle-Mesh (PPPM) method……………………………………...15 2.3.2 Determination of charges – minimization of the electrostatic energy………………...17 2.3.3 Model improvements by Zhou, Wadley, Filhol, and Neurock………………………..21 2.3.4 Summary of the final equations……………………………………….........................23 2.3.5 Electrostatic forces …………………………………………………………………...24 2.4 Non-electrostatic part of the potential: the MEAM potential………………………………...25 2.4.1 Reference Free Modified Embedded Atom Method (RF-MEAM)…...........................25 2.5 Cu/Ta simulations: The EAM + Screened Coulomb potential……………………………….29 2.5.1 Ta-Ta potential…………………………………………………………………..……30 2.5.2 Cu-Cu potential…………………………………………………………………….....32 2.5.3 Cu-Ta potential……………………………………………………………………..…33 2.5.4 Molière-Firsov screened Coulomb potential……………………………………….....33 2.6 Fitting potentials to energy and electrostatic potential field data…………….........................34 2.7 Atoms and their local environment……………………………………………………..……37 3 General performance testing……………………………………………………………………41 3.1 Introduction……………………………………………………………...................................41 3.2 Numerical versus analytical model in the simple case of NaCl……........................................42 3.3 Results of the performance testing…………………………………………............................46 3.4 Discussion…………………………………………………………..…....................................49 3.5 Conclusions…………………………………………………………………………………...50

vii

4 An improved molecular dynamics potential for studying aluminum oxidation Part I – Parameter optimization for the electrostatic part of the potential………………….51 4.1 Introduction…………………………………………………………………………………...52 4.2 The MEAM+CTIP model……………………………………………………………………..53 4.3 Original ZWFN potential and ab initio data…………………………………………………..56 4.4 Novel fitting approach for the Al/O potential – electrostatic part…………………………….59 4.4.1 Fitting procedure………………………………………………………………….......60 4.4.1.1 Crystal structures for fitting………………………………………………....60 4.4.1.2 Details of the fitting procedure……………………………………………...61 4.5 Results and discussion………………………………………………………………………...62 4.6 Summary and conclusions…………………………………………………………………….67 5 An improved molecular dynamics potential for studying aluminum oxidation. Part II – Parameter optimization for the non-electroststic part of the potential……………71 5.1 Introduction…………………………………………………………………………………...71 5.2 Reference Free Modified Embedded Atom Method (RF-MEAM)…………………………...72 5.3 The electrostatic part of the potential…………………………………………………………75 5.4 Fitting the potential to ab initio Al/O data and elastic constants……………………………..76 5.4.1 Potential energy curves…………………………………………………………….....76 5.4.2 Elastic constants……………………………………………………………………....78 5.5 Fitting results………………………………………………………………………………….79 5.6 Oxide stability and beginning of oxide growth results………………………………………..83 5.7 Conclusions…………………………………………………………………………………...86 6 Microstructure of Cu film grown on Ta by large-scale MD simulation……………………...89 6.1 Introduction…………………………………………………………………………………...89 6.2 Computational………………………………………………………………………………...90 6.3 Local symmetry type………………………………………………………………………….92 6.4 Results: Cu film after deposition……………………………………………………………...94 6.4.1 Plane by plane description…………………………………………………………….95 6.4.2 Atomic-scale descriptions………….…………………………………………………99 6.5 Changes in the film during deposition……………………………………………………….104 6.6 Annealing……………………………………………………………………………………106 6.7 Conclusions………………………………………………………………………………….108 7 Summary and conclusions……………………………………………………………………..111 Samenvatting en conclusies………………………………………………………………….......115 Acknowledgements……………………………………………………………………………….119 Curriculum Vitae ………………………………………………………………………………...123 viii

Chapter 1 Introduction ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

1. Introduction

1.1 Aim of the work It has been recognized long ago that molecular dynamics (MD), with its atomic resolution and full dynamical evolution monitoring, is a powerful method for materials modeling. MD could even be considered the ultimate tool, as was pointed out in [1], especially since nonequilibrium conditions are often critical in the synthesis of materials. Nowadays quantum-level or ab initio calculations based on density functional theory (DFT) [2], have become feasible to such an extent that the design and verification of the classical potentials needed for MD have improved spectacularly over the past years. It is the combination of classical MD with DFT that now leads the way and is the focus of the work presented in this thesis. In materials design true understanding begins at the atomic level. It is the concerted behavior of atoms that drives the mesoscale behavior of materials and ultimately determines the required properties. Considerable part of the work here is thus dedicated to this topic. Traditionally, microstructure-properties design has taken place through many experimental iteration cycles. However, these are often extremely costly and time consuming, especially when extreme conditions, rare elements, or specialized treatments should be explored. Moreover, experiments do not always give access to the necessary atomic-scale information. Fortunately, the ongoing rapid growth of computer power and materials theories has now come to a point where a viable additional route is offered by computational modeling [3]. Computational Material Science is one of the fastest growing fields in the materials discipline and can be safely considered as a new key enabling technique for materials design. Designing a material - and especially its microstructure - on the basis of a required property or combination of properties is a grand challenge. What is needed is a thorough understanding of the atomic processes that control the properties of a material and its evolution in time, as well as devising a way to translate this understanding to larger time and space scales and into workable production processes. 1

Chapter 1 Introduction –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– The work presented in this thesis consists of two essentially different parts. The first part is the development of novel potentials and strategies for the MD modeling of aluminum oxide coatings, and it is firmly supported by ab initio methods. The purpose of this part is to understand, and ultimately to exploit in oxide design, the atomic-scale mechanisms of the oxidation of aluminum. The rapid re-oxidation after sustaining mechanical damage makes aluminum a prototype of a self-healing material. The second part of the work is the MD modeling of the deposition of a Cu film on a Ta (100) bcc surface. Here the aim is to understand how exactly the junction of two very dissimilar metals leads to the evolution of a complex microstructure when one metal is deposited on top of the other. The application of this part lies in the interconnect technology in current electronic chip production.

1.2 Al-O potential construction for self-repairing oxide growth Metal-oxide coatings on metal substrates in an oxygen environment are natural self-healing systems [4]. This property is interesting for many applications such as in the mechanical engineering environment to produce longer lasting, better wear resistant cutting tools or to improve the tribological properties of materials in contact. In order to optimize the wear and friction behavior of such coatings, the properties of the oxide film formed at the surface of the tool are important. Hence, much of the composition and microstructure design of these materials is focused on the formation of an oxide film at the surface. In the context of studying self-healing oxides on metals, aluminum is the system of choice, since this metal oxidizes practically “immediately” after being exposed to air. This means that the early part of the process, where already considerable film growth takes place, stays largely out of scope of experimental techniques [5]. Computational methods such as MD and ab initio methods are therefore excellent tools to help filling in the gap formed by this part of the oxidation mechanism. In a comprehensive, full-scale research program several questions are to be addressed, such as how the first O2 molecules dissociate on the metal surface, how deep the initial O atoms diffuse into the metal, which type of oxide is built up across the interface, how the diffusional Al and O atom fluxes are depending on the local composition and structure (ultimately leading to an arrest of the oxide growth), and which atomic elements in the aluminum could be used to control the speeds of these fluxes. In the current work we will set up a general framework and concentrate on a subset of these questions. First principle techniques alone are not yet powerful enough to answer just about any of these questions, primarily because too many atoms or too much time would be needed. Monte Carlo techniques suffer from other limitations, such as in some cases their inability to model nonequilibrium phenomena and in other cases their restriction to crystalline systems with a limited number of microscopic atomic transition probabilities that should be known in advance. Consequently, MD is the only technique left to proceed with these questions at this length scale.

2


a)

b)

c)

d)

Fig 1. Schematic illustration of a “self-healing” oxide coating, showing recovery of the protective oxide layer after damaging the surface by scratching. a) Sample with initial oxide capping layer. b) Damaged area in the oxide layer, where oxygen molecules come in contact with the metal surface; dissociation of the oxygen molecules into atoms is not shown. c) Re-growth of oxide layer requiring metal atoms to diffuse upwards. d) Metal/oxide system, in this case aluminum-oxygen, shown with the oxide surface, the oxide/metal interface, and the metal surface. The charges q on the atoms are zero in the gas and metal phases, close to the valence values in the oxide, and a function of the local atomic coordination, schematically f(CN), in the interface regions.

Given reliable potentials to describe the atomic interactions in the system under study, MD is a powerful technique. However, it is exactly this very issue of reliable potentials that makes the current work on metal oxidation a challenging enterprise. The reasons for this are illustrated in Fig. 1. The pictures schematically show a metal/oxide system in an environment containing oxygen. After suffering damage at the surface, the metal re-oxidizes and forms the protective layer that stops growing at a certain thickness (the self-healing effect). What we see is a sequence of events in which the local metal-oxygen ratio varies, both in time and in space. This presence of oxygen is a crucial difference with purely metallic or metal alloy systems, and it is the principal reason why developing reliable potentials for the current study is not an easy task. The physical interaction between oxygen and metal atoms is the long-ranged ionic Coulomb interaction, which is the result of charge transfer between the oxygen and metal atoms, as they effectively ionize each other when being in each other’s vicinity. On the other hand, in the pure metallic part of the system there is no ionization. What is needed for the MD simulations, therefore, is a potential that lets the atoms respond to the local composition and adjusts the charges on the atoms dynamically. In addition, the calculation of the long-range forces between the ionized atoms themselves also adds to the 3

Chapter 1 Introduction –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– complexity of the potential, compared with the purely metallic case, which is essentially shortranged. And thirdly, the interaction potential should be able to deal with the relatively open crystal structures that are generally found in oxides, which brings up extra requirements for the nonelectrostatic part of the potential, notably angular dependent terms. Without exaggeration it can be said that such a potential belongs to the most complicated potentials ever developed for MD simulations of materials behavior. This work will show how new methods and techniques are used to construct such a potential for the Al-O system. Other metal-oxygen systems, or for example semiconductor-oxygen systems, would profit from treatment by the same approach. This work is not the first MD work on aluminum oxidation or on charge-transfer potentials. In 1994 Streitz and Mintmire [6] introduced a computational method that explicitly includes variable charge transfer between anions and cations. Based on this method a large simulation of the oxidation of a spherical Al nanocluster was performed [7]. Nevertheless, the Streitz-Mintmire scheme had a few flaws, and the potential was improved by Zhou et al. [8], who constructed probably the most advanced Al-O potentials available so far. However, although it is relatively successful in some applications [9, 10], we will show in this thesis that the potential has serious shortcomings, in particular: (1) a much too small charge on O atoms in configurations that represent the beginning of oxidation, (2) a highly incorrect O-O interaction within the oxygen molecule, and (3) the existence of crystal structures with a lower minimum energy than the most stable α-Al2O3 phase. In recent papers considering these topics by Elsener et al. [11, 12], as well as in [8], charges and minimum energies for various Al-O systems were either not discussed as possible sources of error or not mentioned at all. Maybe the reason is that these models were not specifically constructed to describe the beginning of oxidation, so that the requirements could be somewhat relieved. However, here we take a different viewpoint, which is the following. If it would be possible that the atomic charges and the energies of different local aluminum-oxygen environments are incorrectly described by the Al-O potential in use, the MD simulations of oxidation using this potential should not be trusted. The oxidation process is simply too complicated to allow this type of uncertainty. We therefore construct our new potential on a much more rigorous basis. In the first part of the thesis (Ch. 2) a full description of the Al-O potential is given, along with the Particle-Particle-Particle-Mesh method (PPPM), used as long-range Coulomb interaction solver. The potential consists of the Charge Transfer Ionic Potential (CTIP) proposed in [8] in combination with a new Modified Embedded Atom Method potential, the so-called Reference Free Modified Embedded Atom Method (RF-MEAM) [13]. All this has been implemented as Fortran code and was integrated into the MD code camelion [14]. Also in Ch. 2 a summary is presented of the statistical method used for fitting the potential to electrostatic field and configurational energy data calculated by DFT for various aluminum-oxygen structures. The accuracy of the constructed potential model needs validation. Thus, it is tested against the analytical results attainable for simple ionic crystal systems such as NaCl (Ch. 3). Control of phase stability by adjusting parameters of the RF-MEAM part is demonstrated to work well. Ch. 3 has a general character and does not point to any particular system of interest.

4

Chapter 1 Introduction –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Building up the new potential requires a new approach to parameterization. We overcome the problem of using “assumed” charges as fitting targets [8] by treating the charge transfer part separately from the non-electrostatic part and fitting it to the electrostatic potential fields calculated by VASP (Vienna Ab-initio Simulation Package) [15-18] for different Al-O structures (Ch. 4). This approach is similar to the one followed in [19]. The final results show much improvement in the charge values for all the structures included, especially those representing the beginning of the oxidation process. The previous Al-O potential [8] uses an Embedded Atom Method (EAM) potential as its non-electrostatic part. After extending this part to a RF-MEAM potential, which has angledependent terms included, we fit it to the potential energy data calculated by VASP (Ch. 5). As VASP calculates total energies not including the “non-electrostatic” and “electrostatic” energies as separate contributions, the purely electrostatic energy has to be subtracted from the VASP energy first. This purely electrostatic part, in turn, is calculated on the basis of the charge model of which the parameters were determined by the earlier fit to the electrostatic potential fields. Finally, to ensure the stability of the crystal structures of interest, also elastic constants were included in the fit. Several deformed crystals were used for this. Each deformation required separation of electrostatic and non-electrostatic parts. We show that the CTIP+RF-MEAM potential that ultimately emerges from this multistep fitting method fits the large number of target values very well.

1.3 Large scale MD simulation of the Cu film growth on a Ta (100) bcc substrate Metal thin film growth on metallic substrates is of great technological and fundamental interest. Technologically, Cu has excellent electronic properties, but to apply it on submicron scale one needs to address the problem of interdiffusion. In modern IC technology Ta is often applied as a diffusion barrier material between copper, the metallic interconnect material, and the oxide used as insulator. Adhesion and the microstructure of the subsequent Cu fill are affected by the microstructure of the primary deposited Cu film on the Ta diffusion barrier layer. Electromigration resistance of filled Cu is strongly dependent on the interface area (orientation, boundaries) and on the associated diffusion mechanism (volume or grain boundary diffusion). This will lead to specific phenomena on the atomic scale upon further miniaturization of the devices. Fundamental understanding of Cu thin films growth on Ta substrates is important because Cu/Ta is a strongly heterogeneous film/substrate system, exhibiting considerable differences in atomic diameters, shear moduli, cohesion energies, etc, often larger than a factor of two. Using MD simulations we provide new insights in the formation and structure of Cu films grown on Ta substrates. Two aspects were investigated in this work: the microstructure and the wetting/dewetting behavior. Earlier, classical MD results of Cu deposition on bcc-Ta (100) and (110) [20] and on β-Ta (001) [21] were reported. These studies have provided new information, but the simulations were carried out for Ta substrates of only modest size. There was a clear desire to repeat the simulations under 5

Chapter 1 Introduction –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– conditions that would suppress unwanted periodic boundary effects. Also surface wetting and dewetting in the Cu/Ta system was reported very recently in atomistic simulations by Hashibon et al. [22]. This and the experimental investigations of Cu film growth on Ta carried out by Venugopal et al. [23] called for a large-scale MD study of Cu deposition on Ta as well.

10 nm

Fig 2. The third Cu layer deposited on a Ta bcc (100) surface, shown with each atom colored according to the local crystallography type. In the arrow-shaped area in the middle the atoms are actually in the second Cu layer, because the Ta substrate underneath was raised by a monolayer Ta island. The inserted figure in the upper left corner is the third Cu layer after deposition on a much smaller system [20]. The scale indicated is the same for both figures.

In Ch. 6 we present the results of atom-by-atom Cu film deposition on a 100 nm × 100 nm bcc-Ta (100) substrate and analyze with atomic scale precision the interface organization and the microstructural development of the growing film. The substrate was specially constructed to enable an additional study of step-edge effects in the same simulation run. We show all the atomic details provided by the MD model used, yet the system has almost mesoscale dimensions. Formation and 6

Chapter 1 Introduction –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– growth of the interface, the texture, the grains, as well as defect formation and evolution, are observed and explained. As an illustration, a comparison of the current system with respect to the much smaller system studied in [20] is given in Fig. 2. Clearly, a number of phenomena emerge only in the larger system, in particular different grains and grain orientations, grain boundaries of various structures, stacking faults, and various relaxation effects. Apart from providing such detailed information in its own right, this result should also be seen as a warning against performing simulations in too small systems. The Cu on Ta deposition simulations were carried out using a parallelized version of our MD code camelion [14], employing the EAM potential [24] described in Ch. 2. For visualization of the local crystallography in Fig. 2, a method based on spherical harmonics analysis of the interatomic direction vectors was used. This method is explained in Ch. 2. After deposition of the film, simulations were continued at elevated temperatures, in order to investigate the wetting/deweting behavior. Ch. 6 includes the results of these simulations.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

F. J. Humphreys, M. Hatherly, Recrystallization and Related Annealing Phenomena, 2nd edition, Elsevier Ltd, Oxford (2004). E. Kaxiras, Atomic and Electronic Structure of Solids, Cambridge University Press, Cambridge (2003). S. Yip (Ed.), Handbook of materials modelling, Springer (2005). S. van der Zwaag (Ed.), Self Healing Materials, Springer (2008). L. P. H. Jeurgens, W. G. Sloof, F. D. Tichelaar, E. J. Mittemeijer, J. Appl. Phys. 92, 1649 (2002). F. H. Streitz, J. W. Mintmire, Phys. Rev. B 50, 11996 (1994). T. Campbell, R. K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 82, 4866 (1999). X. W. Zhou, H. N. G Wadley, J.-S. Filhol and M.N. Neurock, Phys. Rev. B 69, 035402 (2004). X. W. Zhou, H. N. G. Wadley, J. Phys.: Condens. Matter 17, 3619 (2005). X. W. Zhou, H. N. G. Wadley, D. X. Wang, Comput. Mater. Sci. 39, 794 (2007). A. Elsener, O. Politano, P. M. Derlet, H. Van Swygenhoven, Model. Simul. Mater. Sci. Eng. 16, 025006 (2008). A. Elsener, O. Politano, P. M. Derlet, H. Van Swygenhoven, Acta Mater. 57, 1988 (2009). I. Lazić, B. J. Thijsse, Proc. 4th International Conference on Multiscale Materials Modeling MMM, Ed. A. ElAzab, Department of Scientific Computing, Florida State University, Tallahassee, USA, 454 (2008). http://web.mac.com/barend.thijsse/Site/Home.html (April 2009). G. Kresse, J. Hafner, Phys. Rev. B 47, RC558 (1993). G. Kresse, J. Furthmüller, Phys. Rev. B 54, 11169 (1996). G. Kresse, J. Furthmüller, Comp. Mat. Sci. 6, 15 (1996). G. Kresse, D. Joubert, Phys. Rev. B 59, 1758 (1999). E. Bourasseau, J.B. Millet, L. Mondelain and P.M. Anglade, Mol. Simulat. 31, 705 (2005). T. P. C. Klaver, B. J. Thijsse, J. Comput-Aided Mater. Des. 10, 61 (2003). T. P. C. Klaver, B. J. Thijsse, Mat. Res. Soc. Symp. Proc. 721, 37 (2002). A. Hashibon, A. Y. Lozovoi, Y. Mishin, C. Elsässer, P.Gumbsch, Phys. Rev. B 77, 094131 (2008). V. Venugopal, B. J. Thijsse, Thin Solid Films 517, 5482 (2009). T. P. C. Klaver, Ph.D. Thesis: Atomistic simulation of Cu/Ta thin film deposition and other phenomena, DUP Science, Delft University of Technology, The Netherlands (2004).

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8

Chapter 2 Theory and Methods –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

2. Theory and Methods

2.1 Molecular dynamics Atomistic simulations, in particular classical molecular dynamics (MD) simulations, offer the possibility to study the static and dynamic properties of matter with sub-nanometer and subpicosecond resolution. This implies that details of materials behavior can be studied at levels that are inaccessible by experiment. Also, the conditions under which the atoms are acting can be freely chosen by the user, almost without limitation. It is here where the principal powers of molecular dynamics simulations lie, since complicated processes can be disentangled into elementary ingredients, which in turn allows new insights to be obtained. On larger scales, for example those of interfaces, grain boundaries, grains, diffusion, microstructure formation, and film deposition, modern computers now offer access to spatial and temporal scales not far from micrometers and microseconds, at least if the atomic interaction models are not too complex. The main limitations of MD simulations are that quantum and most electronic properties of matter cannot be studied (although ionization of atoms is treated in this thesis), and that the results depend critically on the potentials that are used to describe the interactions between the atoms. In this thesis two types of potentials are used, one for the aluminum/oxygen system and one for the copper/tantalum system. For the aluminum/oxygen system a complicated interaction scheme is needed, because electronic charge is exchanged between aluminum and oxygen atoms when the structural and chemical composition of their immediate surroundings change. In other words, the potential should be able to describe local ionization. Also, the relatively open structure of oxides requires a potential that contains explicit angular terms. These are necessary to stabilize open structures with respect to more closed packed structures. Therefore as full interaction model a Charge Transfer Ionic Potential (CTIP) was chosen, in combination with a Modified Embedded Atom Method (MEAM) potential. They are described in Secs. 2.2 to 2.4. For the copper/tantalum system a much simpler interaction scheme is permissible, since this system is fully metallic and therefore has no net atomic charges. Here a combination of an Embedded Atom Method (EAM) 9

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– potential and a screened Coulomb potential was chosen. These potentials are described in Sec. 2.5. In Sec. 2.6 the numerical method employed to fit the CTIP and MEAM potentials to structural energies is briefly outlined. The calculation of these energies themselves, by electronic structure computations using Density Functional Theory, and the results of the potential fitting are the subjects of Ch. 4 and Ch. 5. Finally, Sec. 2.7 of this chapter contains a description of a powerful numerical representation of the local crystal symmetry of atoms. Classical molecular dynamics belongs to the “simulation using particles” type of methods [1]. It is based on the integration of the Newton equations of classical mechanics, Fi = mi a i = mi d 2 ri / dt 2 ,

(1)

where Fi is the force on atom i (due to all other atoms in the system), mi its mass, ai its acceleration, ri its position, and t is time. The force Fi is determined by the total potential energy U of the system, which is a function of all N atomic positions, U = U(r1,r2 ,r3 ,...,rN ) , Fi = !" ri U(r1 , r2 , r3 ,K, rN ) = !(#U / #xi , #U / #yi , #U / #zi ) . !

(2)

Several numerical integration methods of the N coupled equations (1) have been developed [2, 3], but one of the most accepted methods, used in this work, too, is the Velocity Verlet algorithm. It belongs to the group of symplectic integrators [4, 5] and it is defined in the next four equations, which describe in succession how the position, velocity, and acceleration of each atom evolve from time t to time t + "t ,

!

!

1 (3) r(t + "t) = r(t) + v(t)"t + a(t)"t 2 , 2 1 1 (4) v(t + "t) = v(t) + a(t)"t , 2 2 (5) a(t + "t) = F(t + "t) /m , 1 1 (6) v(t + !t) = v(t + !t) + a(t + !t)!t . 2 2

! ! Here, v is velocity, Δt is the integration timestep (order fs), and all subscripts i have been omitted for clarity. Note that Eq. (5) relies on Eq. (2) for evaluation. It is, by an overwhelming margin, the most time consuming step of the algorithm. For the CTIP potential also the charge of each atom has to be updated from time t to time t + "t . This is treated separately in Sec. 2.3.2. The MD simulations have been performed by the program camelion [6], developed in the Virtual ! Materials Lab at Delft University of Technology. For the work on Al/O, described in this thesis, the camelion code was extended by a large new part implementing the routines to calculate the CTIP potential energies and forces, described in Sec. 2.3. Camelion therefore is now able to handle ionic systems in a most general way, a feature hitherto unavailable. 10

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Apart from the ionic part, camelion is a fairly standard molecular dynamics code, based on the MEAM potential formalism, including angular screening [7], see Sec. 2.4. Of course MEAM also includes the simpler EAM potentials and pair potentials. The constitutive functions of the potentials are supplied to camelion in the form of external text files containing numerical tables, which makes the program rather flexible. Periodic boundaries can be applied to any combination of the three cartesian directions. Atoms can be automatically introduced into the simulation box under any angle with a selectable kinetic energy and are allowed to leave the system. This makes it possible to model film growth, gas-solid interactions, and bombardment of surfaces. The timestep Δt is variable, dynamically determined in such a way that in every timestep no atom is predicted to move more than a specified distance Δrmax. Hence, Δt is solved from # & 1 max $vi2 (!t)2 + (a i " v i )(!t)3 + ai2 (!t)4 ' < (!rmax )2 . i % ( 4

(7)

The distance prediction includes the acceleration because in certain situations this part far outweighs the velocity part. In the present simulations, Δrmax = 0.02 Å was chosen. Temperature is controlled through a Berendsen thermostat [8], typically with a 0.02 ps time constant, while a Berendsen barostat [8] is available for pressure control. Any number of atoms can be anchored (i.e., bound) to user-selectable equilibrium positions by a harmonic force. This prevents systems from drifting away by the momentum of incoming atoms, yet it interferes less with the natural evolution of the system than rigidly fixated atoms. In the Cu film growth simulations described in this thesis the tantalum atoms in the bottom monolayer were held anchored to bcc-Ta (001) lattice positions with a force constant of 19.4 N/m. Configurations produced by camelion are stored in text files, which are used as input by various analysis and visualization programs. For the MEAM family of potentials, which are relatively short-ranged, radial cutoff distances are applied. This makes it possible to employ speed increasing program devices, such as cell neighbor maps, atom neighbor lists, and triplet lists – all part of camelion. As just mentioned, in this work long-range Coulomb interactions are included, and we have implemented them by other, specialized methods. These are explained in Sec. 2.3.

2.2 Al/O simulations: The CTIP + MEAM potential In this section all expressions pertaining to a potential for atoms in conditions that lead to interatomic ionization (charge transfer) are presented. It will be shown that a split-up of such a potential into two parts, electrostatic and non-electrostatic, is a logical consequence of the general formulation of the energetics. The electrostatic part is based on the Charge Transfer Ionic Potential (CTIP), which in turn is rooted in the electronegativity equilibrization model originally proposed in [9] and later improved in [10]. The non-electrostatic part is a relatively new version [11] of the MEAM.

11

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 2.2.1 General formulation The total potential energy of an ionizable N-atom system is given by N

U = " E i (qi ) + i=1

1 N N " "Vij (rij ,qi,q j ) , 2 i=1 j=1

(8)

j#i

where

1 E i (qi ) = E i (0) + " iqi + J iqi2 , 2

!

(9)

is the energy of an isolated ionizable atom i with a charge qi , expressed to second order. Here, E i (0) is the energy of atom ! i in the unionized state (but see later for additional energy terms in a more refined model), " i is its electronegativity, and J i ( > 0 ) is referenced to as its “atomic hardness” or ,qi ,q j ) is the Coulomb interaction ! self-Coulomb repulsion [12]. The quantity Vij ( rij! energy between atom i with a charge qi and atom j with a charge q j , separated by a distance rij . According to how ! ! Vij ( rij ,qi ,q j ) is chosen, different models for ionized systems can be established. For example, in the ! simplest case, a point-charge atomic model, the Coulomb interaction has the form ! ! ! !

Vij ( rij ,qi ,q j ) = k c

qiq j , rij

(10)

with !

kc =

1 , 4 !" 0

(11)

where ε0 is the vacuum permittivity. In the Streitz-Mintmire model a more realistic atomic shape is assumed [9]. The atomic charge is described in terms of point-like, positively charged nucleus (charge +Zi), and an electron density distribution of charge (–Zi + qi) around the atom. In the other words, the point-like net charge qi of Eq. (10) is replaced by an electron density distribution function " i (r,qi ) around atom i. Here, spherically symmetric orbitals are assumed for simplicity. The distribution function then becomes radial, " i (r,qi ) , with r the distance from the core of atom i, ! (12) " i (r,qi ) = Z i# (r) + (qi $ Z i ) f i (r) , ! where " (r) is the delta function, and Z i is treated as a potential parameter for atoms of chemical ! type i. The radial distribution must satisfy the condition " f i (r)dV = 1, so the whole distribution

!

V

! satisfies

# " (r,q )dV = q . Note ! that, although Zi is a real charge, it does not contribute to the net i

i

i

V

! 12 !

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– charge of atom i. With charge distributed as in Eq. (12), the electrostatic interaction Vij ( rij ,qi ,q j ) between two atoms i and j becomes Vij (rij ,qi ,q j ) = k c

##

Vi V j

" i (ri ,qi ) " j (rj ,q j ) dVi dV j , rvv

(13)

!

where dVi and dV j are the two integrating volume units, ri is the center distance between atom i ! the center distance between atom j and dV j , and rvv is the center distance between and dVi , rj is ! ! !

dVi and dV j . Here the approximation is made that the charge distributions " i (r,qi ) are fixed, i.e., ! ! no polarization effects are taken into account. ! ! Substituting Eq. (12) into Eq. (13) leads to ! !

! Vij (rij ,qi ,q j ) = k c qiq j [ f i | f j ] +

kc qi Z j ([ j | f i ] " [ f i | f j ]) + kc q j Z i ([i | f j ] " [ f i | f j ]) kc Z i Z j ([ f i | f j ] " [i | f j ] " [ j | f i ] +

(14)

1 ) , rij

where the notations [a | f b ] and [ f a | f b ] (a=i,j, b=i,j, a≠b) denote the Coulomb interaction ! integrals f (r ) (15) [a | f b ] = " b b dVb , rav ! ! Vb where rav is the center distance between atom a and elementary volume dVb and ! f (r ) f (r ) (16) [ f a | f b ] = " " a a b b dVa dVb . rvv Va Vb At sufficiently large distance r from the core of an atom, the electron density distribution f i (r) decreases with r. !For reasons of mathematical simplicity, f i (r) was chosen to be a simple exponential, ! "3 (17) f i (r) = i e$2" i r , # ! where the parameter " i controls the spread of the electron distribution around atoms of chemical ! the integrals in Eqs. (15) and (16) can be analytically determined. Eq. type i. With this form of f i (r) (15) yields ! !

[a | f b ] =

1 1 " # b e"2# b rab " e"2# b rab . rab rab

For Eq. (16) there are two cases. If " a = " b = " one obtains

!

13

!

(18)


[ fa | fb ] =

1 1 11 3 1 " (1+ #rab + # 2 rab2 + # 3 rab3 )e"2#rab , rab rab 8 4 6

(19)

while if " a # " b the result is

! [ fa | fb ] =

!

' "2# r 1 $ # a# b4 3# a2# b4 " # b6 "& + e a ab " 2 2 3 3) rab % (# a + # b ) (# a " # b ) rab (# a + # b ) (# a " # b ) ( . $ ' "2# r # b# a4 3# b2# a4 " # a6 + & )e b ab 2 2 rab (# b + # a ) 3 (# b " # a ) 3 ( % (# b + # a ) (# b " # a )

(20)

With this we have obtained all analytical results necessary for calculation of the electrostatic ! (Coulomb) part of Eq. (8). For each chemical element in the system four potential parameters are needed: ! , J, Z, " . However, for the first (or only) element in the system, we can set χ = 0 without loss of generality. This means that in a binary system seven parameters define the Streitz-Mintmire charge transfer model. A problem that we have not yet addressed and which is unavoidable in infinite (or periodic) systems like crystals, is the conditional convergence of the summation 1

!!± r i

j

,

(21)

ij

which, as becomes clear when Eq. (14) is inspected, is the essential part of the second term of Eq. (8). We evaluate this sum by using the particle-particle-particle-mesh (PPPM) method, see Sec. 2.3.1. First the electrostatic and non-electrostatic parts of the potential must be accurately defined. 2.2.2 Separation into electrostatic and non-electrostatic parts Substituting Eqs. (9) and (14) into Eq. (8) and rearranging terms, we obtain U = U nes + U es ,

(22)

with ! N

U nes = " E i (0) + i=1

1 N N 1 k c Z i Z j ([ f i | f j ] # [i | f j ] # [ j | f i ] + ) , " " 2 i=1 j=1 rij j$i

!

14

(23)

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– N 1 U es = # ( " iqi + J iqi2 ) + 2 i=1

1 N N # # (kcqi Z j ([ j | f i ] $ [ f i | f j ]) + kcq j Z i ([i | f j ] $ [ f i | f j ])) + 2 i=1 j=1

(24)

j%i N

N 1 # # kcqiq j [ f i | f j ] 2 i=1 j=1 j%i

As we will see, this split-up of the energy into a non-electrostatic (Eq. 23) and an electrostatic part ! is very convenient. Eq. (23) does not depend of any of the charges qi . Also, in the factor (Eq. 24) 1 1 , so that they cancel each other, and the net ([ f i | f j ] " [i | f j ] " [ j | f i ] + ) all terms include rij rij ! Zj are real charges but they expression is short-ranged. We emphasize once more that Zi and

!

describe the charge distribution in the neutral atoms and do not refer to the net charge if the atoms become ionized. Having derived the formal ! expression Eq. (23) for the non-electrostatic energy, we should next choose a potential that covers the full expression. In this work we will use the MEAM potential for this and present its implementation in detail in Sec. 2.4. First we deal with the electrostatic part, Eq. (24).

2.3 Electrostatic part: The CTIP potential If we rewrite Eq. (24) as follows, N 1 1 N N U es = # ( " iqi + J iqi2 ) + # # ( k c qi Z j ([ j | f i ] $ [ f i | f j ]) + k c q j Z i ([i | f j ] $ [ f i | f j ])) + 2 2 i=1 j=1 i=1 j%i N

1 1 1 N N qq kc qiq j ([ f i | f j ] $ ) + # # k c i j # # 2 i=1 j=1 rij 2 i=1 j=1 rij

(25)

N

j%i

,

j%i

then only the last term is long-ranged. It has now the standard Coulomb form and the PPPM method !can be applied directly. As a reminder, in all other summation terms of Eq. (25) the factors 1/rij

cancel each other. 2.3.1 Particle-Particle-Particle-Mesh (PPPM) method In this section we present here the PPPM method, well known in the literature [1], adjusted and applied to our specific problem. PPPM easily adapts to periodic and non-periodic boundary conditions, and, because of its localized computability, it is convenient for code parallelization. An additional reason to use it is that the computational efficiency of PPPM is O(N log N ) while, e.g., for the most commonly used Ewald Summation it is O(N 3/2 ) [13, 14]. A disadvantage of PPPM is 15

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– that the maximum error cannot be predicted analytically. That in our case this error is nevertheless small we will show in Ch. 3.

Fig. 1. Short-range part (“PP”) of the interaction between atom i and all other atoms that can be taken into account directly, showing the cutoff radius rdir. The mesh is not used here.

=

+

Fig. 2. The rest of the interaction (“PM”). The middle frame shows the interaction of i with the charges attributed to the mesh points. The Poisson equation is used here. The frame on the right shows the correction for self-interaction.

The basic idea of PPPM is explained in Figs. 1 and 2. The grids in the figures represent the mesh, which holds the charge distribution in the middle panel of Fig. 2. Assuming pointlike charges, the interaction between a charged atom i and all other charged atoms j separated by rij, including all periodic images, is divided into two parts by adding and subtracting a charge distribution ! j (r) around every atom j. (This purely computational charge distribution should not be confused with the realistic charge distributions introduced in Eq. (12).) The volume integral of this computational distribution must be equal to qj. The distribution is chosen to be spherically Gaussian, ! j (r) = q j" #3$ #3/2 exp(#r 2 / " 2 ) , and it can be proven that the interaction energy between point charge i and point-charged atom j minus its own charge distribution is qi q j erfc(rij / ! ) / rij . The complementary error function erfc ( rij /" ) = 1# erf ( rij /" ) decreases rapidly with distance, which makes this direct part of the interaction short-ranged, and a cutoff radius rdir can be introduced (Fig. 1), normally on the order of 5 Å. A value of one third of the cutoff radius for σ gave the best results. ! The rest of the interaction (Fig. 2) is calculated by placing point charge i in the electrostatic potential on the mesh, ϕmesh, which is obtained by solving the Poisson equation ! 2" mesh = #$ mesh / % 0 . Here, ! mesh is the total charge of all individual charge distributions ! j (r) plus i’s own charge distribution ρi(r), and the total charge is allocated to the mesh points in two steps: first, starting 16

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– from Dirac delta peaks, qi" (r # ri ) at the original atom positions, charges are assigned to the nearest eight mesh points; and second, from there the charges are distributed further through a diffusionlike algorithm. In this way ! mesh is the same for all atoms i and the Poisson equation needs to be ! For each atom, the spurious interaction between its point charge and its charge solved only once. distribution can then be determined analytically and subtracted (Fig. 2, right). In summary, one obtains N N %r ( qq qq # kc ri j = # kc ri j erfc'& $ij *) + qi (+ mesh (q) , + self (qi )) , (26) ij ij j=1 j=1 j"i

j"i

where ! self (qi ) = 2kc qi / " # is the self-electrostatic potential. The vector notation q indicates the ! set of all charges, q = (q1 , q2 ,..., qN ) . For solving Poisson equation the mesh size should be no more than one quarter of the cutoff radius rdir. If we next insert a factor ½ to compensate for double counting, the last term in Eq. (25) becomes $ rij ' 1 N 1 N N qi q j 1 N N qi q j "" kc r = 2 "" kc r erfc &% # )( + 2 " qi (* mesh (q) + * self (qi )) . 2 i=1 j =1 ij ij i=1 i=1 i=1 j !i

(27)

j !i

Rearranging Eq. (25) then leads to 1 N U es = $ ( 2 ! i qi + J i qi2 + qi (" mesh (q) # " self (qi ))) + 2 i=1 1 N N $$ ( kc qi Z j ([ j | fi ] # [ fi | f j ]) + kc q j Zi ([i | f j ] # [ fi | f j ])) + 2 i=1 j =1

(28)

j %i

N ' ' r ** 1 1 1 kc qi q j ))([ fi | f j ] # ) + erfc ) ij ,,, $ $ 2 i=1 j =1 rij rij ( & ++ ( N

.

j %i

This form of the electrostatic energy of the system is now exactly as it has been implemented in the camelion MD code. The first line contains the single-atom part and the long-range particle-mesh (PM) part, where a mesh must be used. The second two lines contain the short-range interactions, the particle-particle (PP) part. For this part, a reasonable cutoff radius rdir is defined, and the calculations are then performed using various standard MD tricks of the trade (cells, maps, neighbor lists, pair lists etc.) [15]. 2.3.2 Determination of charges – minimization of the electrostatic energy To determine the charge values on the atoms in the system it is necessary to minimize the total electrostatic energy, Eq. (24), with respect to the charges [9, 10]. The underlying idea is as follows. The appearance of charges on the atoms will increase the electrostatic energy because of the ionization of the atoms, Eq. (9), while at the same time the Coulomb interaction energy between the 17

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– atoms, Eq. (14), will decrease the electrostatic energy. The balance between these two will give the minimum energy, thereby defining the optimum state of the system (and the resulting charge set). The system itself is always assumed to be neutral, N

"q

i

=0 .

(29)

i=1

The minimization can in principle be done analytically [10] by matrix inversion [2]. In practice, within a MD code minimization ! is always done numerically, in our case using the Conjugate Gradient method [2], as we will show. N

The extra condition Qcond (q) = " qi = 0 requires a special approach of the procedure to i=1

minimize the electrostatic energy. There are three seemingly reasonable choices: (1) reducing the number of variables by one, i.e. directly applying the condition itself, for example for the last ! q = " N "1 q ; or (2) introducing a Lagrange multiplier and minimizing the charge in the q array: N

#

i=1

i

new system without constraints; or (3) using a Conjugate Gradient method and projecting the search directions for line minimization in q space onto a subspace in which the directions themselves ! preserve the charge neutrality (i.e. slightly modifying the Conjugate Gradient algorithm). In the following each of these approaches is discussed, explaining why the choice made in this work is the third one. ! (1) The problem with the first approach, the direct solution, is that it is very asymmetric. There is no objective reason why to choose one particular charge to be calculated directly using the condition of charge neutrality. Tests showed that results obtained in this way are not satisfactory. It turned out that the atomic charge calculated directly using the condition Qcond (q) is always subject to error agglomeration, and assumes unrealistically high values. This effect is extreme for systems that consist of many atoms. ! (2) The second approach of using a Lagrange multiplier and introducing the Lagrange function F(q, " ) = U es (q) # "Qcond (q) seems to be the natural one. However, this is only partly true, because during the minimization of the function F(q, " ) , which has now no constraints, U es (q) N

must be calculated for cases where condition Qcond (q) = " qi = 0 is not satisfied. This is not possible i=1 ! using PPPM if the computational box is periodic in all directions. Actually, to solve the Poisson ! ! equation numerically using finite differences, with periodic boundary conditions in all three ! satisfied [1]. Otherwise the solution does not converge. In directions, charge neutrality must be cases where at least one computational box direction is non-periodic this approach could be a choice. (3) For the calculations done in this work, the third variant, the projected Conjugate Gradient method is applied. It gives excellent results (Ch. 3) and has no other disadvantages than that one of the steps in Conjugate Gradient algorithm had to be slightly modified. We assume that the modification of original algorithm no longer guarantees an optimal search. Nevertheless, no unreasonably slow convergences were detected.

18

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Next, the algorithm as it is used in this work is presented. As just mentioned, it is basically the Conjugate-Gradient (CG) method [2]. For line minimization, the Brent algorithm (second derivatives not required) is used [2]. The following quantities need to be evaluated in each iteration step k of the method: - The partial derivatives of the electrostatic energy, "U es "qi , (i = 1.. N) . -

The minimization direction dk = (d1k ,d2k ,...,dNk ) in the full charge space. The CG method provides this, see the algorithm below. ! dimensional subspace that preserves The projected direction pk = ( p1k , p2k ,..., ! pNk ) in the (N–1) ! total charge,

! Because

"

N i=1

pik = dik ! (1 N )"

N j =1

d jk , (i = 1.. N) .

(30)

k pik = 0 now holds, moving ! inside charge space along direction p over a

distance " , thus reaching point qi (" ) = qik + "pik , (i = 1.. N) , will automatically maintain charge neutrality for the new charges qi (" ) as long as for the old charges ! ! !The function !

"

N

k i=1 i

q = 0 holds.

!

N N f k (" ) = #U es #" = $ (#!U es #qi )(#qi #" ) = $ (#U es #qi ) pik ,! "qi "# = pik , (31) 1 1

which is used in the line minimization in the Brent algorithm (see the algorithm, below).

!

! Algorithm

Step (1) The initial guess values for qi are denoted as qi0 , (i = 1.. N) and the initial minimization direction 0 0 according to CG is given by di = gi = "#U es #qi q

i

= q i0

, (i = 1.. N) , the steepest descent direction.

The initial projected direction pi0 is then given by Eq. (30). ! ! ! ! Step (2) ! For each iteration step ! ( k = 0,1, 2 ...), the value of α for which the above defined, Eq. (31), function

!

f k (" ) = f k (qk (" )) , qk (" ) = (q1k (" ),q2k (" ),...,qNk (" )) becomes zero, is determined by applying the Brent algorithm (line minimization along the given direction). This value of α is called α0. ! iteration step (k+1) we obtain: Then for the next k +1 k k - The new!charges: qi = qi (" 0 ) = qi + " 0 pi , (i = 1.. N) . N

k +1 k +1 k +1 - The new search direction, according to CG given by di = gi + (" gi 1

gik = "#U es #qi q = q k , (i = 1.. N) . ! i i ! - The new projected direction pik +1 , according to Eq. (30). !

!

19

!

!

"

N

1

gik ) pik , where

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Step (3) k Step (2) is repeated until ! 0 pi , (i = 1.. N) , the updates for qi , become sufficiently small.

End of algorithm !

!

Finally, we formulate the explicit expressions for the partial derivatives "U es /"qi and the function f k (" ) according to our energy model. We also rearrange them so that PPPM can be applied. Starting from Eq. (24) we find ! N N ! "U es (32) = # i + J iqi + & kc Z j ([ j | f i ] $ [ f i | f j ]) + & kc q j [ f i | f j ] . "qi j=1 j=1 j%i

j%i

Substitution of Eq. (32) into Eq. (31) yields ! N

N

N

f k (" ) = $ ( # i + J iqi (" )) pik + $ $ k c pik Z j ([ j | f i ] % [ f i | f j ]) + i=1 N

i=1 j=1 j&i

(33)

N

$ $ k p q (" )[ f c

k i

j

i

| f j] .

i=1 j=1 j&i

In the same way as Eq. (24) was rearranged using Eqs. (27) and (28) we now rearrange Eqs. (32) and (33), ! obtaining

"U es = # i + J iqi + $ mesh (q) % $ self (qi ) + "qi N ' ' r ** q 1 .)) kc Z j ([ j | f i ] % [ f i | f j ]) + kcq j ([ f i | f j ] % r ) + kc r j erfc)( &ij ,+,, , ij ij + j=1 (

(34)

j-i

and N

!

f k (" ) = & ( # i + J iqi (" ) + $ mesh (q) % $ self (qi )) pik + i=1

( ( rij ++ (35) pik q j 1 k k & &** kc pi Z j ([ j | f i ] % [ f i | f j ]) + kc pi q j (" )([ f i | f j ] % r ) + kc r erfc*) ' -,-- . ij ij , i=1 j=1 ) N

N

j.i

Note that Eqs. (28), (34) and (35) include two common terms: ([ j | f i ] " [ f i | f j ]) and ! ([ f | f ] "1/r ) . They depend only on distance between atoms r and parameters " , which are, i j ij ij i

!

!

20

!

!

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– once determined, constants. Values of these expressions could be calculated in advance and stored in a table to save computation time. This was not done in this work. 2.3.3 Model improvements by Zhou, Wadley, Filhol, and Neurock There are two important shortcomings in the Streitz-Mintmire model described above, and they have been improved by Zhou, Wadley, Filhol and Neurock [10]. In this work we adopt this improved model and refer to it as the ZWFN version of the CTIP potential. 1) The first shortcoming of the original potential is the average charge magnitude as a function of isotropic strain of a system. The charge instability that appears can be understood if we consider a simplified system consisting of a cation and an anion, each assumed to be a point charge. In this case the electrostatic energy of the system, using Eq. (24), becomes

U es = !1q1 +

1 1 qq J1q12 + ! 2 q2 + J 2 q22 + kc 1 2 . 2 2 rij

(36)

If the overall system is charge neutral, q2 = !q1 , Eq. (37) simplifies to

$J + J k ' U es = ( "1 # " 2 )q1 + & 1 2 # c )q12 . r12 ( % 2

(37)

This expression has a minimum, i.e. is a concave function of q1, only when (J1 + J 2 ) /2 " kc /r12 > 0 . For any set of ! model parameters J1 and J 2 there is always a critical value r12critical = 2k c /(J1 + J 2 ) below which (J1 + J 2 ) /2 " kc /r12 < 0 and Ues becomes a convex function. When the two ions happen ! to approach each other and cross this distance, U will always decrease as the magnitude of charge es

! ! positive feedback drives the increases. This causes an!even stronger electrostatic attraction, and this ions!closer, until unrealistic conditions develop. ! 2) The second shortcoming of the original potential is that it can only be used to study systems in which a single metal occurs. If more than one metallic element is present then metallic regions will not automatically maintain zero charges. To see this, we set the first derivative of the electrostatic energy (37) with respect to q1 to zero and solve for q1,

!

q1 =

"1 # " 2 . J1 + ! J 2 # 2k c /r12

(38)

It is obvious that q1 becomes zero only when "1 and " 2 are identical. In metallic alloys, however, neighboring atoms can!be different elements with different parameters χ. Consequently, the model may lead to non-zero charges in purely metallic regions, which is unphysical. ! ! ! 21

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Bounding the charges of each atom in the system between values that depend on chemical type, qmin,i < qi < qmax,i , turns out to solve both of these problems. For metals it is reasonable to set M M qmin = 0 , qmax = valence , and for oxygen qOmin = valence , qOmax = 0 . This idea is realized by implementing an extra term in Eq. (28). The extra term must not influence the equations that we have already derived if the charge on the atom is inside the bounds. If this is not the case, however, then see Fig. 1. According to ZWFN this extra ! it should increase the electrostatic ! energy drastically, ! term is

! !

N $$ q # q ' ' $ q #q ' U esextra = *"&&&&1# i min,i ))(qi # qmin,i ) 2 + &&1# max,i i ))(qi # qmax,i ) 2 )) , qi # qmin,i ( % qi # qmax,i ( i=1 %% (

!

(39)

with " a parameter controlling the desired energy penalty. Here we set " equal to 20 eV/e2. It can be! seen that after adding Eq. (39), Eq. (28) will stay quadratic, which is important for implementation of the Conjugate Gradient method. !

a)

b)

Fig 3. Contribution of the extra energy term (of a single atom) to the total system energy. Increasing " makes the bounds for charge excess harder. a) Metallic atom, positive valence. Dots schematically show atoms with charges outside and inside the bounds, see the main text. b) Oxygen or any other negative valence atom.

! Fig. 3 illustrates how the extra energy term solves both shortcomings of the Streit-Mintmire model. In the example of the first instability problem the extra term causes the critical minimal distance to become r12critical = 2k c /(J1 + J 2 + 4" ) . This means that by using the proper value for " we can make

this distance as small as we need. Solution of the second problem, i.e. having more than one type of metal in the system is shown in Fig. 1a. If we assume, for example, that one of the metal type atoms which would not increase !would get a positive charge (blue dot) during the minimization process, ! the energy, another atom should get a negative charge (red dot) because of charge neutrality. But since this would cost much energy, the minimization process would force its charge back to zero – and, because of charge neutrality, also the charge of the other atom. Eq. (39) gives rise to two more derived equations,

22

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– %% q $ q ( ( % q $q ( "U esextra = 2#''''1$ i min,i **(qi $ qmin,i ) + ''1$ max,i i **(qi $ qmax,i )** , "qi & qi $ qmax,i ) && qi $ qmin,i ) )

%% q $ q ( ( % q ( i min,i max,i $ qi ' (" ) = + 2#'''1$ **(qi $ qmin,i ) + ''1$ **(qi $ qmax,i )**di,kp , & qi $ qmax,i ) i=1 && qi $ qmin,i ) )

(40)

N

!

f

extra k

(41)

! In a final note we would like to pay attention to one other shortcoming of the model, namely its nonlocality. If we look at Eq. (38) and assume that in this simplified system we have two atoms with electronegativities of different signs (for example, not two metals), we see that even at infinite distance the atoms will get nonzero charges (of opposite sign). This would suggest that charge transfer has occurred, which is clearly unphysical, and such an infinite transfer range can be overcome by introducing cutoff radii for χ. In our case this is not done, but it was employed in [10], and it was concluded that it had only a minor influence.

2.3.4 Summary of the final equations Collecting everything that was presented in previous three sections (2.3.1-2.3.3), we list the final electrostatic equations for the camelion code for convenience.

U es =

## q " qmin,i 1 N 2! %%%%1 " i ) 2 i=1 $$ qi " qmin,i

& # q " qi (( (qi " qmin,i )2 + %%1 " max,i qi " qmax,i ' $

& & (( (qi " qmax,i )2 ( + ( ' '

1 N 2 * i qi + J i qi2 + qi (+ mesh (q) " + self (qi ))) + ( ) 2 i=1 1 N N )) ( kc qi Z j ([ j | fi ] " [ fi | f j ]) + kc q j Zi ([i | f j ] " [ fi | f j ])) + 2 i=1 j =1

(42)

j ,i

#1 # rij & 1 N N 1 & % k q q erfc + ([ f | f ] " )) c i j % r %$ - (' i j r )(( 2 i=1 j =1 $ ij ij ' j ,i

%% q $ q ( ( % q $q ( "U es = 2#''''1$ i min,i **(qi $ qmin,i ) + ''1$ max,i i **(qi $ qmax,i )** + "qi & qi $ qmax,i ) && qi $ qmin,i ) )

+ i + J iqi + , mesh (q) $ , self (qi ) +

(43)

% % rij (( qj 1 /'' kc Z j ([ j | f i ] $ [ f i | f j ]) + kcq j ([ f i | f j ] $ r ) + kc r erfc'& - *)** ij ij ) j=1 & N

j.i

! 23

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– N %% q $ q ( ( % q $q ( f k (" ) = + 2#''''1$ i min,i **(qi $ qmin,i ) + ''1$ max,i i **(qi $ qmax,i )**di,kp & qi $ qmax,i ) i=1 && qi $ qmin,i ) ) N

+ (,

+ J iqi + - mesh (q) $ - self (qi ))di,kp +

i

i=1 N

N

1

+ + 2 k (d c

i=1 j=1 j.i N

N

++ k d c

p i,k

i=1 j=1 j.i

p i,k

(44)

Z j ([ j | f i ] $ [ f i | f j ]) + d pj,k Z i ([i | f j ] $ [ f i | f j ]))

%1 %r ( 1 ( q j '' erfc' ij * + ([ f i | f j ] $ )** rij ) &/ ) & rij

! 2.3.5 Electrostatic forces In classical MD computer codes forces on atoms are normally calculated directly by evaluating the analytical expressions for the partial derivatives of the total interatomic potential with respect to three independent coordinates of each atom in 3D space. In this work this was done in the same way. Analytical expressions were derived and implemented for the forces of the non-electrostatic interaction as well as of the short-range part of the electrostatic interaction. The long-range electrostatic forces, covered by the mesh of the PPPM method, are evaluated as follows: first, the electrostatic field is numerically calculated in the mesh points; second, these values are interpolated back to the original positions of atoms; and third, the field values are multiplied by the charge on each atom. An interesting remaining point is the following: while the charges generally change as the atoms move, how does this position dependence affect the calculations of forces and stresses? We will repeat the derivation given in [10] and show here in a general way that this position dependence does not affect the force calculations at all. This is a very convenient feature of the CTIP model. As explained earlier, the minimum of the electrostatic part of the energy U es (q,r1,r2 ,...,rN ) with respect to charges q = (q1,q2 ,q3 ,...,qN ) on the atoms, under the condition

" ! !

N

q = 0 , determines these charges. For the current purpose we determine the minimum by

i=1 i

introducing

the

Lagrange multiplier and construct the function µ ! N U(q, µ) = U es (q) " µ# qi = U es (q) . We now search for the unconstrained minimum of this function i

and get

!

"U "U es = # µ = 0 or "qi "qi

!

"U es = µ , (i = 1...N) . "qi

(45)

The remaining equation "U /"µ = 0 is always fulfilled in view of the charge neutrality condition ! N ! ! " qi = 0 . The same condition also leads to equations such as (for example in the x direction for i=1

atom k), !

!

24

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– N

! qi ! N = " ! x ! x " qi = 0 . k k i=1 i=1

(46)

N Finally, the x component of the electrostatic force on atom k is given by Fx, k = ! "U es ! # "U es " qi ,

" xk

which

in

view of

Eqs.

(45)

and (46)

reduces to

Fx, k = !

"U es . " xk

i=1

" qi " xk

This is because

N

N !U es ! qi N ! qi !q = "µ = µ" i = 0 . Consequently, for the force calculation the system behaves as if ! xk i ! xk i=1 i=1 i=1 ! x k

" !q

the charges are constant.

2.4 Non-electrostatic part of the potential: the MEAM potential As explained in Sec. 2.2.2, the formal expression for the non-electrostatic part of the potential is N

U nes = " E i (0) + i=1

1 N N 1 k c Z i Z j ([ f i | f j ] # [i | f j ] # [ j | f i ] + ) , " " 2 i=1 j=1 rij

(47)

j$i

and because of cancellation effects, this expression has no long range 1/r part. At this point we have the full ! freedom to choose any non-electrostatic potential form to make Eq. (47) specific. Possible candidates would be EAM, MEAM, Tersoff, pair potentials, etc. In view of its wide applicability, see e.g. [16, 17, 18], we choose the MEAM potential, and we will use it in a reference free (RF) version for convenience. This means that no reference structures have to be selected for the A-A, AB, and B-B pair interactions (in a binary AB system), and that the universal equation of state [19] is not strictly enforced for these structures. 2.4.1 Reference Free Modified Embedded Atom Method (RF-MEAM) According to the RF-MEAM potential, the (non-electrostatic) energy Unes of a system of N atoms is given by N N

U nes = " "

N 1 S h (r )# (r ) + 2 ij IJ ij IJ ij

i=1 j=1 j!i

" FI ($i ) ,

(48)

i=1

in which i and j are atomic indices, and I and J denote the chemical types of atoms i and j, respectively. In this work I and J can be Al or O. For systems consisting of one chemical type, I and J can be dropped from the equations. In Eq. (48), S is an angular screening factor, h(r) a radial

25

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– cutoff function, φ(r) a pair potential, and F(ρ) an embedding function. In the following we show their mathematical forms and indicate which parameters are needed to define the potential. Embedding function

(

)

FI (!i ) = E0,I yi ln yi + E1,I yi + E2,I yi2 g(yi ) , 2

g(yi ) = 1 ! e yi

/2" 2

,

yi =

!i . nI

(49)

(50)-(51)

For each chemical element four parameters are needed: E0 , E1 , E2 , n . However, for the first (or only) element in the system, we can set n = 1 without loss of generality. The function g(y) is a small modification to suppress the singularity in the derivative of FI(ρi) in ρi = 0. The quantity σ is a small number. The “background electron density at the location of atom i”, ρi, is defined by the following equations, lmax

"i = "i(0)G(#i ) , G(") =

2 , 1+ e#"

" t I(l)#i(l)

!i =

l=1

2

,

2 #i(0)

(52)-(54)

where !

! 2 !i(l) = # # Sij hJJ (rij ) f J(l) (rij )Sik hKK (rik ) fK(l) (rik )P (l) (cos$ jik )

(55)

j"i k"i

and (l)

f J(l ) (r) = pJ(l)e"q J r .

(56)

In Eq. (54), lmax = 3, and in Eq. (55), P(l) is the Legendre polynomial of order l. Note that the MEAM simplifies into the!EAM if lmax is taken as zero, and therefore G(Γ) becomes equal to 1. For each chemical element eleven parameters are needed: t(l) (l =1, 2, 3), p(l) (l = 0, 1, 2, 3), and q(l) (l = 0, 1, 2, 3). However, for the first (or only) element in the system, we can set p(1) = p(2) = p(3) = 1 without loss of generality. Pair potential "#p,ij

2 3 ! IJ (rij ) = "Ep,IJ (1+ #p,ij + c2,IJ #p,ij + c3,IJ #p,ij )e

26

,

(57)

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– $ r ' !p,ij = " p,IJ && ij #1)) . % rp,IJ (

(58)

For each distinct pair of chemical elements five parameters are needed: Ep, c2, c3, αp, rp. In a binary system there are therefore 15 such parameters. At this point it should be mentioned that new in this RF-MEAM format, compared to classical MEAM [7], is the extension of the embedding function by one term (Eq. (49)) and the abandonment of the concept of “reference structures”. The pair potential, Eq. (57) is a parametrized (Morse-like) function rather than a function prescribed by the equations of state of a particular crystal phase. With this, the equations are computationally easier to handle and more powerful than the classical MEAM format. Radial cutoff function %1 ' 5 ' !" z hIJ (rij ) = &e c,IJ ij # bn,IJ zijn ' n=0 '0 (

(zij < 0) (0 $ zij $ 1) ,

(59)

(zij > 1)

with zij =

rij ! rs,IJ rc,IJ ! rs,IJ

.

(60)

The radial cut-off function is an exponentially damped 5th degree polynomial. For each distinct pair of chemical elements three parameters are needed: αc, rs, rc. The coefficients bn are taken such that the cut-off function smoothly decays between r = rs and r = rc. This means that the first and second r-derivatives of the original, non cut-off function are conserved in r = rs, and that the first and second derivatives of the cut-off function are identically zero in r = rc. For this, the following values should be used:

b0 = 1 , b1 = ! c , b2 = 12 ! c2 b3 = !10 ! 6" c ! 23 " c2 , b4 = 15 + 8! c + 23 ! c2

(61)

b5 = !6 ! 3" c ! 12 " c2 Angular screening factor Angular screening is a unique MEAM mechanism, intended to take into account the presence of atoms k in the neighborhood of i-j atomic bonds. It was originally proposed by Baskes [7] and

27

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– works as follows. The i-j interaction is weakened by a factor Sij (a number between 0 and 1), which depends on neighboring atoms k in the following way,

Sij =

" B(y jik ) ,

(62)

k!i, j

$0 (y < 0) & 4 2 B(y) = %[1" (1" y) ] (0 # y # 1) , &1 (y > 1) '

y jik =

!

C jik

C jik ! Cmin,JIK Cmax,JIK ! Cmin,JIK

,

(63)

(64)

1 ! cos2 " jik . = # rij & % ! cos" jik ( cos" jik $ rik '

(65)

For each distinct triplet of chemical elements JIK (distinct apart from exchanging the outer indices) two parameters are needed: Cmin and Cmax. This implies that two parameters are needed for a singleelement system and that twelve parameters are needed for a binary (AB) system, viz. two parameters Cmin and Cmax for each of the triplet combinations AAA, AAB, BAB, ABA, ABB, and BBB. The meaning of Cmin and Cmax is illustrated in Fig. 4. These two numbers define an inner ellipse and an outer ellipse, both centered on the i-j bond. An atom k outside the outer ellipse does not screen the i-j bond at all (Sij =1). An atom k inside the inner ellipse completely screens the i-j bond (Sij =0). An atom k between the two ellipses (as drawn) partially screens the i-j bond (Sij between 0 and 1).

k i

θjik

k

b j

Cmin

a

i

j

Cmax a)

b)

Fig. 4. a) Angular screening of an i-j bond by an atom k. b) Definition of the parameters a and b.

28

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– It can be easily proven that there is a simple relation between the C value of an ellipse and its major and minor axes (see Fig. 4b),

C=

b2 . a2

(66)

Seen from atom i, the atoms k that can possibly screen the i-j interaction are limited to a half-sphere with its center at i and radius $ rij & Cmax r = %r &' ij 2 C #1 max

(Cmax " 2) (Cmax > 2).

(67)

Note that the screening capacity of atom k does not depend on distances rik and rjk. Therefore, in the most general case (different cutoff radii for different pairs), atom k could still screen i-j interaction ! even if it is outside of cutoff areas of atoms i or (and) j. This completes the description of the full CTIP+MEAM potential used in the present work for the Al/O system. Reviewing all parts of the potential, we find that in total 69 parameters are necessary for a complete definition: 7 for CTIP, 7 for the two embedding functions, 19 for the background electronic densities, 15 for the three pair potentials, 9 for the three radial cutoff functions, and 12 for the six possible angular screening situations. In later chapters it will be explained how these parameters were obtained by fitting to various types of data. In Sec. 2.6 general background information on the fitting technique employed is given.

2.5 Cu/Ta simulations: The EAM + Screened Coulomb potential For the large-scale calculations of the purely metallic Cu on Ta system in Ch. 6 an EAM potential was used. Similarly to the case of MEAM, the energy U of a system of N atoms is given by N N

N

U = " " 12# IJ (rij ) + " FI ($i ) , i=1 j=1 j!i

(68)

i=1

but different pair potentials and embedding functions than in MEAM are used. Also, as is usual in EAM potentials, the background electron density is written as a sum of spherical functions only, so instead of Eqs. (52)-(55) we write !i = # fJ (rij ) . j "i

29

(69)

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Note the absence of the angular screening factors and the radial cutoff functions in Eqs. (68) and (69). The reason for this is that in EAM potentials angular screening is not applied, and, for the present Cu/Ta case, the radial cutoff is already built into the pair potential functions φ(r) and the electronic density functions f(r). In Eqs. (68) and (69), I and J can be Ta or Cu. The specific functional forms for the Ta and Cu potentials are presented next. The information was taken from [20]. 2.5.1 Ta-Ta potential Johnson and Oh have introduced a functional form for EAM potentials for bcc metals [21]. The embedding function, electron density around each atom, and the pair potential are given by

(

UF F( ") = #(E C # E1V ) 1# ln( " / " e )

n

)( " / " )

"

f (r) = f e ( r1e /r) ,

n

e

,

(70)

(71)

! 3

2

" (r) = K 3 ( r /r1e #1) + K 2 ( r /r1e #1) + K1 ( r /r1e #1) + K 0 , ! in which !

K3 = "

!

!

K2 = "

15#G $ 3 1 ' & A " S) , 3A + 2 % 4 2 (

(73)

15#G $ 15 3 9 7' & A " S " AS + ) , 3A + 2 % 16 4 8 8( $7 3 ' K1 = "15#G& " S ) , %8 4 (

(74)

(75)

15#G $ 201 27 87 187 ' 1 UF K0 = " A " S " AS + & ) " E1V , 28 56 168 ( 7 ! 3A + 2 % 112

n=

!

9"B #15"G , UF $ 2 ( E C # E1V )

(72)

(76)

(77)

where !S = r2e /r1e , A = 2C44 /(C11 " C12 ) ,

! !

(78)-(79)

G = (3C44 + C11 " C12 ) /5 , B = (C11 + 2C12 ) /3. ! !

30

(80)-(81)

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– In these equations EC is the equilibrium cohesive energy per atom, E1VUF is the unrelaxed monovacancy formation energy, ρe is the equilibrium electron density, fe is a dimensionless element-specific weight factor that cancels out for single-element systems, r1e and r2e are the first ! influence, a value of 6 was and second equilibrium neighbor distances, β is a constant (of little chosen by Johnson and Oh), Ω is the atomic volume, B is the bulk modulus, G is the Voigt average shear modulus, A is the anisotropy factor, and C11, C12 and C44 are the cubic elastic constants. Table I contains the values used for Ta. Table I. Values used for constructing the EAM potential for bcc Ta. The values are the lattice parameter a, atomic volume Ω, the bulk modulus B, the Voigt average shear modulus G, the anisotropy ratio A, the cohesive energy EC, the unrelaxed monovacancy formation energy E1VUF and the cut-off radius rc. a (Å) 3.3026 21.66 ΩB (eV) 7.91 ΩG (eV) A 1.57 EC (eV) 8.089 E1VUF (eV) 2.95 rc (Å) 3.99

To implement Eqs. (68)-(81), a few small alterations were made. First, the radial interaction range around each atom has been limited to the first and second neighbor atoms in the equilibrium Ta bcc crystal only. This was achieved by replacing the electron density function in the interval between rs = r2e + 0.1(r2e " r3e ) and rc = r2e + 0.5(r2e " r3e ) by a smooth cutoff function, f c (r) =

"2 f (r !s ) + (rs " rc ) f '(rs ) ! 3 f (rs ) " (rs " rc ) f '(rs ) (r " rc ) 3 + (r " rc ) 2 . 3 2 (rs " rc ) (rs " rc )

(82)

This function has the properties f c (rs ) = f (rs ), f c '(rs ) = f '(rs ), f c (rc ) = f c '(rs ) = 0 . Therefore the ! corrected function smoothly connects to the original function at rs and then gradually goes to zero at rc. The pair potential has been adapted in a similar manner. Second, the embedding function of Eq. (70) is very steep near ρ = 0, i.e., the derivative ! tends to –∞. As a result of discretization errors this singularity can produce unreliable results. To alleviate this problem, the embedding function has been multiplied by the function h( ") = 1# exp(#0.5( " / " s ) 2 ) ,

(83)

in which ρs was given the value 0.5. This solves the problem of the singularity. There is no ! but only the low-density part of the embedding function is significantly theoretical basis for this, affected. The embedding energy for low densities is small, consequently modifying it by a small fraction has no significant impact. Table I shows the input values that were used in the creation of the Ta-Ta EAM potential. Since Johnson-Oh EAM was derived for low energy situations, different potentials are used for 31

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– short distances. For distances smaller than r1e (2.68 Å) the Johnson-Oh pair potential is stiffened. Instead of φ(r), therefore, Johnson and Oh have proposed to use φs(r), given by 2

" s (r) = " (r) + 4.5(1+ 4 /(A # 0.1))(" (r) # " (r1e ))( r /r1e #1) .

(84)

2.5.2 Cu-Cu potential ! Johnson and Oh have introduced the following functional form for EAM potentials for fcc metals [22, 23]: n

F( ") = a( " / " e ) + b( " / " e ) ,

(85)

f (r) = f old (r) " f c (r) ,

(86)

" (r) = " old (r) # " c (r) ,

(87)

!

where

! !

f old (r) = f e exp("# (r /r1e "1)) ,

(88)

" old (r) = " e exp(#$ (r /r1e #1)) ,

(89)

! f c (r) = f old (rc ) + g(r) f 'old (rc ) /g'(rc ) ,

(90)

" c (r) = " old (rc ) + g(r)" 'old (rc ) /g'(rc ) ,

(91)

! ! g(r) = 1" exp(# (r /r1e " rc /r1e )) .

(92)

! Eqs. (86)-(92) already include a smooth function cutoff, therefore no Table II. Parameter values adaptation as in the!case of the Ta-Ta interaction is necessary. The used in the construction of the embedding function in Eq. (85) has no singularity at ρ = 0, so EAM potential for bcc Cu. β 5.0 multiplying it with Eq. (83) is also not necessary. γ 8.5 Table II lists the parameters used in the construction of the Cuδ 20.0 Cu potential. These values are either generally sensible choices made rc (Å) 4.85 by Johnson and Oh (for those parameters which do not have too great φe (eV) 0.36952 an influence), or were fitted to cohesion energy, vacancy formation a –4.0956 energy, atomic volume, and Voigt average bulk and shear moduli data. b –1.6979 The cutoff radius includes the first, second, and third-nearest neighbors n 0.44217 in the equilibrium Cu fcc lattice. The atomic size appears in the potential not only through the atomic volume to which a number of parameters are fitted, it is also present through r1e (2.55 Å) which in turn influences the equilibrium electron density ρe. Cu has a rather large thermal expansion coefficient at room

32

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– temperature (16.5 × 10–6/K) which would give rise to thermal stresses if Cu is simulated at high temperatures but nevertheless needs a room temperature lattice spacing (conditions that are applied in Ch. 6 of this thesis). Therefore different Cu potentials with different values for r1e were constructed for high temperatures. The 0 K value for r1e was chosen to be slightly smaller, such that at the higher simulation temperatures the Cu lattice has the experimental room temperature lattice constant. It was verified that after making the small changes to r1e the potential still accurately reproduces the other properties to which the potential was fitted. The value for fe was determined to reproduce the heat of formation of a 50-50% ordered CuTa alloy, see next section. 2.5.3 Cu-Ta potential Johnson has presented a model, based on the heat of solution, for a pair potential to describe the interaction between unlike elements in binary alloys [24]. In its original form, the pair potential between atoms of elements A and B is given by % 1 " f (r) f (r) ! AB (r) = $ B ! AA (r) + A !BB (r)' . 2 # fA (r) fB (r) &

(93)

To avoid the singularities that arise when the different elements have different radial cutoff radii (4.85 Å for Cu, 3.99 Å for Ta), we have slightly adapted Johnson’s functional form for Cu-Ta to

1 f Ta2 (r)"Cu (r) + f Cu2 (r)"Ta (r) "CuTa (r) = . 2 #1 &2 % ( f Cu (r) + f Ta (r))( $2 '

(94)

The functional form of eq. (94) has only one free parameter, i.e. the ratio of the weight factors fe,Cu and fe,Ta. This parameter was fitted to the heat of formation of a CsCl-type, ordered 50-50 atom % ! Ta-Cu alloy. The (positive) heat formation of such an alloy is 0.03 eV/atom according to the Miedema model [25]. A value of 0.8924 for fe,Cu/ fe,Ta. (fe,Cu = 0.8924, fe,Ta = 1) results in this heat of formation. 2.5.4 Moliere-Firsov screened Coulomb potential Although not used in this work directly, we also describe, for completeness, the screened Coulomb pair potential with Moliere weight factors and Firsov screening length. This potential should be used at small distances (below approximately 1.5 Å), which occur frequently in high-energy situations such as sputter bombardment. This purely repulsive potential has the form

" (r) =

!

Z1Z 2e 2 & r ) %( + , 4 #$0 r ' a *

33

(95)

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– where

# r& 3 " % ( = * c ie)d i r / a , $ a' 1

3

*c

i

= " (0) = 1 ,

(96)

1

and

!

1/ 3

a = (9" 2 /128) aB ( Z11/ 2 + Z1/2 2 )

#2 / 3

,

(97)

in which Z1 and Z2 are the atomic numbers of the interacting atoms at distance r, e !is the elementary charge, ε0 is the vacuum permittivity, γ is the screening function, a is the screening length, ci and di are constants, and aB is the Bohr radius. The Moliere values for ci and di given in Table III.

Table III. Moliere constants in the screening function i ci di 1 0.35 0.3 2 0.55 1.2 3 0.10 6.0

Since the EAM potential is based on a combination of a pair potential and an electron density and the Screened Coulomb potential is a pure pair potential, a method has to be devised to go smoothly from one potential to the other. One way to do this is to let the electron density function f(r) fall off to zero when the distance r from the nucleus of the atom decreases towards the range where the Screened Coulomb pair potential takes effect. This is done by multiplying f(r) by a Fermi-Diraclike function, g(r) = 1/(1+ exp((rz " r) /#rz )) ,

(98)

in which rz is a constant that determines where the function is equal to 1/2 and Δrz is a constant that ! the function rises from 0 to 1. In this way the electron density and hence the determines how steeply embedding energy are practically zero near the atom core. The Screened Coulomb pair potentials can be smoothly spline-connected to the EAM potentials.

2.6 Fitting potentials to energy and electrostatic potential field data In this work the electrostatic part (CTIP) of the potential is fitted to electrostatic potential field data (Ch. 4) and the non-electrostatic part (RF-MEAM) to energy data (Ch. 5) of Al/O atomic structures. For the fitting we have used a simulated annealing technique [2]. The essentials of the fitting method and its implementation in a computer program called ff are briefly explained here. Basically the problem is to fit a function y = f(x,a) to given datapoints xi, yi (i = 1 ... N), each with an uncertainty σi in the value of yi. The symbol a is shorthand notation for the array of parameters a1, a2, ... am that occur in the function f(x). The objective of fitting is to determine the parameter values a that make the function f fit the data as closely as possible, or, to be more precise (and referring to the standard operational mode of ff), to minimize the quantity R2, given by

34


R2 =

!2 , N "m

(99)

in which N is the number of datapoints and m the number of fitparameters. The usual definition of χ2 applies here, N

! 2 = " zi2 ,

(100)

i=1

where zi denotes the weighted fit-residual

zi =

yi ! f (xi , a) . "i

(101)

Here we list the meaning of the variables in the present contexts of fitting an electrostatic potential (CTIP) to electrostatic potential field data and fitting a non-electrostatic potential (RF-MEAM) to energy data. In the electrostatic case, the parameter xi is simply an index that denotes a particular field point in the 3D space of a certain binary crystal structure; yi denotes the value of the electrostatic potential field in this point i; and f(xi,a) is the electrostatic potential field in this point calculated by the CTIP potential with values a for the potential parameters. In the non-electrostatic case the parameter xi is an index that denotes a particular atomic structure such as a certain molecule or crystal structure (possibly binary), which may or may not contain defects or strains; yi denotes the non-electrostatic potential energy (per atom) of this structure i, as calculated by ab initio methods –or as known from experiments. This nonelectrostatic potential energy is the total potential energy from which the electrostatic part has been subtracted; and f(xi,a) is the non-electrostatic energy of structure i as calculated using the RFMEAM potential with values a for the potential parameters. As indicated above, the number σi is in principle the uncertainty of yi, but in practice it is often used as a user-selectable parameter to control the importance of structure i in the fit. The statistical weight of datapoint i is given by 1/σi2, which shows how the importance of structure i can be adjusted. Statistical theory tells us that, if the fit is correct, i.e., if there are no remaining systematic deviations, the expectation value of R is equal to 1, with standard deviation 1/√(2(N–m)), provided that the σ’s have been given their true values. As these true values are often not known, or not used (as just mentioned), the correctness of the fit has to be assessed by an independent test. For this we use the statistic ψ2, given by [26]

35

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– N

!2 =

" zi2 i=1

.

N

(102)

1 (zi+1 # zi )2 " 2 i=1

This number measures any leftover systematic deviations in the fit, at least for as far as the sequential order of the datapoints (i = 1 ... N) has some meaningful significance. Note that there is a “wraparound”, in the sense that zN+1 is in fact z1. If the fit is correct, i.e. if there are no systematic deviations, the expectation value of ψ2 is equal to 1, with standard deviation 1/√(N–m+2). Significantly larger values of ψ2 than 1 point to an incorrect fit. Note that, because ψ2 is a ratio of z2 values, the influence of the σ’s is minimized. In other words, even if the σ’s are not the true uncertainties, ψ2 is fairly insensitive to this and accordingly is a powerful diagnostic of the quality of the fit. This is its great strength. The stimulated annealing fitting technique used in this work is based on the Downhill Simplex Method [27], which is a widely accepted method to find the minimum of a multidimensional function (here m-dimensional) via a downhill search that does not need the derivatives of the function f with respect to its parameters aj (j = 1 ... m). However, to allow the fitting procedure to find deeper minima that cannot be reached by a downhill search alone because they lie in a different valley of the χ2 surface, simulated annealing has been added. This is implemented as follows. Each single value of ! 2 / (n " m) that is used in the search trajectory of the a values is replaced by an augmented version,

!2 (1 " T ln # ) , N "m

(103)

where ε is a uniform random number on (0,1] and T is called the “temperature” in the ongoing fit. This temperature gradually decreases as the fit proceeds through its many iterations. Being essential to the annealing method, the random number in Eq. (103) causes the downhill method in fact to move uphill frequently, thereby allowing the fit to visit various valleys in parameter space before ending up in a hopefully deep minimum. In our code ff, the temperature program can be selected according to a desired intensity of annealing,

Tk = T0 exp(!" k) ,

(104)

where k is the iteration number, T0 the starting temperature, and α the cooling rate. High values of T0 and low values of α cause intense annealing, which leads to an extensive exploration of the χ2 surface. Typical moderate values are T0 = 0.1 and α = (20 m)–1. During fitting, each parameter aj can be free, bounded, or frozen at its starting value. In the bounded case, the parameter aj has an associated parameter Aj, and it is this parameter Aj that is handled by

36

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– the program as a regular parameter and allowed to vary freely. In each iteration aj is then calculated from the value of Aj according to the type of bound imposed. In case of a lower bound, lj, one has a j = l j + (a 0j ! l j )exp(A j / (a 0j ! l j )) ,

(105)

in which aj0 is the starting value of aj. In case of an upper bound, uj, one has a j = u j ! (u j ! a 0j )exp(!A j / (u j ! a 0j )) ,

(106)

and in the case of both an upper and a lower bound, one has #u j ! h j exp(!A j / h j ) (A j " 0) , aj = $ % l j + h j exp(A j / h j ) (A j < 0)

(107)

with h j = (u j ! l j ) / 2 . The default minimization method of ff is the well-known least squares method, Eqs. (99)-(101). This effectively assumes that the errors in the datapoints have a Gaussian probability distribution. As an alternative, least absolute values, Σ |zi|, can be chosen. This assumes a double negative exponential distribution of the errors. A third possibility is to minimize Σ ln(1 + zi2/2). Since these methods impose a smaller penalty on outliers, they are often closer to experimental practice. The technical term is robust fitting.

2.7 Atoms and their local environment In this work we characterize the local environment of an atom in terms of its so-called Atomic Environment Type (AET). This characterization, which is actually a set of numbers, specifies enough of the environment of an atom to be able to calculate its energies and hence to allow fitting a chosen potential to the energies of one or more of such atoms in particular AETs. In the case of the electrostatic part of the potential, it is the electrostatic potential field between the atoms rather than the energies of the atoms that matters. This is the first Fig. 5. Atomic Environment purpose of AETs. The second purpose is to compare the AET of a Type (AET) of an atom in the given atom in a given configuration with the AETs of atoms in selected fcc crystal structure (nearest “standard” crystal structures. If the atomic AET is close enough to one neighbors only). of the standard AETs (in terms of the first few spherical harmonic coefficients, to be explained below), the atom can be considered as “having” locally that particular crystal structure. This makes it possible to color atoms in visualizations of simulation results in a way that can be conveniently called a local crystal structure map. 37

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– The Atomic Environment Type (AET) of an atom (i) is defined by the positions of all atoms (j) that can be considered essential neighbors of i (cf. the work of Villars and Daams, e.g. [28]). Only relative positions matter, and therefore the AET does not change if the group of i and j atoms is rotated or uniformly stretched or compressed. As an illustration, the colored atoms in Fig. 5 form the AET of the white central atom in an ideal fcc crystal (nearest neighbors only). Villars and Daams have found that only 20 different AETs are needed to describe 80% of all AETs (over 200) that occur in more than 200000 natural binary compounds. Nature apparently has its clear favorites. For a comprehensive review, see the PAULING database [29]. A convenient way to express the AET of an atom (i) is in terms of the rotationally invariant parameters wi(l), defined as 2

4! l w = Y(m(l )) (" ij , #ij ) , $ $ 2l +1 m=%l j (l ) i

l = 0,1, 2...

(108)

(l ) where Y(m ) is the l, m spherical harmonic function and "ij , #ij are the polar and azimuthal angles of

the ij-vector with repect to the z-axis, respectively. Together these parameters form a convenient “numerical fingerprint” of the AET of the atom (only the symmetry type, not the nearest neighbor ! ! distance). Note that w (0) = z 2 , where z is the number of neighbors. Examples of common AETs, for atoms in ideal crystals and small molecules, are given in Table IV. One sees how well the w(l) number sequences distinguish between the different AETs. ! Table IV. Values w(l) for an atom in different atomic environment types (AETs). (M) denotes a molecule, z is the coordination number. Note that for an atom in the bcc structure 2nd nearest neighbors are included in Eq. (108). dim -er (M) z w(0) w(1) w(2) w(3) w(4) w(5) w(6)

1 1 1 1 1 1 1 1

equilat. triangle (M) 2 4 3 1.75 1.13 1.42 2.18 2.65

Square (M)

tetrahedron (M)

2 4 2 1 2 2.75 2 1.38

3 9 6 2.25 0.375 1.27 3.54 4.94

diamond, next to vac. 3 9 1 1 5.45 3.07 1 4.16

diamond

sc

4 16 0 0 8.89 4.15 0 6.32

6 36 0 0 0 21 0 4.50

fcc (111) surface 9 81 6 2.25 0.375 3.89 3.54 28.7

fcc, next to vac. 11 121 1 1 1 5.38 1 40.6

fcc

hcp

icosahedral

bcc

12 144 0 0 0 5.25 0 47.5

12 144 0 0 0.833 1.36 9.12 33.8

12 144 0 0 0 0 0 63.4

14 196 0 0 0 0.259 0 51.1

There is also a direct connection between the numbers wi(l) and the contributions !i(l ) to the square 2

of the background electron density in the MEAM potential, Eq. (55). In this sense the MEAM potential is strongly coupled to the angular environment of an atom, which is one of its strong points. One can easily see this connection by realizing that Eq. (108) is mathematically equivalent to the form 38

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– wi(l ) = # # P (l ) (cos! jik ),

(109)

l = 0,1, 2,K

k "i j "i

with P(l) the Legendre polynomial of order l. Eq. (109) is in turn equivalent to !i(l ) , Eq. (55), when 2

angular screening and distance-dependent factors are omitted. The numbers w(l) bear a close resemblance to the numbers s(l) already considered in [7]. The relations are as follows: w (0) = s(0), w (1) = s(1), w (2) = 32 s(2), w (3) = 52 (s(3) " 35 s(1) ) .

!

The “symmetry-index” of an atom, produced by the program party and used for visualization of the local crystal symmetry is not based on the numbers w(l) but rather on the numbers Q(l) defined as wi(l ) Q ! . z

(110)

(l ) i

This choice was inspired by the work of Steinhardt et al. [30]. For each atom in a configuration the numbers Qi(l) are calculated for l = 2, 4, 6, 8, 10, 12 and compared (on a root mean square basis) to the corresponding numbers for atoms in common crystal structures. To be more precise, for an atom in a perfect, simple reference structure such as diamond cubic, sc, fcc, hcp, bcc, icosohedal or dimer, the values of Qref(2! ) , λ = 1,2, ..., 6, are shown in Table V. For an atom in a general structure, we are interested in the differences ΔS between the actual angular environment and the angular environment of an atom in each of these reference structures,

1 6 (2 " ) 2 (Qi(2 " ) # Qref ) . $ 6 " =1

!Si =

(111)

The reference structure that has the minimum difference ΔS with the numbes for the actual atom is “attributed” to this atom as local crystal structure, provided that this minimum ΔS is smaller than 0.11 (this number was obtained by numerical experimentation). This attributed crystal structure is subsequently used for coloring atoms. Table V. Numbers Q (2 ! ) for an atom in different reference structures: dimer, diamond, sc, icosohedral, fcc, hcp, bcc. The number z is the coordination number of the atom. 2λ Q (2 ! ) dimer

diamond

sc

icosohedral

fcc

hcp

bcc

2 4 6 8 10 12

1 1 1 1 1 1

0 0.509 0.629 0.213 0.650 0.415

0 0.764 0.354 0.718 0.411 0.696

0 0 0.663 0 0.363 0.585

0 0.191 0.575 0.404 0.013 0.600

0 0.097 0.485 0.317 0.010 0.565

0 0.036 0.511 0.429 0.195 0.405

z

1

4

6

12

12

12

14

39

Chapter 2 Theory and Methods ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

R. W. Hockney, J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York (1981). W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd edition, Cambridge University Press, Cambridge (1992). C. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New York (1971). T. Schlick, Molecular Modeling and Simulation, Springer, Berlin, 2002. M. E. Tuckerman, G. J. Martyna, J. Phys. Chem. B, 104, 159 (2000). http://web.mac.com/barend.thijsse/Site/Home.html (April 2009). M. I. Baskes, Phys. Rev. B 46, 2727 (1992). H. J. C. Berendsen, J. P. M Postma, W.F. Van Gunsteren, A. DiNola, J. R. Haak, J. Chem. Phys. 81, 3684 (1984). F. H. Streitz, J. W. Mintmire, Phys. Rev. B 50, 11996 (1994). X. W. Zhou, H. N. G Wadley, J.-S. Filhol and M.N. Neurock, Phys. Rev. B 69, 035402 (2004). I. Lazić, B. J. Thijsse, Proc. 4th International Conference on Multiscale Materials Modeling MMM, Ed. A. ElAzab, Department of Scientific Computing, Florida State University, Tallahassee, USA, 454 (2008). R. G. Parr, R. G. Pearson, J. Am. Chem. Soc. 105, 7512 (1983). P. Ewald, Ann. Phys. 64, 253 (1921). R. Fletcher and C. M. Reeves, Computer J. (UK) 7, 149 (1964). M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford (1987). K. Mae, Thin Solid Films 395, 235 (2001). C-L. Kuo, P. Clancy, Surf. Sci. 551, 39 (2004). H. Dong, L. Fan, K-s. Moon, C. P. Wong, M. I. Baskes, Model. Simul. Mater. Sci. Eng. 13, 1279 (2005). J. H. Rose, J.R. Smith, F. Guinea, and J. Ferrante, Phys. Rev. B 29, 2963 (1984). T. P. C. Klaver, Ph.D. Thesis: Atomistic simulation of Cu/Ta thin film deposition and other phenomena, DUP Science, Delft University of Technology, The Netherlands (2004). R. A. Johnson, D. J. Oh, J. Mater. Res. 4, 1195 (1989). D. J. Oh, R. A. Johnson, J. Mater. Res. 3, 471 (1988). R. A. Johnson, D. J. Oh, Embedded atom method for close-packed metals, in ‘Atomistic Simulations of Materials: Beyond Pair Potentials’, eds. V. Vitek, D. J. Srolovitz, Jan 1989, p233-238. R. A. Johnson, Phys. Rev. B 41, 9717 (1990). F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema, and A. K. Niessen, Cohesion in metals, ed. F. R. de Boer and D. Pettifor, North Holland Physics Publishing, p. 544 (1989). B. J. Thijsse, M. A. Hollanders, and J. Hendrikse, Comput. Phys. 12, 393 (1998). J. A. Nelder and R. Mead, Computer Journal 7, 308 (1965). J. Daams, P. Villars, Eng. Appl. Artif. Intell. 13, 507 (2000). PAULING FILE, Binaries Edition CD-ROM, ASM (2002). P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B 28, 784 (1983).

40

Chapter 3 General Performance Testing –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

3. General Performance Testing

Abstract New methods for modeling initial oxide film growth using MD (Molecular Dynamics) simulations are explored, in order to study the atomic mechanism of self-healing oxidation in metal/oxide systems, in particular in Al/Al2O3. Computationally, this is a challenging undertaking. A new version of the MEAM (Modified Embedded Atom Method) potential is proposed and extended by a variable charge ionic potential model from the literature. This makes it possible to include angular forces, which are needed for relatively open crystal structures such as metal oxides, as well as localized and dynamic charge transfer between Al and O, which is necessary to handle timedependent local composition variations. As Coulomb solver we use the PPPM method (particleparticle-particle-mesh). In this work the first results are reported. Tests of the ionic potential model in combination with the PPPM solver yield excellent results. Simulations of oxygen atom arrival at an Al surface are presented.

3.1 Introduction The surface oxides on aluminum and aluminum alloys can be called “self-healing” in that they quickly re-form after scratching. The initial stage of oxidation is very rapid, much faster than for a more noble metal such as e.g. Ru (Fig. 1) and is therefore difficult to follow experimentally. Here simulations would be able to extend time scales and resolution. Such self-healing phenomena of metal oxides using Molecular Dynamics (MD) simulations require sophisticated interatomic potentials of the underlying metal-oxygen systems, which is a challenging undertaking. The main target of our investigation is Al2O3. For oxides and metal/metal-oxide systems in general, extra 41

Chapter 3 General Performance Testing ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– difficulties appear because of the presence of ionic bonds, giving rise to long-range Coulomb interactions. Moreover, at interfaces, surfaces and defects, the atomic charges cannot be considered fixed, and models that allow charge transfer should be applied. Finally, aluminum oxide has a relatively open structure, which suggests that angular terms in the interatomic potentials could play an important role. For these reasons we are developing a potential for Al/Al2O3 that combines a charge-transfer potential for the electrostatic interactions with the Modified Embedded Atom Method (MEAM) [1] for the non-electrostatic interactions. Such a potential would also be applicable to the relatively open oxides of e.g. Si and to the mildly ionic III-V and II-VI semiconductors. This paper reports on the potential under development and on the implementation of the Coulomb solver. Previous MD work on Al/Al2O3 has been done using MEAM but without considering charges explicitly [2], and using charge transfer ionic potentials but in combination with EAM, which does not include angular dependencies [3][4]. Here we propose to use the Charge Transfer Ionic Potential (CTIP) published in [4] in combination with a novel version of the MEAM. Unlike previous work, we use the iterative Particle-Particle-Particle-Mesh (PPPM) method as Coulomb solver [5]. PPPM easily adapts to periodic and nonperiodic boundary conditions, and, because of its localized computability, it is convenient for code parallelization. An additional reason is that the computational efficiency of PPPM is O(N log N) while, e.g., for Ewald Summation it is O(N 3 / 2 ) [6]. A disadvantage of PPPM is that the maximal error can not be predicted analytically. !

!

Fig. 1. Oxide growth measured by in-situ ellipsometry on oxidizing single crystals of aluminum and ruthenium. Formation of the initial, chemisorbed oxide layer is easily noticeable for ruthenium but not for aluminum. Very likely it does take place also in aluminum but very fast.

3.2 Numerical versus analytical model in the simple case of NaCl In our MEAM+CTIP model, the energy U of a system of N atoms is the sum of a non-electrostatic and an electrostatic part, U = U nes +U es , where the non-electrostatic MEAM part is given by

42

Chapter 3 General Performance Testing ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– N N

N

U nes = " " 12# IJ (rij ) + " FI (xi ) i=1 j=1 j!i

(1)

i=1

in which upper case I, J denote the chemical types of the atoms i, j. The pair potential and embedding functions for each of the chemical types or type combinations are given by

!(r) = "E p (1+ # + c p# 2 + d p# 2 )e"# ,

# = $ p ( r / rp "1) , (2)

F(x) = Ae x ln x + Be x + Ce x ,

(3)

where the p and e subsripts denote adjustable parameters. The quantity xi is proportional to the square of what in (M)EAM terminology is called the “background electron density at atom i” and contains all dependencies on the i-angle in the ijk atom triplets, 3

xi = $ t I(l) $ $ pJ(l)e l=0

!qJ(l ) rij

(l )

pK(l)e!qK

rik

P (l) (cos" jik ) . (4)

j#i k#i

Here, t(l), p(l), q(l) are fitparameters for each chemical type, and P(l) is the Legendre polynomial. For simplicity, Eqs. (1-4) are given without angular and radial cutoff functions. New in this MEAM format is that the embedding function has been extended by two terms, that the concept of “reference structure” has been abandoned, and that the pair potential is a parametrized (Morselike) function rather than a function determined by a prescribed equation of state of a particular crystal phase. Eqs. (1-4) are more flexible than the classical format, without changing much in the underlying physical picture. With these expressions it proved possible to represent the well-known Stillinger-Weber and Tersoff-III silicon potentials to a very high degree of accuracy [7], which suggests that they will be appropriate to the current Al/Al2O3 case as well. In our implementation of the electrostatic CTIP part, the long-range Coulomb interactions are handled by the PPPM method, which is briefly explained in Figs. 2 and 3. The grids on the figures represent the mesh, which holds the charge distribution in the middle panel of Fig. 3. Assuming pointlike charges, the interaction between a charged atom i and all other charged atom j separated by rij, including all periodic images, is divided into two parts by adding and subtracting a charge distribution ! j (r) around every atom j. The volume integral of this distribution must be equal to qj. If it is chosen to be Gaussian, ! j (r) = q j" #3$ #3/2 exp(#r 2 / " 2 ) , it can be proven that the interaction energy between point charge i and point-charged atom j minus its own charge distribution is qi q j erfc(rij / ! ) / rij .

43


Fig. 2. Short-range part (“PP”) of the interaction between atom i and all other atoms, which is taken into account directly, showing the cutoff radius. The mesh is not used here.

Fig. 3. The rest of the interaction (“PM”). The middle frame shows the interaction of i with the charges attributed to the mesh points. The Poisson equation is used here. The frame on the right shows the correction for self-interaction.

The complementary error function decreases rapidly with distance, which means that this part of the interaction has become short-ranged and a cutoff radius can be used (Fig. 2), normally on the order of 5 Å. The rest of the interaction (Fig. 3) is calculated by placing point charge i in the electrostatic potential on the mesh, solved by the Poisson equation ! 2" mesh = #$ mesh / % 0 . Here, ! mesh is the total charge of all individual charge distributions ! j (r) , plus i’s own charge

distribution, allocated to the mesh points through a diffusion-like algorithm. In this way ! mesh is the same for all atoms i and the Poisson equation needs to be solved only once. For each atom, the spurious interaction between its point charge and its charge distribution can then be subtracted analytically (Fig. 2, right). In summary, one obtains N

# kc j=1 j"i

N %r ( qiq j qq = # k c i j erfc' ij * + qi (+ mesh (q) , + self (qi )) rij rij &$ ) j=1

(5)

j"i

where kc = 1 / 4 !" 0 , and ! self (qi ) = 2kc qi / " # is the self-electrostatic potential. The vector ! notation indicates the set of all charges. The mesh size should be no more than one quarter of the cutoff radius.

44

Chapter 3 General Performance Testing ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– According to the CTIP model the charges in the system are not constant. Every atom is allowed to become ionized by charge transfer to and from other atoms. The ionization energy E iionization (qi ) = " iqi + J iqi2 /2 , with ! i the electronegativity and Ji the electrostatic hardness of atom

i,

is

compensated

by

the

appearance

of

Coulomb

electrostatic

energy

1 E iCoulomb = kc " qiq j /rij . The charges that atoms obtain for a certain atomic arrangement are 2 ! determined by searching for the minimum of total electrostatic energy Ues (ionization plus Coulomb) with the condition of total system charge neutrality. This energy is given by ! N # & 1 N N 1 U es = )% " iqi + J iqi2 ( + ) ) ( k c qi Z j ([ j | f i ] * [ f i | f j ]) + k c q j Z i ([i | f j ] * [ f i | f j ])) + $ ' 2 i=1 j=1 2 i=1 j+i

(6)

## q * q & & # q *q & 1 kc qiq j [ f i | f j ] + ),%%%%1* i min,i (((qi * qmin,i ) 2 + %%1* max,i i (((qi * qmax,i ) 2 (( ) ) 2 i=1 j=1 qi * qmin,i ' $ qi * qmax,i ' i=1 $$ ' N

N

N

j+i

where notations [a | f b ] and [ f a | f b ] (a=i,j; b=i,j; a≠b) denote the Coulomb interaction integrals !

fb (rb , qb ) f (r ,q ) f (r ,q ) dVb and [ f a | f b ] = k c " " a a a b b b dVa dVb . This is according to the rvv rav Va Vb Vb ! ! used charge model, in which each atom has a pointlike charge part and a distributed part: " iatom (r) = Z i# (r) + (qi $ Z i ) f i (r) . For mathematical simplicity the distributed part of the atomic ! charge is chosen to be f i (r) = " i3 exp(#2" i r) / $ . The last term in Eq. (6) is an extra term that softly [a | fb ] = kc !

bounds the charges between the values qmin and qmax ( ! > 0 ) to account for chemical valence. Parameters that need to be specified for each type of atom in the system are then: " i , J i , Z i , " i ,

!

!. qmin , and qmax ! Eq. (6) ! After applying PPPM (Eq. (1)) to we obtain ! ! ! ! !

!

$$ q # q ' 1 ' $ q #q ' 1 2"&&&&1# i min,i ))(qi # qmin,i ) 2 + &&1# max,i i ))(qi # qmax,i ) 2 )) + * 2 + iqi + J iqi2 + qi , (- mesh (q) # - self (qi )) + * 2 i=1 %% qi # qmin,i ( % qi # qmax,i ( ( 2 i=1 N

U es =

N

(

)

(7)

N N N N $1 $r ' 1 1 1 ' k c qi Z j ([ j | f i ] # [ f i | f j ]) + k c q j Z i ([i | f j ] # [ f i | f j ])) + * * kc qiq j && erfc& ij ) + ([ f i | f j ] # ))) * * ( 2 i=1 j=1 2 i=1 j=1 rij ( %/ ( % rij j.i

!

j.i

The minimization of Ues together with the condition

"

N

q = 0 is done using the Conjugate-

i=1 i

Gradient method [8]. For the line minimization in the charge space the Brent algorithm was used [8]. ! In order to test the accuracy of the CTIP/PPPM implementation, we use the fact that the electrostatic energy for the simple case of, e.g. a NaCl crystal type, can also be determined and minimized analytically, in terms of the Madelung constant,

45

Chapter 3 General Performance Testing ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– N # & 1 N N 1 U es (q) = )% " iqi + J iqi2 ( + ) ) ( k c qi Z j ([ j | f i ] * [ f i | f j ]) + k c q j Z i ([i | f j ] * [ f i | f j ])) + $ ' 2 i=1 j=1 2 i=1 j+i N

(8)

N

1 1 N NaCl q 2 kc qiq j ([ f i | f j ] * ) * , madelung ) ) 2 i=1 j=1 rij 2 r0 j+i

!

NaCl where qi = q if atom i is Na and qi = "q if atom i is Cl, " madelung is the Madelung constant for the

NaCl crystal type, and r0 nearest neighbor distance. The double summation terms include only short-range functions and converge very fast. In this formula the charge bounding energy term is ! ! not included!for simplicity. ! ! ! 3.3 Results of the performance testing (1) Results of how well PPPM works are given in Table I. The basic tests of Madelung constant calculations were done for several artificially made but convenient structures. If a unit point charge is placed inside the heavily drawn box (at fractional coordinates X, Y, Z) in Fig. 4 and a mirroring operation is applied, infinite three-dimensional structures as shown in Fig. 4 can be generated. If the charge is placed in the middle of the box we obtain exactly the NaCl crystal type, but by varying the coordinates of its position inside the box we obtain other crystal types. For all of these we can calculate the Madelung constant analytically. A comparison of the exact, analytical result and the PPPM result for the Madelung constant is shown in Table I.

Fig. 4. Generating the crystal structure samples for the Madelung constant test.

46

Chapter 3 General Performance Testing ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Table I. Madelung constants for different structures from Fig. 4 calculated analytically and using PPPM. Madelung constant X Y Z Exact PPPM 0.2 0.5 0.3 2.83447 2.83447 0.6 0.6 0.1 5.10754 5.10755 0.4 0.9 0.8 5.27626 5.27628 0.1 0.1 0.7 6.47665 6.47664 0.5 0.5 0.5 1.74755 1.74756 0.8 0.2 0.5 3.26069 3.26071 0.7 0.9 0.8 5.29997 5.29998 0.9 0.7 0.1 6.47665 6.47664 0.6 0.5 0.4 1.85929 1.85934

(2) Values of charges and potential energies per atom for the NaCl structure in which one allows charge transfer can be calculated exactly (Eq. 8) and compared with the minimization carried out by PPPM (Eq. (7)). Results are given in Fig. 5. The curves “exact” and “PPPM ( " = 0 )” overlap almost perfectly. As expected, the PPPM calculations with bound charges (in our case ! = 20 eV/e2 ) yield different curves in the region where charge bounding is actually taking

!

place. The parameters used for these calculations are shown in Table II. These parameters are just test parameters, not literature values or values obtained by fitting. The cut-off radius for all shortcut"off range terms in PPPM method was chosen to be rpppm = 5.4 Å.

!

Figure 5. Charges q (left) and electrostatic potential energy per atom Ues/N (right) for the NaCl crystal structure type after energy minimization calculated analytically and using PPPM for different nearest neighbor distances. The inserts show the deviation between PPPM (ω = 0) and the exact method.

47

Chapter 3 General Performance Testing ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Table II. Test parameters for Na and Cl atomic types Element

qmin

qmax

"[eV ]

J[eV ]

" [Å-1]

Z

Na Cl

0 -1

1 0

-3.4 2.0

10 14

1.0 2.1

0.6 0.0

!

!

!

!

!

!

(3) The effect of angular forces is illustrated by a simplified calculation of the III-V compound AlP. Using a realistic MEAM potential, the cubic ZnS phase is the stable crystal phase. By changing only one parameter value in the MEAM potential, namely t(3) in Eq. 4 , which controls the energy associated with the third spherical harmonic describing an atom’s geometrical environment, the cubic CuAu phase can be made the stable crystal phase (Fig. 6).

Figure 6. Energies of two phases of a simplified model of AlP as a function of nearest-neighbor distance. By simply changing a single parameter in the MEAM potential, the relative stability of the two phases can be tuned (left and right). Charge transfer varies from approximately 0.25 e at small R to 0.13 e at large R.

(4) The first simulation results of an oxygen atom approaching an Al surface kept at 375 K are shown in Fig. 7. The oxygen atom was directed at the Al sample at normal incidence, with an initial kinetic energy of 0.0125 eV, which corresponds to the average thermal speed at 300 K. For this simulation, the EAM potential and parameterization from [4] were used, i.e. a potential without the angular terms l = 1, 2, 3 in Eq. (4). After impact, the oxygen atom reaches the second Al atomic monolayer from the surface and stays there as an interstitial. As expected, no permanent change was produced in the Al lattice structure. The amount of charge that the oxygen atom has acquired as interstitial is –0.54 e, which is far away from the minimum value of –2 e, and this charge is almost totally transferred to the six closest neighboring Al atoms (around +0.08 e each).

48


Figure 7. Oxygen atom approaching Al surface held at 375 K. a) Oxygen atom reaching the first Al plane. b) Final oxygen position as octahedral interstitial. Upper two figures: view along the Al direction. Lower two figures: top view along the Al direction. The squares indicate the same Al unit cell. Colors represent different charge ranges: light blue (–0.02 e, +0.02 e), dark blue (+0.07 e, 0.25 e), magenta (–0.55 e, –0.45 e).

3.4 Discussion Results of calculating Madelung constant using PPPM, given in Table I, are in very good agreement with analytically determined values for all structures generated as described above (Fig. 4). In agreement with this, the results of using PPPM as a long-range Coulomb solver in minimization process are also very satisfactory (Fig. 5); the deviations using PPPM in comparison with the exact calculations do not exceed 8 meV/atom for the electrostatic potential energy and are less than 0.009 e for the charges in the NaCl crystal type. In the range of nearest neighbor distances where the equilibrium positions are expected, between 2 and 3.5 Å, these errors are even less than 1 meV/atom and 0.001 e. Using an MEAM instead of EAM potential, the possibility of fine tuning of nonelectrostatic angular dependent part of the potential interaction should allow us better fitting to experimental and ab initio data for open structures as Al2O3. Finally, the approach of a single oxygen atom towards the surface of a pure Al crystal gives us some insight in very beginning of process of aluminum oxidation. Further work is needed to understand why the interstitial oxygen atom in Fig. 7 is only partially charged, while one could expect a full charge of –2 e. This may be due to a flaw in the charge transfer model, although it correctly leads to fully charged atoms in Al2O3.

49

Chapter 3 General Performance Testing ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 3.5 Conclusions In this work the first steps towards building a sophisticated enough simulation system for metaloxide/metal-alloy systems and some first results are reported. The combination of a charge transfer ionic potential (CTIP) [4] together with PPPM as long-range Coulomb interaction solver was tested against analytical calculations for simple system such as the NaCl crystal type. The results for charges and energies are excellent. Also, a modified version of the MEAM potential is proposed, one that is designed for more flexibility then the original one. Control of the phase stability by adjusting parameters of the angular dependent terms in the model has been demonstrated to work well, even together with atomic charges and charge transfer in the system. A first simulation on oxygen atom approaching an Al surface yields realistic results on initial oxidation. These first tests will be followed by further simulations that combine MEAM and CTIP in the studying of self-healing effect in Al/Al2O3.

Acknowledgments Author would like to thank Prof. dr. Barend J. Thijsse, dr. Ir. Marcel H. F. Sluiter, Dipl. Ing. Darko Simonović, and dr. Ir. Wim G. Sloof from the Department of Material Science and Engineering, Delft University of Technology, for discussions and critical review of the manuscript. This research is supported by the Netherlands Foundation for Fundamental Research on Matter (FOM) and the Netherlands Institute for Metal Research (NIMR) within the project 02EMM31.

References [1] [2] [3] [4] [5] [6] [7] [8]

M. I. Baskes, Phys. Rev. B 46, 2727 (1992). M. I. Baskes, Modified Embedded Atom Method Calculations of interfaces, Sandia National Laboratories Livermore, CA, Edited by S. Nishijima and H. Onodera, (1996). T. Campbell, R.K. Kalia, A. Nakano, P. Vashishta, S. Ogata, and S. Rodgers, Phys. Rev. Lett. 82, 4866 (1999). X. W. Zhou, H. N. G. Wadley, J. S. Filhol, M.N. Neurock, Phys. Rev. B 69, 035402 (2004). R. W. Hockney, J. W. Eastwood, Computer Simulation Using Particles, (McGraw-Hill, New York, 1981). R. Fletcher and C. M. Reeves, Computer J. (UK) 7, 149 (1964). M. Timonova, I. Lazic, and B.J. Thijsse, to be published. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes, Chapter 10, Cambridge University Press, 2nd edition, (1992).

50

Chapter 4 Al/O potential construction - Electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

4. An improved molecular dynamics potential for studying aluminum oxidation. Part I - Parameter optimization for the electrostatic part of the potential

Abstract We present an improved embedded atom + charge-transfer potential for molecular dynamics studies of the self-healing capability of aluminum oxide layers, i.e. the quick regrowth after damaging. The potential is based on the potential of Zhou et al. (X. W. Zhou, H. N. G Wadley, J.-S. Filhol and M.N. Neurock, Phys. Rev. B 69, 035402 (2004)), but it differs in two important respects. First, we present ab initio calculations of several Al/O structures, together with simulation results of initial oxidation of aluminum, which show that the potential of Zhou et al. needs to be improved. This motivated a new determination of the potential parameters. Second, we have added angular interactions to the non-electrostatic part of the potential, in the form of a revised version of the Modified Embedded Atom Method, in order to obtain a finer control over the energy differences between local atomic environments and to improve the description of the relatively open oxide structures. In this chapter, which is the first of a series of two, we present the results of fitting the electrostatic charge-transfer part of the potential to the ab inito results of the electrostatic potential field in different Al/O crystal structures. We find that with the new values of the charge-transfer parameters, the potential field and especially the atomic charges are significantly better represented in Al/O structures that are particularly important for the initial oxidation of an aluminum surface: αalumina, γ-alumina, and a single interstitial oxygen atom in fcc Al. The results of fitting the nonelectrostatic part of the potential to the ab initio energies reduced by the currently determined electrostatic energies will be presented in the next chapter, together with a performance analysis of the full potential.

51

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 4.1. Introduction Surface oxides on aluminum and aluminum alloys can be called “self-healing” in that they quickly re-form after being damaged. The initial stage of oxidation is very rapid and therefore difficult to follow experimentally [1]. Here atomistic simulations can help to provide insight into processes occurring at small time and length scales. Studying such self-healing phenomena using molecular dynamics (MD) simulations, however, requires sophisticated interatomic potentials of the underlying metal-oxygen systems. For oxides and metal/metal-oxide systems the computational costs of MD simulations are much higher than those for purely metallic systems, because of the presence of ionic bonds, which give rise to long-range Coulomb interactions. Moreover, at interfaces, surfaces, and defects, the atomic charges cannot be considered fixed, and MD models that allow charge transfer should be applied. Finally, aluminum oxide has a relatively open structure, which suggests that angular terms in the interatomic potentials play an important role. For these reasons we develop a potential that combines a Charge Transfer Ionic Potential (CTIP) [2] for the electrostatic interactions with a Modified Embedded Atom Method (MEAM) potential [3] for the non-electrostatic interactions. We aim at a fairly general potential model, one that would also be applicable to other technologically important materials, such as the relatively open oxides of Si and to the mildly ionic III-V and II-VI semiconductors. This chapter is the first of two reports on the development of this potential for the Al-O system and on the application of a novel fitting process [4] of the potential to ab initio computed data for AlxOy compounds. In the fitting process, the parameters of the electrostatic and nonelectrostatic parts of the potential are treated as separate groups, and these parts are presented in two publications. This work was undertaken to overcome some of the limitations of earlier work: MD simulations on Al/Al2O3 have been performed using MEAM but without considering charges explicitly [5], and also using charge transfer ionic potentials but in combination with EAM, which does not include angular dependencies [2, 6]. Here we present a new version of the MEAM, in combination with the CTIP published in ref. [2] but with completely re-fitted parameters. Unlike previous work, we use the iterative Particle-Particle-Particle-Mesh (PPPM) method as Coulomb solver [7]. PPPM easily adapts to periodic and nonperiodic boundary conditions, and, because of its localized computability, it is convenient for code parallelization. An additional reason is that the computational efficiency of PPPM is O(N log N) while, e.g., for Ewald Summation it is O(N 3 / 2 ) [8, 9]. A minor disadvantage of PPPM is that the maximal error can not be predicted analytically. For the ab initio energies we have used the Vienna ab initio simulation package (VASP) [10-13]. In Sec. 2 the new MEAM+CTIP potential is presented, along with a short summary of the ! ! PPPM method. In Sec. 3 the results of a few simulations using the existing EAM+CTIP potential [2] are discussed, together with an ab initio data collection for different AlxOy compounds. We show various reasons why the best potential so far [2] needs to be improved. Sec. 4 describes the novel fitting method of the electrostatic part of the MEAM+CTIP model, and the results are discussed in Sec. 5. Summary and conclusions follow in Sec. 6.

52

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 4.2 The MEAM+CTIP model According to our MEAM+CTIP potential, the energy U of a system of N atoms is the sum of a nonelectrostatic and an electrostatic part, U = U nes +U es , where the non-electrostatic MEAM part is given by N N N 1 U nes = $ $ "ij (rij ) + $ Fi (x i ) . i=1 j=1 2 i=1

(1)

j#i

Here the embedding functions F(x) for each of the chemical types (in this work Al and O) and the pair potential functions Φ(r) for each of the type combinations (AlAl, AlO, and OO) are given by ! "(r) = #E p (1+ $ + c p$ 2 + d p$ 3 )e#$ ,

$ = % p ( r /rp #1) ,

F(x) = Ae x ln x + Ce x ,

(2)

(3)

! where the subscripts p and e denote fitparameters. The quantity xi is proportional to the square of ! what in (M)EAM terminology is called the “background electron density at atom i” and contains all dependencies on the i-angle in the ijk atom triplets, 3

x i = % t i(l ) % % p(lj )e l= 0

"q (j l ) rij

(l)

p(lk )e"q k

rik

P (l ) (cos# jik ) .

(4)

j$i k$i

Here, t(l), p(l), q(l) (l = 0, 1, 2, 3) are fitparameters for each chemical type (except t(0) = 1), and P(l) is ! polynomial of order l. For simplicity, Eqs. (1-4) are given without angular and radial the Legendre cutoff functions. New in this MEAM format, compared to the classical MEAM format [3], is that the embedding function has been extended by one term, that the concept of “reference structure” has been abandoned, and that the pair potential is a parametrized (Morse-like) function rather than a function determined by a prescribed equation of state of a particular crystal phase. Eqs. (1-4) are computationally easier to handle and more powerful than the classical format, without changing much in the underlying physical picture. With these expressions, for example, it proved possible to represent the well-known Stillinger-Weber and Tersoff-III silicon potentials to a very high degree of accuracy [14], which suggests that they will be appropriate for the current Al-O system as well. For the electrostatic CTIP part we use the potential model proposed by Zhou and coworkers [2]. In our implementation the long-ranged Coulomb interactions are handled by the PPPM method [7] rather than the Ewald summation method [8]. A brief explanation of the PPPM method follows here, see also Fig. 1. Assuming pointlike charges for simplicity, the interaction between a charged atom i and all other charged atoms j separated by rij including all periodic images, is divided into two parts by adding and subtracting a charge distribution ! j (r) around every atom j. The volume integral of this distribution must be equal to qj. As it is chosen to be Gaussian, 53

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– ! j (r) = q j" #3$ #3/2 exp(#r 2 / " 2 ) , it can be shown that the interaction energy between point charge i and point-charged atom j minus j’s own charge distribution is kc qiq j erfc(rij /" ) /rij , where

kc = 1 / 4 !" 0 . The complementary error function decreases rapidly with distance, which means that this part of the interaction is short-ranged, and a cutoff radius can be used, normally on the order of ! results. The rest of the interaction 5 Å. A value of one third of the cutoff radius for σ gave the best is calculated by considering point charge i placed in a mesh containing an electrostatic potential determined by the Poisson equation ! 2" mesh = #$ mesh / % 0 . Here, ! mesh is the total charge of all individual charge distributions ! j (r) , plus i’s own charge distribution ρi(r), allocated to the mesh points in two steps: first, staring from Dirac peaks, qi" (r # ri ) , at the original atom positions charges are assigned to the nearest eight mesh points and second, from there the charges are distributed further through a diffusion-like algorithm. In this way ! mesh is the same for all atoms i, and the Poisson equation needs to be solved only!once. For each atom, the spurious interaction between its point charge and its own charge distribution can then be determined analytically (self-electrostatic potential below) and subtracted.

a)

b) Fig. 1. Schematic illustration of the PPPM algorithm. a) Arbitrary atom i with charge qi shown surrounded by other charged atoms j. For simplicity the charges are shown as positive delta-functions. b) Atoms j shown with additional charge distributions +ρj(r) and –ρj(r). c) The short-range particle-particle part (PP) of the electrostatic interaction between atom i and all other atoms is calculated directly, between qi and the charges qj–ρj(r) that lie within the cutoff radius rdir. The mesh is not used here. d) Atom i and the remaining charge distributions with which i interacts. e) Atom i shown with the remaining charge distributions discretely represented at the mesh points, also including i’s own additional charge distribution +ρi(r). The interaction between particle i and the mesh (“PM”) is calculated by solving the Poisson equation on the mesh and multiplying the determined electrostatic potential field by the charge qi. The different gray scales indicate the two steps of the diffusion-like algorithm allocating the charge distributions to the mesh points. f) Self-interaction of qi with –ρi(r) as final correction. The total electrostatic energy of the system is computed by repeating steps a) to f) for all atoms and adding the results. The Poisson equation has to be solved only once.

54

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– In summary, one obtains for the Coulomb energy of atom i,

E iCoulomb =

%r ( 1 qq 1 N qiq j 1 N kc = # k c i j erfc' ij * + qi (+ mesh (q) , + self (qi )) , # 2 j=1 rij 2 j=1 rij &$ ) 2 j"i

(5)

j"i

in which the factor 1/2 compensates for double counting, the vector notation q is used to denote the ! set of all charges, q = (q1,q2 ,q3 ,...,qN ) , and ! self (qi ) = 2kc qi / " # is the self-electrostatic potential. The mesh size should be no more than one quarter of the cutoff radius. According to the CTIP model the charges in the system are not constant. Every atom is ! become ionized by charge transfer to and from other atoms. The ionization energy allowed to

E iionization (qi ) = " iqi + J iqi2 /2 ,

(6)

with ! i the electronegativity and Ji the electrostatic hardness of atom i, is accompanied by the ! appearance of the Coulomb energy. The charges that atoms obtain for a certain atomic arrangement are determined by searching for the minimum of the total electrostatic energy Ues (ionization plus Coulomb) under the condition of total system charge neutrality. This energy is given by N # & 1 N N 1 U es = )% " iqi + J iqi2 ( + ) ) k c (qi Z j ([ j | f i ] * [ f i | f j ]) + q j Z i ([i | f j ] * [ f i | f j ])) + $ ' 2 i=1 j=1 2 i=1 j+i

(7)

## q * q & & # q *q & 1 kc qiq j [ f i | f j ] + ),%%%%1* i min,i (((qi * qmin,i ) 2 + %%1* max,i i (((qi * qmax,i ) 2 (( , ) ) 2 i=1 j=1 qi * qmin,i ' $ qi * qmax,i ' i=1 $$ ' N

N

N

j+i

!where the notations [a | f b ] and [ f a | f b ] (a=i,j; b=i,j; a≠b) denote the Coulomb interaction integrals

f b (rb ) f (r ) f (r ) dVb and [ f a | f b ] = " " a a b b dVa dVb . Here, dVa and dVb are the two rav rvv Vb Va Vb ! r and r are the center distances between atom a and dV , and atom b integrating!volume units, a b a a and dVb! and dVb respectively, rav is the center distance between atom ! , and rvv the center distance ! ! b . This is according to the charge model actually used [2], in which each atom between dVa and dV ! part and a!distributed does not have a!single ! pointlike charge as suggested so far, but!a pointlike ! ! ! !for each of the chemical types. part, " iatom (r)!= Z i# (r) + (qi $ Z i ) f i (r) , where Zi is a model parameter !For mathematical ! simplicity the distributed part of the atomic charge has been chosen to be 3 f i (r) = " i exp(#2" i r) / $ , with " i also a model parameter. The last term in Eq. (7) is an extra term ! that softly bounds the charges between the values qmin and qmax to account for chemical valence [a | f b ] =

!

"

( ! > 0 ). The constant ω was kept fixed at a value of 20.0 eV/e2. After applying PPPM (Eq. (5)) to Eq. (7) we obtain ! ! ! 55


U es =

$$ q # q ' 1 N ' $ q #q ' 1 N 2"&&&&1# i min,i ))(qi # qmin,i ) 2 + &&1# max,i i ))(qi # qmax,i ) 2 )) + * (2 + iqi + J iqi2 + qi (, mesh (q) # , self (qi ))) + * 2 i=1 %% qi # qmin,i ( % qi # qmax,i ( ( 2 i=1

(8)

$1 $r ' 1 1 1 ' kc (qi Z j ([ j | f i ] # [ f i | f j ]) + q j Z i ([i | f j ] # [ f i | f j ])) + * * k c qiq j && erfc& ij ) + ([ f i | f j ] # ))) . * * 2 i=1 j=1 2 i=1 j=1 rij ( %. ( % rij N

N

N

j-i

!

N

j-i

The minimization of Ues together with the condition

"

N

q = 0 is performed using the conjugate

i=1 i

gradient method [15]. For the line minimization in charge space the Brent algorithm is used [15]. The Poisson equation is solved by the successive overrelaxation (SOR) method [15]. The accuracy of the CTIP/PPPM implementation and the!effect of the angular forces in the CTIP+MEAM potential have been discussed earlier [1, 16]. It is important to note at this point that the non-electrostatic and electrostatic parts of the potential are completely separate and independent contributions to the total energy of the system. The parameters for the electrostatic part do not influence the non-electrostatic part in any way. This property is exploited in this work. Hence, the parameters that we will need to find in this work are the electrostatic parameters J i , Z i , " i , qmin , and qmax for Al and O, and the electronegativity difference "# O$Al defined as ! ! ! !

! = # O $ # Al . "# O$Al

(9)

! Because of overall charge neutrality, one of the electronegativities is redundant, so that in this case of a binary system only the!difference is an independent parameter. The soft bounds on the atomic Al Al charges were kept fixed at their chemical valence values, qmin = 0 , qmax = 3 and qOmin = "2 , qOmax = 0 , leaving seven electrostatic parameters to be fitted to ab initio data.

! 4. 3 Original ZWFN potential and ab inito data

!

!

!

Before explaining the method for determining optimal electrostatic parameter values for our potential we first briefly discuss results obtained with the original Al/O potential published in [2], further referred to as the ZWFN (Zhou, Wadley, Filhol, and Neurock) potential. Several results on aluminum oxidation using this potential have been reported earlier [2, 17, 18]. Although these results appeared to be physically realistic, we decided that the correctness of the charge transfer model needed to be verified in more detail. Hence new simulations were performed. Results on oxide growth on an Al (111) surface from an atomic oxygen vapor, using the ZWFN potential are shown in Fig. 2, which presents three stages of the oxidation process. To be in accordance with the experiments reported in ref. [19], the overall system temperature was controlled at 375 K. The timestep used in the simulations was 0.177 fs, and the charges were calculated every 20 timesteps. With this we gained some calculation speed while not influencing the resulting charges much. In average during 20 timesteps atoms were moving around 1.5 pm. 56


Fig. 2. Initial oxide growth on an Al (111) surface. Color scales and sphere sizes represent atomic types (blue, big = Al; yellow, small = O). a) Starting configuration, shown shortly after the introduction of the first 12 oxygen atoms, with thermal initial energy and randomly directed toward the surface. The bold arrow shows the level at which the oxygen atoms are introduced. b) Configuration after 0.6 × 1015 impacted O atoms per cm2. c) Configuration after 1.2 × 1015 impacted O atoms per cm2. In the figure, d indicates the increase of the surface height (3.2 Å) and D is a measure of the oxide thickness (9.4 Å). The two horizontal arrows indicate the deepest level at which oxygen atoms are found.

In Fig. 2 one clearly sees that the O atoms are not getting deep into the aluminum. The deepest atoms are located between the second and third monolayers of the Al crystal. At the same time, Al atoms have started to diffuse upward, forming the oxide layer on the top. This is in good agreement with experimental results [19] and theoretical predictions [20]. Three issues, however, were found to be unrealistic: (1) The negative charges established on the oxygen atoms, shown in Fig. 3, are at most –0.71 e, much too low compared to the valence value –2 e. (2) After approximately two monolayers of amorphous oxide formation the γ-Al2O3 crystal structure should appear (according to experiment), but it did not in the simulation, as Fig. 3 demonstrates. (3) In reality oxygen molecules rather than oxygen atoms approach the surface. In ZWFN potential, however, the O-O interaction was found to be unrealistic: a much too low O2 dissociation energy (0.023 eV, while the experimental value is 2.575 eV [21]) and a much too large O-O equilibrium Fig. 3. Configuration of Fig. 2c with the color scale showing distance (3.97 Å, while the experimental atomic charge value is 1.21 Å [21]).

57

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– In addition to these issues, a comparison of different aluminum oxide crystal structures, representing Al and O atoms in various local environments, reveals more shortcomings of the ZWFN potential. Table I shows the ab initio energy minima and the corresponding lattice constants calculated by VASP together with the values calculated by the ZWFN potential for different AlxOy crystal structures. (In the energy minimization process, the ratios of the three lattice constants of each crystal structure were kept constant.) The last two columns list the atomic charges determined by the ZWFN potential. Note that the determination of charges using ab initio calculations is not very well defined. The Bader analysis [22] is one of the possible ways, but it is not always successful or sufficiently precise. However, total energies can always be compared between ab initio and classical calculations. Table I. Minimum energies and corresponding lattice constants for different AlxOy crystal structures calculated using ab initio and ZWFN MD calculations. In the energy minimization process, the ratios of the three lattice constants of each crystal structure were kept constant. One, two, and three lattice constants denote, respectively, cubic (a), tetragonal (a, c), and orthorhombic (a, b, c) unit cells. The last two columns show the charges on the atoms according to the ZWFN potential. An asterisk denotes a value averaged over non-equivalent atoms. Compounds in parentheses are prototype crystal structures. Experimental values for αAl2O3, based on various sources, are shown in square brackets. Crystal structure Ab initio minimum ZWFN MD minimum Charge on Al Charge on O atom [e] atom[e] Energy Lattice Energy Lattice (ZWFN MD) (ZWFN MD) [eV] const. [Å] [eV] const. [Å] AlO2 (CaF2) α-Al2O3

-5.38 -6.60 [-6.46]

γ-Al2O3 – Fd3m

-6.53

γ-Al2O3 – I41/amd

-6.51

AlO (ZnS) AlO (NaCl) AlO (CsCl) AlO (CuAu) Al20O16 (I41/amd, no vacancies) Al2O (Co2 B)

-5.12 -4.55 -4.05 -3.54 -3.75

Al3O (Ni3P)

-3.65

Al4O (Fe4 C) O in fcc Al (Al32O) fcc Al

-3.99 -3.58 -3.46

-3.79

4.67 4.76 12.99 [4.76] [12.99] 7.91 23.73 11.20 5.60 23.56 4.41 4.33 2.68 3.44 6.79 9.46 5.49 4.62 9.47 4.71 4.18 4.05 4.05

-5.43 -6.40

4.94 4.76 12.99

2.53 2.85

-1.26 -1.90

-6.24

7.80 23.40 11.04 5.52 23.22 4.06 3.99 2.51 3.26 5.99 8.35 5.27 4.43 8.98 4.47 4.18 4.05 4.05

2.89 *

-1.93 *

2.84 *

-1.89 *

2.03 2.02 1.67 0.97 1.45 *

-2.03 -2.02 -1.67 -0.97 -1.82 *

0.44 *

-0.88 *

0.35 *

-1.05 *

0.23 * 0.03 * 0.0

-0.91 * -0.96 * 0.0

-6.23

-6.66 -7.05 -6.22 -4.97 -5.44 -4.55 -4.62 -4.08 -3.64 -3.58

58

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– The VASP calculations used the projector augmented wave (PAW) method [23]. The exchange correlation potential was of the generalized gradient approximation (GGA) type as formulated by Perdew and Wang [24]. Integrations in reciprocal space were preformed by sampling with Monkhorst-Pack grids. The product of the number of atoms and the k-points used in all calculations was constant, around 3000. Precision was set to “medium”. In all calculations the electronic wave functions were expanded in terms of plane waves up to a cutoff kinetic energy of 400 eV. The convergence criterion for energy was 1meV. Table I shows that the ZWFN energies for AlO (ZnS structure) and AlO (NaCl structure) are much lower than those computed with ab initio calculations, and –even more disturbing–, that they are lower than the energy of the most stable Al2O3 crystal structure, corundum (α-Al2O3). The implication of this is that in MD simulations a corundum crystal, allowed to sample phase space at elevated temperatures, might collapse into some other structure other than α-Al2O3, which would be unrealistic. As expected, the ab initio results do not suffer from this inconsistency (α-Al2O3 has the lowest energy). In addition, a structure of particular importance for this work, one oxygen atom in pure Al, has an unphysically low charge transfer as determined by the ZWFN potential (only –0.96 e on the O atom). Ab initio estimates using the Bader method, however, suggest a maximum possible charge transfer, the oxygen valence value (–2 e). This single-oxygen configuration is especially important to be reproduced as realistically as possible, because it represents the very beginning of the oxidation process. In the present calculations a single oxygen atom in pure Al is modeled by a 32atom fcc Al supercell containing an O atom at an octahedral interstitial position. Comparing the ab initio and ZWFN lattice constants one notices that the differences are on the order of 0.3 Å, with the ZWFN potential in most cases predicting a too small unit cell. For structures in which the Al/O ratios are close to or less than that in α-Al2O3 the ZWFN energy values show better agreement with the ab initio results than for large Al/O ratios (AlO2 (CaF2 structure) is an example). Also the values of the charges are more reasonable. This may not be surprising because the ZWFN parameter set was obtained by fitting to certain properties of αAl2O3 (lattice constants, “assumed” charges on the atoms [25], cohesive energy and elastic constants). Nevertheless, the overall picture indicates that the ZWFN potential is not accurate enough to study self-healing phenomena in detail. This conclusion has motivated the current work.

4.4 Novel fitting approach for the Al/O potential – electrostatic part In this work we set out to upgrade the potential from EAM+CTIP to MEAM+CTIP and simultaneously determine a new set of parameters for the Al/O system. As indicated above, the MEAM is able to control the energies associated with atomic triplet angles and thus offers a finer control over the energy differences between competing structures. Developing this new potential, by fitting its parameters to ab initio energies, is a major task. However, it can be simplified by following a two-step strategy analogous to the one published in ref. [4]. First the seven electrostatic parameters of the CTIP model "# O$Al , J Al , Z Al , " Al , JO , ZO , "O are determined, and subsequently

!

!

!

!

59 ! ! !

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– the remaining parameters of the MEAM model (of which there are 60 in total, namely 39 parameters in Eqs. (2-4) (after removing redundancies) and 21 parameters in the cutoff functions (not shown) [26]). We leave the second part of this procedure to a forthcoming paper and will treat the electrostatic part here. Although conceptually transparent in a classical context, the separation of the nonelectrostatic and electrostatic parts of the energy in ab initio calculations is not possible in such an unambiguous way as it is in classical MD models. Fitting directly to atomic charges is no viable option, because, as mentioned above, charges cannot be reliably extracted from first principles calculations. Instead it is much better to fit to the electrostatic potential field " es (r) , which is a welldefined quantity in both ab initio and classical calculations. The original idea comes from ref. [4], in which it was applied to liquid HF. Applying it to a solid system with variable binary composition, such as AlxOy, is new. Hence, we determine the seven ! electrostatic CTIP parameters, such that the electrostatic potential field calculated using the MD model, optimally fits the ab initio electrostatic potential field, available on a mesh, in a large number of points outside the atomic cores. The regions inside the atomic cores should be avoided because here the ab initio results cannot be trusted [4]. As in ref. [4] we consider a spherical region within a radius of 1.5 Å not safe. From our MD program CAMELION the electrostatic potential field value " es (r) is obtained as N

" es (r) = " mesh (q,r) + % kc (Z i # qi )($ i + i=1

N ' r # ri * 1 1 #2$ / r#r )e i i + % k c qi erfc) ,, r # ri r # ri ! ( & + i=1

(10)

following the same PPPM approach as in the case of the potential energy. ! The parameters ΔχO-Al, J Al , and JO are implicitly included in the determination of charges during minimization. The other four parameters take part in the minimization as well, but they also appear explicitly in Eq. (10). The electrostatic potential field is fitted to the ab initio data for AlxOy crystal structures of various types. ! ! Because the ab initio field can only be determined up to a constant, and this constant may be different for different structure types, we have to consider these constants as additional fit (but not model) parameters C(type) . Instead of fitting " MD (q,r,type) directly to ab initio field " VASP (q,r,type) , we fit " MD (q,r,type) + C(type) to the ab initio field. 4.4.1 Fitting procedure

!

!

! ! Using Eq. (10), the electrostatic potential field is calculated in several hundreds randomly chosen points outside the atomic cores of atoms in different AlxOy systems, to serve as fitting targets. The positions of these points are chosen so that they belong to the VASP output field mesh.

4.4.1.1 Crystal structures for fitting The only stable crystalline compound in the Al-O system is α-Al2O3. Nevertheless, in order to sample different local geometries of Al and O atoms, many of which occur during initial oxidation of aluminum, any other crystal structure AlxOy can be selected for fitting the electrostatic potential field, as long as the same atomic positions are used in the classical and in the first principle

60

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– calculations. Ideally, after a successful fit to a few different structures, the potential should be able to fit any structure with the parameter values obtained. However, this is seldom the outcome, since a classical potential is a simplification of the quantum mechanical truth, and therefore compromises will have to be sought. For the present work the primary requirement must be that the most relevant crystal structures for oxidation should be described accurately by the potential. Pure Al systems and pure O systems do not include ionic bonds and therefore do not play a role in fitting the electrostatic part of the potential. Candidate Al/O structures are the ones collected in Table I, covering Al/O ratios from 1/2 to more than 4 (top to bottom). Clearly, there is considerable room for improving the agreement between the ab initio and classical results, as we already mentioned in Sec. 2.1. Here we do not fit the total energies, because that would also involve the non-electrostatic part, but we concentrate on the electrostatic potential field and inspect the resulting charges. As will be shown in the next section, most of the structures in Table I were included in fitting the field, but higher priority was given to three structures of special interest for the recovery of a damaged aluminum oxide layer. These are: (1) α-Al2O3, alpha alumina or corundum, the most stable structure; (2) γ-Al2O3, the crystal structure experimentally observed to grow on Al (111) surfaces and also favored by crystallographic and thermodynamic analysis [27]; (3) one oxygen atom in the Al bulk, approximated as Al32O, the structure that can serve as a prototype of the very beginning of the oxide growth. Of these three structures, γ-Al2O3 needs a few more comments, since in the literature there is no complete agreement about the positions of the atoms in the unit cell. The unit cell itself is also the subject of debate. Two possible crystal structures have been proposed as reference configurations: Fd3m and I41/amd [28]. In order to accommodate an Al/O ratio of 2/3, both unit cells have to include vacancies, of which the randomness of their positions is under much discussion [29]. Both crystal structures are included in Table I, and we have adopted the atomic positions that were recently determined to be the most energetically favorable of all possible vacancy combinations, using first principles calculations [30]. 4.4.1.2 Details of the fitting procedure Several fits were performed employing the simulated annealing method [15], using different starting parameter values, different sampling points of the electrostatic potential field and including different structures into the fitting: Fit 1: Only to α-Al2O3, starting from the ZWFN parameters. A total of 300 randomly chosen field points outside a 1.5 Å radius from each atomic core were used as the fitting targets. Fit 2: Only to γ-Al2O3 based on Fd3m crystal structure, starting from the ZWFN parameters and from the parameter set determined in fit 1. The fit was performed in 500 field points. Fit 3: To α-Al2O3 (300 points) and γ-Al2O3 (500 points) simultaneously, starting from the ZWFN parameter set. Fit 4: To α-Al2O3 (300 points), γ-Al2O3 (500 points) and the single O atom in fcc Al system (200 points) simultaneously, starting from the ZWFN parameter set. As a consistency check, a duplicate fit was executed, initialized from the parameter values obtained in fit 3. The result was essentially the same.

61

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Fit 5: To α-Al2O3 (300 points) and AlO in the NaCl, CsCl, CuAu, ZnS structures (200 points each), starting from parameter set determined in fit 1. This fit was executed mainly to test the effect of including more exotic instead of more relevant structures. All data points were given equal weights in the fitting. Although the great advantage of simulated annealing fit methods is that no derivatives of the objective function are needed, there is a price to pay in terms of computational speed. The long-ranged electrostatic interactions (included in our minimization process for the determination of charges, unlike the approach followed in [4]) make the fitting procedure relatively costly. On a 2.4 GHz single processor machine, a typical fit including three different crystal structures takes three days. Each extra crystal structure in the fit introduces one extra minimization procedure call and, as mentioned before, one new fitparameter C(type).

4.5 Results and discussion Fig. 4a shows the ab initio results for the electrostatic potential field in sequentially arranged mesh points between the atoms in the α-Al2O3, γ-Al2O3, and single O in fcc Al structures. The points shown are not just those used during the fitting but all mesh points available as output of VASP. Fig. 4b displays the difference between the VASP results and the values calculated in the same points with the ZWFN potential. Note the difference in vertical scales. The root mean square n

differences for the three cases, defined as RMS =

$ ("

MD k

# " VASP + C(type)) 2 /n , where n is the k

k=1

number of displayed points of the potential field, are 0.613 V, 0.873 V, and 0.720 V, respectively. Interestingly, the agreement for α-Al2O3, to the properties of which the ZWFN potential was fitted, ! ! two structures. Fig. 5 shows the field differences after the four is not much better than for the other most important fits described in the previous section, Table II lists the corresponding RMS values (including fit 5), and Table III shows the atomic charges according to the MD model resulting from two of the five fits. Table II. Root mean square electrostatic potential field differences between MD and ab initio data for α-Al2O3, γ-Al2O3 and single O atom in fcc Al. Values in italics denote structures included in the fit. RMS difference [eV/e] α-Al2O3 γ-Al2O3 O in Al ZWFN Fit 1 Fit 2 Fit 3 Fit 4 Fit 5

0.613 0.230 0.529 0.274 0.296 0.327

0.873 2.442 0.791 0.928 0.919 1.415

62

0.720 1.067 1.264 1.064 1.050 0.700


Fig. 4. Electrostatic potential field multiplied by elementary charge (–e) at mesh points in 3D space 1.5 Å away from atom centers. The horizontal axis has no other meaning than numbering the mesh points in which the field is shown. a) left: calculated ab initio for α-Al2O3; middle: calculated ab initio for γ-Al2O3; right: calculated ab initio for a single interstitial O atom in fcc Al (Al32O). b) Difference between ab initio fields and fields calculated with the ZWFN potential. Note the difference of vertical scales for the three structures.

Fig. 5a displays the result for fit 1. The RMS value for α-Al2O3 has improved from 0.613 V to 0.230 V, and the MD charges for α-Al2O3 come out as +3.00 e and –2.00 e for aluminum and oxygen, respectively. In fact, as Table III shows, the atomic charges of all structures have improved after fit 1, notably the charge on the single oxygen atom in the aluminum matrix (–2.08 e), which is now close to its full valence value. All this is an encouraging result, given the fact that the charges themselves were not fitted to, and it gives credibility to the current fitting approach. However, as Fig. 5a and Table II show, the results of the potential field of the other two structures, γ and single-O, which were left out of the fit, have considerably deteriorated. This clearly underlines that for a comprehensive description of oxidation it is not sufficient to consider only the α phase. Summarizing this result, compared to the ZWFN data, fit 1 has considerably improved the potential field for the α phase and the charges in all phases. The remaining task therefore is to improve the potential field for γ and single-O without sacrificing too much of the α phase and the charges. As a first indication of whether this is possible, Fig. 5b shows the result of fitting only to the γ phase. Both α-Al2O3 and γAl2O3 show improvement with respect to the ZWFN values but not by much (Table II). The single O case, however, is still fairly unrealistic, with a further deteriorated potential field and a charge of only –1.51 e on the O atom, significantly below its valence value.

63


Fig. 5. Electrostatic potential field differences shown as in Fig. 4b, after fitting the MD expression (Eq. (10)) in four different ways, see also Table II. a) Fit 1, only to α-Al2O3. b) Fit 2, only to γ-Al2O3. c) Fit 3, to α-Al2O3 and γ-Al2O3. d) Fit 4, to α-Al2O3, γ-Al2O3, and single O in fcc Al. Note the difference of vertical scales for the three structures.

At least this fit shows that fitting to the γ phase does not necessarily imply that the correspondence for the α phase becomes worse. The relatively high RMS value of the γ phase suggests that the electrostatic field in this structure is intrinsically difficult to fit with the current CTIP model. In the non-electrostatic part of the potential such disagreements should be compensated, when it comes to the energy of the structures.

64

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Table III. Charges on the atoms according to the CTIP model after fitting. Only results of fit 1 (to α-Al2O3) and fit 3 (to α-Al2O3 and γ-Al2O3) are shown. An asterisk denotes a value averaged over non-equivalent atoms. Fit 1, to α-Al2O3 Fit 3, to α-Al2O3 and γ-Al2O3 Crystal structure AlO2 (CaF2) α-Al2O3

Charge on Al [e] 2.48 3.00

Charge on O [e] -1.24 -2.00

Charge on Al [e] 2.02 2.90

Charge on O [e] -1.01 -1.93

γ-Al2O3 - Fd3m *

2.58

-1.72

2.74

-1.83

γ-Al2O3 – I41/amd

2.44

-1.63

2.74

-1.83

AlO (ZnS) AlO (NaCl) AlO (CsCl) AlO (CuAu) Al20O16 (γ-Al2O3 no vacancies, I41/amd) * Al2O (Co2 B) * Al3O (Ni3P) * Al4O (Fe4 C) * O in fcc Al, Al32O * fcc Al

2.11 2.08 2.06 1.97 1.61

-2.11 -2.08 -2.06 -1.97 -2.02

2.09 2.07 2.04 1.46 1.63

-2.09 -2.07 -2.04 -1.46 -2.04

1.04 0.71 0.54 0.07 0.0

-2.07 -2.12 -2.14 -2.08 0.0

1.02 0.69 0.53 0.06 0.0

-2.04 -2.07 -2.11 -2.04 0.0

Fig. 5c shows the result of fit 3, the simultaneous fit to α-Al2O3 and γ-Al2O3 starting from the ZWFN parameters. We find that the potential field for the α phase and the single O phase are quite acceptable, and that, compared to fit 1, the γ phase is now well described. Also, Table III shows that the charges of all phases are physically realistic. This shows that fit 3 fulfills most of the requirements, which makes it the strongest candidate for the optimal result. The result of simultaneously fitting to all three phases α-Al2O3, γ-Al2O3, and single O (fit 4) is shown in Fig. 5d. Compared to fit 3, this fit improves the γ and single O phases only very slightly, while the α phase deteriorates by a much larger percentage. The price to pay is too high. Finally in Table II the last row shows the results of including other structures into the fitting (fit 5). Compared to fit 3, only the single O phase has improved. As in the previous case, the price to pay in terms of the result for the α phase, and here also for the γ phase, is too high. Fits 4 and 5 must be regarded suboptimal. From the fit results we conclude that fit 3 is the best compromise between the three important Al/O structures on the one hand, and between potential field and charges on the other. When one computes the mean RMS fit error for the three phases, by averaging the values in Table II, one finds that fit 3 has the smallest value of all fits (0.755 eV/e), but also that, surprisingly, the original non-fitted ZWFN value is still lower (0.735 eV/e). Keeping in mind that fit 3 yielded “good” atomic charges and ZWFN “bad” atomic charges, this means that fitting the potential field of the α phase has a particularly favorable effect on the correctness of the charges in the present CTIP model, more so than to any of the other phases. To give an indication of how well the

65

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– potential performs overall other structures we have also calculated the mean RMS fit error for all structures listed in Table I. The result is 0.789 eV/e for the ZWFN potential and 1.147 eV/e for fit 3. The fact that the average fit errors for all structures are higher than those for the three important structures only, is mainly due to non-cubic, large Al/O ratio structures as Al2O (Co2B), Al3O (Ni3P) and Al4O (Fe4C), which are less relevant for the present work on oxidation of fcc Al. We consider these errors therefore acceptable. Tables IV and V list the parameter values that resulted from fits 1-4, along with the ZWFN values and (for ΔχO-Al, JAl, and JO) experimental values. Experimental values were taken from ref. [31] as χ = (I + A) and J = (I – A), with I the ionization energy and A the electron affinity. Table IV shows that compared to the ZWFN, the fits produce approximately the same ΔχO-Al values, higher JAl values, and lower JO values. The increase of the difference between the J values is consistent with the increase of the charge transfer in the single O system, as intended. For both atomic types the electron cloud has a radius ξ–1 on the order of 1 Å, which is not unreasonable. The values for Z, the pointlike part of the electronic charge, are very small, for oxygen even totally negligible. Compared to the experimental values, ΔχO-Al is of the correct magnitude, JAl is significantly greater, and JO is significantly smaller. Table IV. Electrostatic parameters resulting from fits 1-4. The second row (Bounds) shows the ranges within which the parameters were restricted during the fitting. The bottom row shows the experimental values for the Mulliken electronegativity difference [31, 32, 33] and electrostatic hardness [31] for Al and O (see main text for definitions). The last column lists the grades (good +, reasonable ±, bad -) for three most important requirements for the new potential: α = correctness of α-Al2O3 potential field, γ = correctness of γ-Al2O3 potential field q = correctness of charge transfer. Fit 3 is considered the best fit. JAl [eV/e2]

Bounds

ΔχAl-O [eV/e] [-∞,∞]

ZAl [e]

JO [eV/e2]

[-∞,∞]

ξAl [Å-1] [0, ∞]

[0, 13]

[-∞,∞]

ZWFN

5.402

10.216

0.968

0.561

13.992

Fit 1 Fit 2

5.123 4.245

10.853 16.008

0.675 0.987

1.765 0.633

6.829 4.860

Fit 3 Fit 4 Exp.

5.146 6.519 4.320

13.709 13.108 5.540

1.126 1.075 -

0.380 0.364 -

7.125 8.104 12.160

ξO [Å-1] [0, ∞] 2.144 1.143 1.871 1.339 1.302 -

ZO [e]

αγq

[0, 8] 0.0

±+±

0.0 0.0

+-+ ±+-

0.0 0.0 -

+++ +++

The values for the parameters C are given for completness in Table V. One notes that for fit 3 the values are not far from typical work function values, as should be expected. Overall we can say that the fits produce parameter values that are not unrealistic, given the relative simplicity of the CTIP charge model. This brings us to the point where the non-electrostatic MEAM part of the MEAM+CTIP potential can be added to the fitting procedure. By calculating ab initio energies of all AlxOy crystal structures listed in Table I for a range of lattice parameters and subtracting the electrostatic energy predicted by the CTIP model using the parameters that result from fit 3, a dataset is obtained to which the MEAM energy equations, Eqs. (1)-(4), can be fitted. Data for pure Al and pure O should 66

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– also be added to this dataset. The results of this non-electrostatic part of the Al/O potential development, together with a performance analysis of the full potential, will be presented in the next chapter (Ch. 5). Table V. Parameters C(type) (Eq. (9)) resulting from fits 1-4. Values in italics denote structures included in the fit. Values in roman were not fitted but calculated as the average difference of the ab initio and CTIP potential fields. C(α-Al2O3) C(γ-Al2O3) C(O in Al) [eV/e] [eV/e] [eV/e] ZWFN Fit 1 Fit 2 Fit 3 Fit 4

4.15 4.91 6.18 7.81 7.49

4.29 7.34 6.11 7.23 6.89

8.64 18.71 5.56 5.29 5.30

4.6. Summary and conclusions We have presented an MEAM+CTIP potential that is an advanced version of the ZWFN potential [2], i.e. extended by angle-dependent interaction terms in the embedded atom formalism. For more flexibility and easier computations, reference structures are not used in the MEAM part, and the embedding function contains a quadratic term in addition to the usual logarithmic term. This MEAM+CTIP potential is intended for molecular dynamics studies of systems having directional bonding and variable charges on the atoms, depending on their local environments. Oxidation of aluminum is the application targeted in this work. Ab initio calculations of many different AlxOy crystal structures were made, and it is shown that the ZWFN potential with its published parameters has several shortcomings, notably in the predicted relative cohesion energies of different structures and the values of the atomic charges. For reliable MD studies a better potential was deemed necessary. To facilitate the development, the non-electrostatic (MEAM) and electrostatic (CTIP) parts of the potential were treated separately. Leaving the non-electrostatic part to a second paper, we have fitted the electrostatic part to the ab initio electrostatic potential field in several AlxOy structures, attributing most weight to the α and γ alumina phases and to fcc Al containing a single interstitial O atom. These are the structures of particular importance for initial and ongoing oxidation of aluminum. It was found that chemically correct atomic charges could be obtained by this method, i.e. close to the valence values –2 e for the oxygen atoms and +3 e for the aluminum atoms, in cases where this is to be expected (e.g. α-Al2O3 and single O2– in Al), whereas the charges themselves were not fitted. The optimal set of potential parameters was extracted from a fit in which the potential field of α alumina was much better fitted than by the ZWFN potential, sacrificing part of the fit quality of the γ alumina and single O phases. The mean RMS fit error of this fit is 0.755 eV/e for the three most important structures (1.147 eV/e for all structures listed in Table I), showing that the charge transfer model applied in this work performs satisfactorily. 67

Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– To complete the development of the potential, the MEAM energy equations will next be fitted to ab initio energies for all AlxOy crystal structures from which the energies predicted by the current CTIP model have been subtracted. The results will be presented in the next chapter.

Acknowledgments Author would like to thank Prof. dr. Barend J. Thisse (Tu Delft, The Netherlands), dr. PierreMatthieu Anglade (UCL Leuven, Belgium), dr. Ir. Marcel H. F. Sluiter and Dipl. Ing. Darko Simonović, (both at TU Delft, The Netherlands) for discussions and critical review of the manuscript. This research is supported by the Netherlands Foundation for Fundamental Research on Matter (FOM) and the Netherlands Institute for Metal Research (NIMR) within the project 02EMM31.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

I. Lazić, M. A. Ernst, B. J. Thijsse, Proc. First International Conference in Self Healing Materials, CD, Editors A. J. M. Schmets and S. van der Zwaag, Springer, Dordrecht, The Netherlands (2007). X. W. Zhou, H. N. G Wadley, J. S. Filhol and M. N. Neurock, Phys. Rev. B 69, 035402 (2004). M. I. Baskes, Phys. Rev. B 46, 2727 (1992). E. Bourasseau, J. B. Millet, L. Mondelain and P. M. Anglade, Mol. Simulat. 31, 705 (2005). M. I. Baskes, Modified Embedded Atom Method Calculations of interfaces, Sandia National Laboratories Livermore, CA, Edited by S. Nishijima and H. Onodera (1996). T. Campbell, R.K. Kalia, A. Nakano, P. Vashishta, S. Ogata, and S. Rodgers, Phys. Rev. Lett. 82, 4866 (1999). R. W. Hockney, J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York (1981). P. Ewald, Ann. Phys. 64, 253 (1921). R. Fletcher and C. M. Reeves, Computer J. (UK) 7, 149 (1964). G. Kresse, J. Hafner, Phys. Rev. B 47, RC558 (1993). G. Kresse, J. Furthmüller, Phys. Rev. B 54, 11169 (1996). G. Kresse, J. Furthmüller, Comp. Mat. Sci. 6, 15 (1996). G. Kresse, D. Joubert, Phys. Rev. B 59, 1758 (1999). M. Timonova, I. Lazic, and B. J. Thijsse, to be published W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, Cambridge University Press, 2nd edition (1992). I. Lazić, B. J. Thijsse, Mater. Res. Soc. Symp. Proc. 978E, Warrendale, PA, 0978-GG08-05 (2007). X. W. Zhou and H. N. G. Wadley, J. Phys.: Condens. Matter 17, 3619 (2005). X. W. Zhou, H. N. G. Wadley, D. X. Wang, Comput. Mater. Sci. 39, 794 (2007). L. P. H. Jeurgens, W. G. Sloof, F. D. Tichelaar, E. J. Mittemeijer, J. Appl. Phys. 92, 1649 (2002). Yu. F. Zhukovskii, P. W. M. Jacobs, M. Causa, J. Phys. Chem. Solids 64, 1317 (2003). R. C. Weast (ed.), Handbook of chemistry and physics, CRC Press, Boca Raton (1993). R. F. W. Bader, Atoms in Molecules-A Quantum Theory, Oxford University Press, Oxford (1990). P. E. Blochl, Phys. Rev. B 50, 17953 (1994). J. P. Perdew, Y. Wang, Phys. Rev. B 45, 13244 (1992). F. H. Streitz, J. W. Mintmire, Phys. Rev. B 50, 11996 (1994). B. J. Thijsse, Nucl. Instrum. Methods Phys. Res., Sect. B 288, 198 (2005). L. P. H. Jeurgens, W. G. Sloof, F. D. Tichelaar, and E. J. Mittemeijer, Phys. Rev. B 62, 4707 (2000).

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Chapter 4 Al/O potential construction - Electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– [28] [29] [30] [31] [32] [33]

G. Paglia, C. E. Buckley, A. L. Rohl, B. A. Hunter, R. D. Hart, J. V. Hanna, L. T. Byrne, Phys. Rev. B 68, 144110 (2003). B. Ealet, M. H. Elyakhoufi, E. Gillet and M. Ricci, Thin Solid Films 250, 92 (1994). G. Paglia, A. L. Rohl, C. E. Buckley, J. D. Gale, Phys. Rev. B 71, 224115 (2005). R. G. Parr, R. G. Pearson, J. Am. Chem. Soc. 105, 7512 (1983). R. S. Mulliken, J. Chem. Phys. 2, 782 (1934). M. E. McHenry, R. C. O’Handley, K. H. Johnson, Phys. Rev. B 35, 7 (1987).

69


70

Chapter 5 Al/O potential construction - Non-electrostatic part –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

5. An improved molecular dynamics potential for studying aluminum oxidation. Part II - Parameter optimization for the non-electrostatic part of the potential and overall result

Abstract Molecular Dynamics simulations of the self-healing capacity of Al oxide coatings, e.g. after scratching, demand the best possible description of Al oxide growth. Only with a fully developed and well-calibrated potential, the factors that control the diffusion of Al and O under locally varying Al/O ratios can be studied. To obtain this, a new “reference free” version of the Modified Embedded Atom Method (RF-MEAM) potential is developed, together with a Charge Transfer Ionic Potential (CTIP). This allows a highly advanced description of atomic interaction: angular forces for the relatively low-coordinated crystal structures such as metal oxides, and dynamic charge transfer between Al and O to handle instantaneous local composition variations. As Coulomb solver we use the Particle-Particle-Particle-Mesh (PPPM) method. We present the most complete Al-O potential up to now, being the result of fitting RF-MEAM+CTIP to a large ab initio database of Al-O structural energies using a recent method which separates the electrostatic and non-electrostatic fitting steps. In this chapter the focus lies on the non-electrostatic part of the potential, being the final part of the two-step fitting strategy.

5.1 Introduction Studying the very rapid regrowth of Al oxide on an Al substrate after damaging is a challenging task. Experimental techniques are often inadequate because the very beginning of the oxidation process stays out of their scope [1]. First principles (ab initio) computational techniques based on density functional theory (DFT) [2] are severely limited by the relatively small number of atoms they can 71

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– handle, and are not able to capture the diffusion effects that drive atoms to the area of growth. Monte Carlo methods [3] are only suitable for equilibrium processes, which do not occur at the beginning of an oxidation or in a recovery process. Using molecular dynamics (MD) for this purpose therefore seems a natural choice. In Ch.4 we have argued that a sophisticated interatomic potential is required for a proper MD description of the oxidation of a metal, and we have introduced the RF-MEAM+CTIP potential for this purpose. This potential is a combination of the Reference Free Modified Embedded Atom Method potential [4] for the non-electrostatic part of the interactions and the Charge Transfer Ionic Potential [5] for the electrostatic part. It was also shown in Ch. 4 that earlier models [5, 6] as well as their applications [7, 8, 9] suffered from several important shortcomings, because they were based on oversimplified physics and inadequate validation of the potential. Also even in the most recent work by Elsener et al. [10, 11], ionic charges and minimum energies for various Al-O systems were left out of the analysis. We therefore decided that the potential model for Al-O should be significantly improved (to RF-MEAM+CTIP) and fitted in a new and systematic way to a large collection of calculated and experimental data for the Al-O system. The fitting technique employed for this is a two-step process [12]. In Ch. 4 the electrostatic part of the potential, the CTIP, was determined. In this chapter we report on the second step of the process, the fitting of the nonelectrostatic part of the potential (RF-MEAM). We will also combine the two parts into the full potential and present a few MD results obtained by this potential: the stability of α-Al2O3, surface relaxation of an Al-terminated α-Al2O3 surface, and an O atom impacting on Al (111) and Alterminated α-Al2O3 surfaces with thermal speed. In Secs. 5.2 and 5.3 the mathematical formulations of the non-electrostatic and electrostatic parts of the potential, respectively, are given in condensed form for reference. Sec. 5.4 covers the fitting procedures, and in Sec. 5.5 the results are presented. Finally, in Sec. 5.6 the MD results are shown, and Sec. 5.7 contains the conclusions.

5.2 Reference Free Modified Embedded Atom Method (RF-MEAM) The energy U of a system of N atoms for the full RF-MEAM+CTIP model is the sum of a nonelectrostatic and an electrostatic part, U = U nes +U es , where U nes is the RF-MEAM potential. According to the RF-MEAM, the non-electrostatic energy Unes of a system of N atoms is given by N N

U nes = " "

N 1 S h (r )# (r ) + 2 ij IJ ij IJ ij

i=1 j=1 j!i

" FI ($i ) ,

(1)

i=1

in which i and j are atomic indices, and I and J denote the chemical types of atoms i and j, respectively. In this work I and J can be Al or O. For systems consisting of one chemical type, I and

72

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– J can be dropped from the equations. In Eq. (1), S is an angular screening factor, h(r) a radial cutoff function, φ(r) a pair potential, and F(ρ) an embedding function. In the following we show the mathematical forms of the functions in Eq. (1) and indicate which parameters are needed to define the potential. Embedding function FI ( !i ) = E0, I yi ln yi + E1, I yi + E2, I yi2 g ( yi ) ,

(

)

2

g(yi ) = 1 ! e yi

/2" 2

, yi =

!i . nI

(2)

(3)-(4)

For each chemical element four parameters are needed: E0 , E1 , E2 , n . However, for the first (or only) element in the system, we can set n = 1 without loss of generality. The function g(y) is a small modification to suppress the singularity in the derivative of FI(ρi) at ρi = 0. The quantity σ is a small number. The “background electron density at the location of atom i”, ρi, is defined by the following equations, lmax

"i = "i(0)G(#i ) , G(") =

2 , !i = 1+ e#"

" t I(l)#i(l)

2

l=1

#i(0)

,

2

(5)-(7)

where !

! 2 !i(l) = # # Sij hJJ (rij ) f J(l) (rij )Sik hKK (rik ) fK(l) (rik )P (l) (cos$ jik )

(8)

j"i k"i

and (l)

f J(l ) (r) = pJ(l)e"q J r .

(9)

In Eq. (7), lmax = 3, and in Eq. (8), P(l) is the Legendre polynomial of order l. Note that the MEAM simplifies into the EAM if!lmax is taken as zero, and therefore G(Γ) becomes equal to 1. For each chemical element eleven parameters are needed: t(l) (l =1, 2, 3), p(l) (l = 0, 1, 2, 3), and q(l) (l = 0, 1, 2, 3). However, for the first (or only) element in the system, we can set p(1) = p(2) = p(3) = 1 without loss of generality. Pair potential "#p,ij

2 3 ! IJ (rij ) = "Ep,IJ (1+ #p,ij + c2,IJ #p,ij + c3,IJ #p,ij )e

73

,

(10)

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– $ r ' !p,ij = " p,IJ && ij #1)) . (11) % rp,IJ ( For each distinct pair of chemical elements five parameters are needed: Ep, c2, c3, αp, rp. In a binary system there are therefore 15 such parameters. At this point it should be mentioned that new in this RF-MEAM format, compared to classical MEAM [13, 14], is the extension of the embedding function by one term (Eq. (2)) and the abandonment of the concept of “reference structures”. The pair potential, Eq. (10) is a parameterized (Morse-like) function rather than a function prescribed by the equations of state of a particular crystal phase. With this, the equations are computationally easier to handle and more powerful than the classical MEAM format. Radial cutoff function %1 ' 5 ' !" z hIJ (rij ) = &e c,IJ ij # bn,IJ zijn ' n=0 '0 (

(zij < 0) (0 $ zij $ 1) ,

(12)

(zij > 1)

with zij =

rij ! rs,IJ rc,IJ ! rs,IJ

.

(13)

The radial cut-off function is an exponentially damped 5th degree polynomial. For each distinct pair of chemical elements three parameters are needed: αc, rs, rc. The coefficients bn are taken such that the cut-off function smoothly decays between r = rs and r = rc. This means that the first and second r-derivatives of the original, non cut-off function are conserved in r = rs, and that the first and second derivatives of the cut-off function are identically zero in r = rc. For this, the following values should be used:

b0 = 1 , b1 = ! c , b2 = 12 ! c2 , b3 = !10 ! 6" c ! 23 " c2 , b4 = 15 + 8! c + 23 ! c2 ,

(14)

b5 = !6 ! 3" c ! 12 " c2 . Angular screening factor Angular screening is a unique MEAM mechanism, intended to take into account the presence of atoms k in the neighborhood of i-j atomic bonds. It was originally proposed by Baskes [15] and works as follows. The i-j interaction is weakened by a factor Sij (a number between 0 and 1), which depends on neighboring atoms k in the following way, 74


$0 (y < 0) & 4 2 B(y) = %[1" (1" y) ] (0 # y # 1) , &1 (y > 1) '

" B(y jik ) ,

Sij =

k!i, j

y jik =

C jik ! Cmin,JIK C ! !C max,JIK

,

min,JIK

C jik =

1 ! cos2 " jik . # rij & % ! cos" jik ( cos" jik $ rik '

(15)- (16)

(17)-(18)

For each distinct triplet of chemical elements JIK (distinct apart from exchanging the outer indices) two parameters are needed: Cmin and Cmax. This implies that two parameters are needed for a singleelement system and that twelve parameters are needed for a binary (AB) system, viz. two parameters Cmin and Cmax for each of the triplet combinations AAA, AAB, BAB, ABA, ABB, and BBB. For even more detailed description see Ch. 2.

5.3 The electrostatic part of the potential To complete the description of the RF-MEAM+CTIP potential we briefly show the electrostatic (CTIP) part, described in more detail in Ch. 4. The electrostatic energy Ues consists of an atomic ionization term, a Coulomb pair interaction term, and a charge bounding term [5]. The long-range Coulomb term is treated by the Particle-Particle-Particle-Mesh term (see Ch. 4). With this we obtain

U es =

' 1 N $ q #q ' 1 N $$ qi # qmin,i ' 2"&&&&1# ))(qi # qmin,i ) 2 + &&1# max,i i ))(qi # qmax,i ) 2 )) + * (2 + iqi + J iqi2 + qi (, mesh (q) # , self (qi ))) + * 2 i=1 %% qi # qmin,i ( % qi # qmax,i ( ( 2 i=1

(19)

$1 $r ' 1 1 1 ' kc (qi Z j ([ j | f i ] # [ f i | f j ]) + q j Z i ([i | f j ] # [ f i | f j ])) + * * k c qiq j && erfc& ij ) + ([ f i | f j ] # ))) , * * 2 i=1 j=1 2 i=1 j=1 rij ( %. ( % rij N

N

N

j-i

!

j-i

where χ and J are electronegativity and electrostatic hardness, and [a | f b ] and [ f a | f b ] denote the f (r ) f (r ) f (r ) Coulomb integrals [a | f b ] = " b b dVb and [ f a | f b ] = " " a a b b dVa dVb , with dVa and dVb rav rvv Vb Va Vb ! a and dV ; r between atom the two integrating volume units; r the center distance between atom ! a

!

N

a

b

b and dVb ; rav between atom a and dVb ; and rvv between dVa and dVb . This!is according ! to [5, 6], ! pointlike charge, but a pointlike part and a distributed part, in which ! an atom does not have a single ! for each!chemical type. The distributed " iatom (r) = Z i# (r) + (qi $ Z!i ) f i (r) with Zi is a model parameter ! ! 3 ! part ! has the form f i (r) = " i !exp(#2" i r) ! / $ , with "!also a parameter. ! The first term in Eq. (19) softly i limits an atom’s charge between qmin and qmax to account for chemical valence (ω = 20.0 eV/e2). The

!

!

vector q = (q1,q2 ,q3 ,...,qN ) symbolizes all charges, ! self (qi ) = 2kc qi / " # is the self-electrostatic ! ( k = 1 / 4 !" ) and " (q) !is the long-ranged Coulomb part determined by solving the potential c 0 mesh ! ! Poisson equation within the PPPM approach [16]. The charges on the atoms are determined by !

75

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– N minimizing the total electrostatic energy Ues under the condition " qi = 0 , using the conjugate i=1

gradient method [17]. The parameters in this part of the potential are χ, J, Z, and ξ for each of the chemical types of the atoms, with the stipulation that one of the electronegativities χ can be set to ! zero without loss of generality. For two chemical types, therefore, only the difference Δχ counts.

5.4 Fitting the potential to ab initio Al/O data and elastic constants Reviewing all parts of the RF-MEAM+CTIP potential, we find that in total 69 parameters are necessary for a complete definition: 7 for CTIP, 7 for the two embedding functions, 19 for the background electronic densities, 15 for the three pair potentials, 9 for the three radial cutoff functions, and 12 for the six possible angular screening situations. Exploiting the property that the non-electrostatic part, Eq. (1), and the electrostatic part, Eq. (19), are completely independent, the seven CTIP parameters were already determined in Ch. 4. Their values will be presented together with the other parameters in Sec. 5.5. Here we will describe how the remaining 62 RF-MEAM parameters are determined. For fitting Eq. (1) to the non-electrostatic energies of a variety of relaxed and deformed crystal structures, a simulated annealing technique [17] was used, of which the essentials are given in Ch. 2. The starting values of the parameters of the fit were taken from the existing ZWFN model for all parameters that are also defined for the present potential. In some cases these values had to be scaled in view of a different normalization of the functions. The starting values of the essentially new parameters (angular terms) were set either to zero or to values found in the literature for MEAM models. Fits were repeated many times in order to sample different random number sequences and to explore the effects of different annealing programs. We have also made ample use of strategically changing the weights of the datapoints in the fit to arrive at the result that was ultimately considered the best. In total 365 datapoints were used as fitting targets. All RF-MEAM parameters except those of the O-O interaction (pair potential, embedding function, background density, and radial cut-off function) were varied simultaneously. The O-O interaction parameters were kept as in ref. [18] because they describe the size and the energetics of the O2 molecule properly. Two groups of fitting targets were included in the fit, the “potential energy curves” and the elastic energies associated with elastic deformations. The potential energy curves are derived from ab initio calculated energies of different AlxOy crystal structures for a range of lattice constants. The elastic energies are based on the five elastic constants of α-Al2O3 [5] and the three elastic constants of fcc-Al [19, 20] as determined experimentally. In the following the fitting procedures will be explained in more detail. 5.4.1 Potential energy curves Table I lists the different AlxOy crystal structures involved in the current non-electrostatic fitting procedure. For each structure, ab initio energies were calculated for a range of lattice constants, or alternatively, a range of nearest neighbor distances R, in each system, together forming the potential energy curves. As target values, 20 to 30 different values of R were chosen for each structure, with 76

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– an emphasis on values close to the energy minima. To get an impression of the datapoints used in the fit, the reader may consult Fig. 1. Prior to making available the calculated energies for fitting by Eq. (1), the electrostatic energies were first subtracted from the ab initio values. This is necessary because DFT calculations do not recognize separate “electrostatic” and “non-electrostatic” contributions to the total energies. The electrostatic energy values, in turn, were calculated by Eq. (19), with the parameters earlier obtained (fit3, Table IV in Ch. 4). This splitting of the fitting procedure is essential in our two-step technique. In this way control of the charge transfer optimization was obtained without having to assume charges, while the best accuracy for the total energies could be maintained. Table I. Minimum energies and corresponding lattice constants for different AlxOy crystal structures according to ab initio calculations and the ZWFN potential, and according to the final fit of the RF-MEAM+CTIP potential presented in this work. In the energy minimization process of all crystal structures, the ratios of the three lattice constants were kept constant. One and two lattice constants denote, respectively, cubic and tetragonal unit cells. Experimental values for α-Al2O3, based on various sources, are shown in square brackets. The γ-Al2O3–Fd3m structure was not included in the fit of the non-electrostatic part. The weight 1/σ is the inverse of the expected uncertainty of the datapoints in the range 0.98 Rmin to 1.10 Rmin, where Rmin is the nearest neighbor distance corresponding to the energy minimum (these are relative values only). The root mean square (rms) fit error is given for datapoints of Figs. 1a, 1b, 1d within the range 0.95 Rmin to 1.20 Rmin. The top-to-bottom order of the structures in the table is from low to high Al/O molar ratio. Crystal structure Ab initio calculations ZWFN potential This work RF-MEAM+CTIP potential Minimum Lattice Minimum Lattice Minimum Lattice weight rms energy constant(s) energy constant(s) energy constant(s) fit 1/σ [eV] [Å] [eV] [Å] [eV] [Å] error [eV] AlO2 (CaF2) -5.38 4.67 -5.43 4.94 -5.36 4.72 100 0.011 -6.60 4.76 -6.40 4.76 -6.60 4.76 100 0.019 α-Al2O3 [-6.46] 12.99 [-6.46] 12.99 [-6.46] 12.99 [4.76] [4.76] [4.76] [12.99] [12.99] [12.99] -6.53 7.91 -6.24 7.80 -5.14 8.54 Not Not γ-Al2O3 – Fd3m 23.73 23.40 25.63 fitted fitted AlO (ZnS) -5.12 4.41 -6.66 4.06 -6.53 4.41 2 1.167 AlO (NaCl) -4.55 4.33 -7.05 3.99 -4.71 4.46 10 0.127 AlO (CsCl) -4.05 2.68 -6.22 2.51 -4.53 2.68 10 0.260 AlO (CuAu) -3.54 3.44 -4.97 3.26 -3.02 3.54 10 0.617 Al20O16 (I41/amd, -3.75 6.79 -5.44 5.99 -3.89 6.99 10 0.211 no vacancies) 9.46 8.35 9.74 Al2O (Co2 B) -3.79 5.49 -4.55 5.27 -3.88 5.54 2 0.056 4.62 4.43 4.66 Al3O (Ni3P) -3.65 9.47 -4.62 8.98 -3.96 9.66 1 0.230 4.71 4.47 4.80 Al4O (Fe4 C) -3.99 4.18 -4.08 4.18 -4.05 4.30 100 0.076 O in fcc Al, -3.58 4.05 -3.64 4.05 -3.57 4.09 100 0.015 Al32O fcc Al -3.46 4.05 -3.58 4.05 -3.46 4.05 100 0.005 hcp Al -3.44 4.05 -3.56 4.05 -3.46 4.05 100 0.052 bcc Al -3.37 3.25 -3.39 3.03 -3.37 3.20 100 0.032 sc Al -3.13 2.77 -3.44 2.72 -3.02 2.78 10 0.065 dc Al -2.79 6.11 -3.02 5.69 -2.68 6.23 10 0.126

77

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– For all ab initio calculations the VASP program was used [21]. The VASP calculations used the projector augmented wave (PAW) method [22]. The exchange correlation potential was of the generalized gradient approximation (GGA) type as formulated by Perdew and Wang [23]. Integrations in reciprocal space were preformed by sampling with Monkhorst-Pack grids. The product of the number of atoms and the k-points used in all calculations was constant, around 3000. Precision was set to “medium”. In all calculations the electronic wave functions were expanded in terms of plane waves up to a cutoff kinetic energy of 400 eV. The convergence criterion for energy was 1meV. 5.4.2 Elastic constants The elastic constants of α-Al2O3 and pure fcc-Al were included in the fit via the calculation of the elastic energies of homogeneously strained crystals [24]. For cubic and hexagonal crystals the elastic energies per unit volume are given by [25]

1 1 elas E cubic V = C11 ("12 + "22 + "32 ) + C12 ("1"2 + "2"3 + "3"1 ) + C44 ("42 + "52 + "62 ) , 2 2

!

elas Ehex V=

(20)

1" 1 1 # C11 (!12 + ! 22 + ! 62 ) + C12 (2!1! 2 $ ! 62 ) + C13 (2!1! 3 + 2! 2! 3 ) + C33! 32 + C44 (! 42 + ! 52 ) & , (21) % 2' 2 2 (

where V is the volume of the system, Cij, (i,j=1..6) is the element of the elastic constant tensor and εi the element of the strain or deformation tensor, both expressed in Voigt notation. Table II. Applied deformations and corresponding elastic constants for α-Al2O3 and fcc-Al used in the fitting procdure. In all eight cases the elastic energies were calculated for εi = –0.02, –0.01, +0.01, and +0.02. Structure α-Al2O3

Elastic constant C11

Applied deformation Stretching in x dir. (ε1≠ 0)

α-Al2O3

C33

Stretching in z dir. (ε3≠ 0)

α-Al2O3

(C11+ C33)/2 - C13

Lin. comb. x, z dir. (ε1=-ε3≠ 0)

α-Al2O3

C66 = (C11 - C12)/2

Shear x, y dir. (ε6≠ 0)

α-Al2O3

C44

Shear x, z (ε5≠ 0)

fcc-Al

C11

Stretching in x dir. (ε1≠ 0)

fcc-Al

C’ = (C11 - C12)/2

Lin. comb. x, z dir. (ε1=-ε2≠ 0)

fcc-Al

C44

Shear x, z (ε5≠ 0)

Using Eqs. (20) and (21) and experimental or other values of the elastic constants values, in combination with the proper deformations of the crystal (choice of a strain), one can calculate the elastic energies of certain strained states and include them as target values in fitting process. Different deformations address different elastic constants or their linear combinations in Eqs. (20) and (21). Table II contains the various deformations used to probe all elastic constants, Table III contains the experimental values of the elastic constants used to calculate the target elastic energy values in the fit [5, 19, 20]. Prior to fitting Eq. (1), the electrostatic energy of each deformed 78

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– structure, calculated using Eq. (19) and the parameters from Ch.4, was subtracted. It should be noted that for the hexagonal crystal α-Al2O3 not all deformations can be accomplished in a rectangular box with periodic boundary conditions. Instead, a big crystal without periodic boundaries was subjected to the shearing strains ε6 and ε5 for C12 and C44, respectively, see Table II, and only the energies of the atoms in the middle of the crystal were considered for the fit. Table III. Elastic constants: target and result values. C66 in the case of αAl2O3 is called C’ for fcc-Al. They are both equal to (C11 – C12)/2 and are listed for completeness. Target value [eV/Å3] This work RF-MEAM+CTIP potential [eV/Å3] Elastic α-Al2O3 fcc-Al α-Al2O3 fcc-Al constant [19, 20] [5] C11 C12 C13 C33 C44 C66 or C’

3.108 1.025 0.701 3.119 0.921 1.046

0.714 0.386 0.197 0.164

6.139 4.039 2.167 1.679 1.145 1.050

0.853 0.235 0.106 0.309

5.5 Fitting results As happens frequently in potential construction, the fitting results were quite sensitive to the statistical weights given to the target energy values (see Ch.2 for an outline of the role of statistical weights in fitting). Initially we have given zero weights to the elastic deformation energies (Table II), so that we have only fitted the potential energy curves. All binary structures in Table I were included. The resulting fit was quite good [26]. However, the elastic constants in this case turned out to be rather incorrect, and the α-Al2O3 system was not stable (see also Sec. 5.6). On increasing the weights of the elastic energies in the fit, it was observed that the elastic constants improved at the expense of the potential energy curves. This was no surprise, of course, but the effect is somewhat stronger than was hoped for, indicating that the already complicated RFMEAM+CTIP potential is still not powerful enough to describe the full phase space of AlxOy as explored here precisely. However, as oxidation of aluminum is our prime objective rather than homogeneous deformation, we have decided to attribute a greater importance to the correctness of the potential energy curves, which sample a great variety of local atomic environments, than to the correctness of the elastic constants. After many attempts we have chosen the final fit to be a compromise based on the weight assignments listed in Table I for the datapoints in the range 0.98 Rmin to 1.10 Rmin, where Rmin is the nearest neighbor distance corresponding to the total energy minimum (i.e. the minimum of the electrostatic plus non-electrostatic energies). Weights are here chosen as 1/σi, where σi is the “expected uncertainty” assigned to the data point i, see Sec 2.6. Data points outside of this range were given increasingly smaller weights. The elastic energies (being small values) have been given a weight 104. 79


(a)

(b)

(c)

(d)

Fig. 1. Fit of the non-electrostatic RF-MEAM part of the potential. a) AlxOy structures with relatively low minimum energies. Data points are ab initio energies from which the electrostatic energy has been subtracted. Lines show the final fit established in this work b) Same as but for AlxOy structures with relatively high minimum energies. c) Selected results from a) and b) shown as total energies, i.e. with the electrostatic energies added. The best-fitting results were selected. Data points are ab initio energies. Some data points were hidden for increased visibility of the fit qualities d) Pure Al structures. There is no electrostatic part in this case.

The results of the final fit are shown in Fig. 1 (potential energy curves) and are given numerically in Table I (energies, lattice constants, and fit errors), Table III (elastic constants), Table IV (charges), and Table V (potential parameters). Figs. 1a, 1b, and 1d show the direct fit results to the nonelectrostatic energies, where each datapont is a target energy value derived from the ab initio energy (see Sec. 5.4.1). For clarity Fig. 1c shows selected total energy curves, i.e. a selection of the data from Figs. 1a and 1b but with the electrostatic energies added. The minima of these total energies, 80

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– together with the corresponding lattice constants, are given in Table I, where the results of the ZWFN potential are included for comparison. All crystal structures are included in this Table. In Fig. 1d, where only results for pure aluminum crystals are shown, the total energy is the same as the nonelectrostatic energy. Table IV. Charges on the Al and O atoms for different AlxOy crystal structures according to the ZWFN potential and the final fit of the RF-MEAM+CTIP potential presented in this work. Results are given for the lattice constants that correspond to the minimum energies (Table I). Asterisks denote compounds for which the charge values are averaged over non-equivalent atom positions. Crystal structure

AlO2 (CaF2) α-Al2O3 γ-Al2O3 - Fd3m * AlO (ZnS) AlO (NaCl) AlO (CsCl) AlO (CuAu) Al20O16 (I41/amd) * Al2O (Co2 B) * Al3O (Ni3P) * Al4O (Fe4 C) * O in fcc Al (Al32O) * fcc Al

ZWFN potential qAl [e] 2.53 2.85 2.89 2.03 2.02 1.67 0.97 1.45 0.44 0.35 0.23 0.03 0

qO [e] -1.26 -1.90 -1.93 -2.03 -2.02 -1.67 -0.97 -1.82 -0.88 -1.05 -0.91 -0.96 0

This work RF-MEAM+CTIP potential qAl qO [e] [e] 2.02 -1.01 2.91 -1.94 2.18 -1.46 2.09 -2.09 2.05 -2.05 2.04 -2.04 1.30 -1.30 1.62 -2.02 1.02 -2.04 0.68 -2.05 0.53 -2.10 0.06 -2.03 0 0

Table I shows that the minimum energies, the corresponding lattice constants, and the mid-range rms fit errors of the current RFMEAM+CTIP potential are generally in very good agreement with the ab initio data. Somewhat less agreement was found for AlO (CsCl), Al20O16, and Al3O (Ni3P), and the least agreement was obtained for γ-Al2O3 and AlO (ZnS). It should be remarked that γ-Al2O3 was not included in the fit, because the structure is too complex to be handled by our fitting program. In all cases but one (γ-Al2O3) the present potential describes the AlxOy system and the pure Al crystals better than the ZWFN potential, which is a major result of this work. It should be noted that using an MEAM potential instead of an EAM potential is an essential improvement of the potential. Only with angular forces and angular screening active in the fitting it turned out that α-Al2O3 could be made the lowest-energy structure, as is the case experimentally. In contrast, according to the ZWFN potential the AlO (ZnS) and AlO (NaCl) structures have even lower energies than α-Al2O3. The elastic constants of α-Al2O3 and fcc-Al are shown in Table III. As can be clearly seen, the agreement with experiment (the target values) is rather poor. We have just explained that improving the agreement would be easily possible, by changing the statistical weights, but that then the potential energy curves would suffer. It would actually not be unrealistic to generate and use a second parameter set for special deformation simulations. This is work for the future. Table IV lists the charges on the Al and O atoms in the various crystal structures at their minimum total energy. The values are very reasonable and in some cases certainly better than the ZWFN values. Notable the single oxygen atom in fcc Al must have a full valence charge of –2 instead of –0.91. 81

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Table V. Full set of parameters for the RF-MEAM+CTIP potential optimized in this work, referred to in the text as “final fit”. Asterisks denote values that were held fixed without loss of generality, as explained in the main text. The CTIP parameters were taken from fit 3 (Table IV) of Ch. 4. All RF-MEAM values in the O and O-O columns and the two O-O-O values for Cmin and Cmax were taken from [18] and held constant in the fit. CTIP Parameter \ Element Al O 5.146 0* χ [eV/e] J [eV/e2] 13.709 7.125 Z [e] 0.380 0.0 -1 1.126 1.339 ξ [Å ] RF-MEAM Embedding Function Parameter \ Element Al O n 1* 0.889 E0 [eV] 1.858 2.046 E1 [eV] 0.0 0.0 E2 [eV] -0.302 0.0 Background Electron Density Parameter \ Element Al O p(0) 2.964 5.037 p(1) 1* 0.013 p(2) 1* 103.841 p(3) 1* 0.163 t(1) -0.030 270.915 t(2) 0.344 87.924 (3) t -0.497 -12.961 q(0) [Å-1] 0.856 1.909 q(1) [Å-1] 3.244 1.868 q(2) [Å-1] -0.028 1.711 (3) -1 q [Å ] 1.368 1.256 Pair Potential Parameter \ Pair Al-Al O-O Al-O Ep [eV] 1.401 5.589 1.154 4.873 3.457 5.206 αp rp [Å] 2.469 1.105 2.148 c2 0.061 -0.537 -0.011 c3 0.0 0.0 0.0 Radial Cut-off Function Parameter \ Pair Al-Al O-O Al-O rs [Å] 4.45 1.40 4.45 rc [Å] 4.50 1.50 4.50 0.0 0.0 0.0 αc Angular Screening Triplet \ Parameter Cmin Cmax Al-Al-Al 1.474 2.304 Al-Al-O 0.489 0.906 Al-O-Al 0.640 0.761 Al-O-O 0.022 2.288 O-Al-O 0.064 0.774 O-O-O 1.0 2.80

82

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 5.6 Oxide stability and beginning of oxide growth The first simulations done with the new potential were tests of the bulk stability of α-Al2O3 and the relaxation of the Al-terminated α-Al2O3 surface, and simulations of an O atom impinging on the Al (111) and Al-terminated α-Al2O3 surfaces with thermal speed.

[0001]

!

[1120] [0001]

! !

[1120] a)

b)

Fig. 2. Stability of α-Al2O3. Blue (bigger) – Al atoms, orange (smaller) – O atoms. Top view (top) and side view (bottom) are shown. Crystallographic directions are indicated. a) After 1 ps constant-volume simulation at 375 K using the earlier obtained model without elastic constants in the fit. Disordering is clearly visible. b) Same as a) but using the potential parameters from the final fit of this work, with elastic constants included. Results obtained at much higher temperature (3500 K) and longer time (5 ps) are indistinguishable from those shown in b), indicating the stability of the α-Al2O3 crystal using the parameters determined in this work.

!

83

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– The results of the simulations for α-Al2O3 crystal stability are shown in Fig. 2. The earlier obtained potential without elastic constants in the fit [26] is observed to cause unphysical disordering already after 1 ps at 375 K (Fig. 2a), while with the current potential with elastic constants in the fit the αAl2O3 crystal is stable under the same conditions (Fig. 2b). In fact, with the current potential the crystal remains stable at a much higher temperature (3500 K) and after a longer time (5 ps). Except noticeable thermal vibrations the picture of that system would look the same as Fig. 2b. One oxygen atom approaching an Al (111) surface is the first and simplest Al-O/Al simulation. It gives us an insight into the very early state of aluminum oxide growth. The result is shown in Fig. 3. The oxygen atom is introduced normally to the surface with an initial kinetic energy of 40 meV, which corresponds to a temperature of 375 K. Two moments are shown. The local charge transfer between the O atom and the surrounding Al atoms is clearly seen (colors represent charges). As the O atom is approaching, the intensity of the charge transfer increases, as expected: the magnitude of the charge on the oxygen atom increases from qO = –1.02 e closely above the surface (Fig. 3a) to qO = –1.92 e just inside the surface layer (Fig. 3b), where the oxygen atom has penetrated into the Al bulk. As in the case of the ZWFN model, no surface reconstructions were detected after the system had reached equilibrium. The crucial difference with the ZWFN model is that with the current potential the final values of the charges established on the oxygen and the six nearest neighbor aluminum atoms are excellent: qO = –2.03 e and qAl = 0.29 e (note that in Tab. IV the Al charge was averaged over all 32 Al atoms in the box).

O

O

b)

a)

Fig. 3. Oxygen atom approaching an Al (111) surface at 375 K. The incident direction of the O atom is normal to the surface (arrow) and its initial kinetic energy has thermal magnitude. a) Oxygen atom just above the surface (charge qO = –1.02 e). b) Oxygen atom in the subsurface layer (charge qO = –1.92 e). Colors represent charge: light blue – zero, dark blue – positive, magenta – negative.

84

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– [0001]

O

O

O

O

a)

b)

[1010]

!

!

[0001]

[1120]

!

!

Fig. 4. Oxygen atom approaching an Al terminated α-Al2O3 (0001) surface at 375 K. The incident direction of the O atom is normal to the surface (arrow) and its initial kinetic energy has thermal magnitude. Colors represent charge: blue – positive, magenta – negative, yellow – zero. Bigger spheres – Al, smaller spheres – O. a) According to ZWFN model. b) According to RF-MEAM+CTIP model determined in this work.

The final simulation is of the same type, but this time the O atom approaches an α-Al2O3 surface. The idea was to see the difference in surface chemistry between this and the pure Al surface O approach. From the several possible surfaces of α-Al2O3 the Al terminated (0001) surface was chosen, i.e. the one with a first layer of aluminum atoms above the oxygen atoms. Before the O atom was introduced normal to the surface (Fig. 4), the surface was relaxed. Upon relaxation the aluminum atoms at the surface moved into the bulk, but the vertical displacement was noticeably less with the new potential (Fig. 4b) than with the ZWFN potential (Fig. 4a). Surface reconstructions of the alumina were not detected in either case. For both potential models, contrary to the previous case where the O atom was directed towards a pure Al surface, the O atom did not penetrate into αAl2O3 crystal but remained captured at the surface, above an Al atom. This is a physically very credible result, because in practice an aluminum oxide layer prevents further oxidation of aluminum, thus one would not expect oxygen atoms to penetrate easily through alumina. The charges on the oxygen atoms in the final state are qO = 0.05 e for ZWFN and qO = 0.0024 e for the present potential. 85

Chapter 5 Al/O potential construction - Non-electrostatic part ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Both models show that a single O atom approaching a perfectly flat and defectless alpha aluminum oxide will not experience any significant charge transfer. This means that it will interact with the surface only through much weaker non-electrostatic forces. This is opposite to the case of the O atom approaching a pure Al surface, as we have seen, were charge transfer is very strong and the interaction thus has a strongly ionic nature.

5.7 Conclusion In this chapter we have accomplished the construction of a RF-MEAM+CTIP potential model for the Al-O system. Here the principal focus was the non-electrostatic part of the potential, but together with the material from Ch. 4, the whole construction procedure, starting from scratch, is explained, and initial simulations with this potential are reported. As non-electrostatic part of the potential a MEAM potential was used (i.e., with angular forces), which turned out to be a crucial improvement over the EAM potential used in the earlier ZWFN model. The potential has been fitted to a large database of ab initio energies of different AlxOy crystals with a range of lattice constants and to elastic energies of deformed α-Al2O3 and fcc-Al representing their elastic constants. A compromise between optimizing the fit to the structural energies and optimizing the fit to the elastic deformation energies was chosen such that more importance was attributed to the structural energies. Nevertheless, the second data set, the elastic constants, was necessary to ensure structural stability. Also, atomic charges were no longer “assumed” target fit values, as is sometimes employed, but they are the result of the parameter optimization of the electrostatic part of the model (Ch. 4). The simulations on which this chapter reports, a few tests and several basic runs, clearly show the following points: (1) The present RF-MEAM+CTIP potential and the dual fitting strategy (electrostatic and non-electrostatic parts separately) together make it possible to construct a complete description of a metal/metal-oxide system from scratch. In our case it is the Al/Al-O system. (2) The presently constructed model yields correct charges on the atoms in the important crystal structures as well as at the beginning of the oxidation process. (3) All tested binary structures have higher minimum energies than corundum (α-Al2O3), as is physically correct, and agree much better with ab initio values than with earlier potentials. (4) α-Al2O3 and Al-fcc crystals are stable, also at elevated temperatures. (5) O-O interaction is correct as well as O interaction with a pure Al (111) surface and an Al terminated α-Al2O3 (0001) surface. Thermal oxygen penetrates into Al but not into α-Al2O3. (6) In general, the present model describes the very beginning of the Al oxidation process much better than earlier models and it is therefore suitable for a study of Al oxide self-repair.

References [1] [2] [3] [4]

L. P. H. Jeurgens, W. G. Sloof, F. D. Tichelaar, E. J. Mittemeijer, J. Appl. Phys. 92, 1649 (2002). E. Kaxiras, Atomic and Electronic Structure of Solids, Cambridge University Press, Cambridge (2003). M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford (1987). I. Lazić, M. A. Ernst, B. J. Thijsse, Proc. First International Conference in Self Healing Materials, CD, Editors A. J. M. Schmets and S. van der Zwaag, Springer, Dordrecht, The Netherlands (2007).

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X. W. Zhou, H. N. G Wadley, J. S. Filhol and M. N. Neurock, Phys. Rev. B 69, 035402 (2004). F. H. Streitz, J. W. Mintmire, Phys. Rev. B 50, 11996 (1994). T. Campbell, R. K. Kalia, A. Nakano, P. Vashishta, Phys. Rev. Lett. 82, 4866 (1999). X. W. Zhou, H. N. G. Wadley, J. Phys.: Condens. Matter 17, 3619 (2005). X. W. Zhou, H. N. G. Wadley, D. X. Wang, Comput. Mater. Sci. 39, 794 (2007). A. Elsener, O. Politano, P. M. Derlet, H. Van Swygenhoven, Model. Simul. Mater. Sci. Eng. 16, 025006 (2008). A. Elsener, O. Politano, P. M. Derlet, H. Van Swygenhoven, Acta Mater. 57, 1988 (2009). E. Bourasseau, J.B. Millet, L. Mondelain and P.M. Anglade, Mol. Simulat. 31, 705 (2005). M. I. Baskes, Phys. Rev. B 46, 2727 (1992). B. J. Thijsse, Nucl. Instrum. Methods Phys. Res. B 288, 198 (2005). M. I. Baskes, Mater. Chem. Phys. 50 (1997) 152. R. W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York (1981). W. H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C, Cambridge University Press, 2nd edition (1992). M. I. Baskes, Modified Embedded Atom Method Calculations of Interfaces, Sandia National Laboratories Livermore, CA, Edited by S. Nishijima and H. Onodera (1996). G. N. Kamm, G. A. Alers, J. Appl. Phys 35, 327 (1964). G. V. Sin’ko, N. A. Smirnov, J. Phys.: Condens. Matter 14, 6989 (2002). G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996). P. E. Blochl, Phys. Rev. B 50, 17953 (1994). J. P. Perdew, Y. Wang, Phys. Rev. B 45, 13244 (1992). M. Finnis, Interatomic forces in condensed matter, Oxford University Press, Oxford (2003). J. F. Nye, Physical Properties of Crystals, Their Representation by Tensors and Matrices, Oxford University Press, Oxford (1985). I. Lazić, B. J. Thijsse, Proc. 4th International Conference on Multiscale Materials Modeling MMM, Ed. A. ElAzab, Department of Scientific Computing, Florida State University, Tallahassee, USA, 454 (2008).

87


88

Chapter 6 Copper on Tantalum ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

6. Microstructure of a Cu film grown on Ta (100) by large-scale Molecular Dynamics simulations

Abstract Molecular Dynamics simulations using Embedded Atom Method potentials were carried out to study the growth and subsequent annealing of a 6.3 monolayer thick Cu film on a bcc-Ta (100) substrate of a very large area (100 nm × 100 nm). The purpose was to obtain, for this typical example of a crystallographically incompatible system, atomic level insight into the microstructural evolution of the film/substrate interface and the film itself, with the limiting influence of in-plane boundary conditions kept to a minimum. It is found that the first Cu plane grows heteroepitaxially on the bcc-Ta (100) surface. The second Cu plane is the most interesting plane; it grows in the form of a pattern of 26-atom misfit supercells which after prolonged film deposition relax energetically, by breaking up in groups and forming 30 Å wide in-plane island strips separated by fcc (1/6) vectors. The distorted monolayer represents a way of enabling epitaxy between very different crystal structures without introducing misfit dislocations. The third and higher Cu planes are fcc (111) planes. The Cu film is polycrystalline, starting from the second plane up, with two different in-plane crystal orientations. The influence of a one-monolayer Ta terrace on the substrate is minimal. Upon annealing the film at elevated temperatures, a tendency towards island coarsening and agglomeration is observed, but a full 3D island formation such as found experimentally is not observed. Compared to an earlier simulation on a smaller substrate area (18 nm × 18 nm), the current simulations have produced a much richer microstructure. The present results therefore warn against the use of too small simulation domains when complex strain fields can be expected. 6.1 Introduction Understanding the growth of Cu thin films on Ta substrates is fundamentally as well as technologically important. The fundamental importance is due to the fact that Cu/Ta is a prime example of a strongly heterogeneous film/substrate system: in equilibrium Cu has the fcc and Ta the 89

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– bcc crystal structure, the atomic diameters differ by a factor 1.12, the bulk and shear moduli both by a factor 1.43, the cohesive energies by a factor 2.29, and the melting temperatures by a factor 2.92, in all cases Ta having the greater value. The technological importance lies in modern IC interconnects. Tantalum is a diffusion barrier material, and the microstructure of the Cu film affects its adhesion to Ta and the microstructure of the subsequent Cu fill (electromigration resistance). Atomistic simulations can provide new insights in the formation and structure of Cu films grown on Ta substrates. In earlier work, classical Molecular Dynamics (MD) results of Cu deposition on bcc-Ta (100) and (110) [1] and β-Ta (001) [2] were reported (β-Ta is a metastable form of Ta that is sometimes found in thin films). While much new information was obtained from these simulations, the Ta substrates were only modest in size (18 nm × 18 nm). Therefore the in-plane periodic boundary conditions may very well have altered certain growth phenomena in the film, especially those involving extended stress fields. For instance, it was observed that the Cu film growing on the bcc-Ta (100) was monocrystalline from the third layer onwards. This might be an indication that the system was too small, at least so small that different in-plane grain orientations – and therefore grain boundaries – were artificially suppressed. It was therefore decided to continue the work and simulate a much larger system. With boundary condition effects reduced by an order of magnitude and with step edges of various orientation added to the substrate, it was expected that a richer and more realistic variety of crystal defects could be studied. In this work we report simulations of Cu growth on a bcc-Ta (100) substrate of an unusually large size, 100 nm × 100 nm. Prior to Cu growth, a single-monolayer Ta terrace was added on top of the substrate in order to be able to study possible step-edge effects in the same simulation run. We will discuss the host of phenomena that emerged in the successive atomic Cu planes during and after the deposition: heteroepitaxy, grain boundaries, dislocations, step-edge effects, point defects, and texture. It will be shown that after a few initial planes exhibiting different types of morphology, a textured polycrystalline fcc-Cu (111) film develops. Analysis of the symmetry types of the local atomic angular environments turned out to be a very successful tool in this study, and we will explain how this is done. Very recently, atomistic simulations of surface wetting and de-wetting in the Cu/Ta system were reported by Hashibon et al. [3]. A number of differences with the current work make it difficult to compare the results. Most notably, Hashibon et al. have placed a Cu film against a Ta substrate, rather than performing an actual deposition simulation, which is a rather unphysical approach. Also, their Ta substrate was (110) oriented and much smaller in size, approximately 40 times smaller in area, and they have used another potential than ours.

6.2 Computational The MD simulations were performed using the parallelized version of the Delft Molecular Dynamics code camelion and applying Johnson-Oh Embedded Atom Method (EAM) potentials for Cu [4, 5] and Ta [6]. The Cu-Cu interaction has a cut-off radius of 4.83 Å, which includes the equilibrium first, second, and third nearest neighbor distances (2.56, 3.61 and 4.43 Å). The Ta-Ta interaction has a cut-off radius of 3.99 Å, which includes the equilibrium first and second nearest

90

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– neighbor distances (2.86 and 3.30 Å). For the Cu-Ta cross potential we have used the unlike-atom pair potential form by Johnson [7], in a slightly modified form. This potential was fitted to the (positive) heat of formation of a B2 (CsCl-type) Ta-Cu alloy, 0.03 eV/atom. This value was determined according to the Miedema model [8], although more recent ab initio calculations show that the actual value is probably ∼0.1 eV/atom higher. Nevertheless, in either case the interface will behave energetically as a rather inert region. Also, although the potential contains no explicit angular interaction terms, it was found that it describes the β phase of Ta, which has a rather complex structure, quite well [9, 10]. Further details of the potential are given in Ch. 2. In the MD simulations, the Cu atoms were deposited onto a seven monolayers (ML) thick, 100 nm × 100 nm bcc-Ta (100) Ta substrate, with in-plane periodic boundary conditions. The inplane number of unit cells was 303 × 303 = 91809. On the substrate a wing-shaped singlemonolayer Ta terrace was added, covering 44.7% of the surface (Fig. 1(a)). In total the substrate contained 683 678 atoms. The volume and temperature were kept constant in the simulation. The incident deposition angle of the Cu atoms was 15° off-normal, the same angle as experimentally used in Cu film growth on Ta for thermal helium desorption experiments [11]. The azimuthal incident direction of each Cu atom was chosen randomly, to avoid a specific relation to the crystal structure of the substrate, as was the initial position of each atom in the top of the simulation box. The atoms arrived with a kinetic energy of 0.17 eV, a typical magnitude for electron-beam evaporated atoms. In total 1 016 035 Cu atoms were deposited. As we will see, the first deposited monolayer of Cu can be counted as 92000 atoms and all subsequent monolayers as 174000 atoms. Both numbers are approximate. The full deposition run has produced a 6.3 ML thick Cu film, covering a vertical range of more than seven planes (Figs. 1(a) and (b)).

a)

b)

Fig. 1. (a) Schematic side and top views of the bcc-Ta (100) substrate with terrace and the first eight Cu planes. Plane numbers as used in this work are indicated. In reality the substrate contains three more Ta planes. (b) Top view of the final Cu film, after 6.3 ML deposition, atoms colored according to height. The red atoms are in plane Cu 9.

91

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– In order to finish the simulation in a feasible amount of time, the deposition rate in the simulation was taken as 1 Å/ns, nine orders of magnitude faster than in typical experimental conditions (1 Å/s). In order to permit the system nevertheless a realistic opportunity for thermal relaxation and surface diffusion in the simulations, we have adopted the elementary method of giving the film and substrate atoms a higher mean kinetic energy in the simulations than in the experiments (300 K). To compensate the deposition rate fully (at least, for as far as time and temperature can be considered exchangeable), a deposition temperature of 1000 K would be necessary. This can be seen from the simple transition-state-theory estimate N = v0t exp(–Q/kT) for the number of jumps N of activation energy Q that an atom will execute in time t at temperature T. Taking v0 = 1013 s–1 and t equal to, say, half the monolayer deposition time in the experiment (1 s at 300 K), an atom will execute jumps with Q < 0.78 eV at last once in the experiment (Nexp > 1). In the simulations, where t =1 ns, one would need T = 1000 K to reach the same situation (Nsim > 1). Note, however, that such a 1000 K simulation would still reproduce a 300 K experiment with systematic distortion of the kinetics. For example, the ratio Nsim / Nexp is less than 1 for all Q smaller that 0.78 eV, which means that the relative rates of microscopic events essential for relaxation will very likely be affected. More importantly, the simulation temperature should not be too close to the melting point of the simulated materials, or else undercooled liquids (glasses) may form. Since Cu has a melting temperature of 1356 K (1365 K with the Oh-Johnson potential used), such problems have indeed been observed in trial deposition simulations at 1000 K. As it was found that all dubious high-temperature effects had disappeared from the simulations at 750 K, it was decided to run the full simulations with the film/substrate system thermostat set at 750 K. The exact effects of the limited Cu mobility compared to a 1000 K simulation are hard to estimate, since on a growing surface many different activation energies are in operation. Yet, in order to give an impression of the kinetic downscaling from 1000 K to 750 K, we mention that in a simulation of Cu deposition onto Cu at 1000 K, the Cu atoms move up to ~8 nm away from the positions where they are adsorbed. At 750 K, this distance is only ~3 nm. There are two more simulation details to mention. When the velocity distribution is nonMaxwellian (e.g. during and immediately after adsorption of an atom), temperature control is suspended until the distribution is Maxwellian again. The temperature control therefore does not interfere with the way in which the energy of adsorbing atoms is dissipated. Furthermore, carrying out the simulation at 750 K instead of 300 K with unmodified potentials would create unphysical thermal stresses between Ta and Cu, since the thermal expansion coefficient of fcc Cu (16.5 × 10– 6 /K) is 2.6 times larger than that of bcc Ta (6.3 × 10–6/K). Therefore the Cu-Cu potential was slightly adapted to make the equilibrium lattice spacing at 750 K equal to the experimental value at 300 K. We did not change the Ta potential because of its substantially lower expansion coefficient.

6.3 Local symmetry type In order to attribute a local crystal symmetry to every individual atom in the system, we have made use of a particular quantification of the angular environment of each atom i. This was inspired by work of Steinhardt et al. [12]. First, six numbers Qi(2 " ) , λ = 1, 2, ..., 6, are calculated for atom i,

!

92


(2 ! ) i

Q

1 = Zi

Zi

Zi

## P k

(2 ! )

(cos" kij ) , " =1, 2, …, 6,

(1)

j

! 2 " , θkij is the triplet angle subtended at atom i where P (2 " ) is the Legendre polynomial of degree formed by the vectors from i to its neighboring atoms k and j, and Zi is the coordination number of i. Note that the lengths of the vectors play no role other than selecting or excluding the atoms k and j ! as nearest neighbors. The maximum nearest ! neighbor distances were chosen as 3.27 Å for Cu-Cu, 3.46 Å for Cu-Ta, and 3.66 Å for Ta-Ta. For atoms in perfect fcc, hcp, bcc, diamond cubic and icosahedral environments, the values of Q (2 ! ) are shown in Table I. Table I. Numbers Q (2 ! ) for an atom in different angular environments: fcc, hcp, bcc, diamond, icosahedral. Z is the coordination number of the atom. In colored figures, the local symmetry types are colored as indicated; an unclassified symmetry type appears as magenta. 2λ Q (2 ! ) fcc

hcp

bcc

diamond

icosohedral

2 4 6 8 10 12

0 0.191 0.575 0.404 0.013 0.600

0 0.097 0.485 0.317 0.010 0.565

0 0.036 0.511 0.429 0.195 0.405

0 0.509 0.629 0.213 0.650 0.415

0 0 0.663 0 0.363 0.585

Z

12

12

14

4

12

color

green

orange

blue

yellow

cyan

These five structures are the candidate structures used in this work. To every atom i in the system under study one of these is assigned as the local symmetry type, provided that the actual environment of the atom and the environment in the assigned candidate structure are not too different. For atom i we therefore calculate the RMS differences of the first six even orders Qi(2 " ) (2 ! ) with the corresponding Qstruc of each of these candidate structures,

!Si, struc

1 6 (2 " ) 2 = (Qi(2 " ) # Qstruc ) . $ 6 " =1

! (2)

The structure for which ΔS has the smallest value is attributed to atom i as its local symmetry type, provided that this ΔS is smaller than 0.11. This critical bound was determined by numerical experimentation. If ΔS is greater, the local crystal symmetry of i is recorded as “unclassified”. In this way we can divide up the full system into fcc, hcp, bcc regions, etc, which after visualization are useful for detecting different crystal defects, as we will see. Note that local symmetry type is an atomic property derived from the neighbors of the atom in three dimensions, not just from the

93

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– neighbors that lie in the same plane of deposition. In this work we will show that the local symmetry type is a very useful tool for clarifying the microstructure of the film, although the 3D character of the local symmetry type calls for cautious interpretation of in-plane domains in some cases.

6.4 Results: the Cu film after deposition Before presenting the results of the deposition simulations, we note that all results in this work are shown for 0 K configurations, i.e. for configurations copied from the ongoing run and subsequently rapidly quenched to 0 K. This was done to suppress thermal noise, which especially affects the determination of the local crystal symmetry of the atoms. Table II. Data of the atomic planes after film deposition, top to bottom, see also Figs. 1 and 2. NTa, NCu, and Ntot are the actual number of atoms in the plane, nCu is the hypothetical number of Cu atoms in the plane if the areas outside the terrace area would be representative for the whole plane. Plane NTa NCu Ntot RMS thickness of Distance to next, nCu plane (Å) lower plane (Å) Cu 10 0 60 60 0.41 1.97 ∼0 Cu 9

0

3194

3194

0.13

1.95

∼0

Cu 8 Cu 7 Cu 6 Cu 5 Cu 4 Cu 3 Cu 2 Ta terrace Cu 1 Ta 0 Ta –1 Ta –2 Ta –3 Ta –4 Ta –5 Ta –6

0 0 0 1 7 23 37 40977 2 91777 91809 91809 91809 91809 91809 91809

34879 107387 163842 173678 173838 172725 135499 31 50816 32 0 0 0 0 0 0

34879 107387 163842 173679 173845 172748 135536 41008 50818 91809 91809 91809 91809 91809 91809 91809

0.17 0.21 0.21 0.21 0.20 0.21 0.10 0.07 0.08 0.09 0.06 0.05 0.04 0.04 0.03 0.02

1.90 1.93 2.00 2.01 2.03 1.91 0.94 0.81 0.91 1.72 1.64 1.66 1.66 1.65 1.60

7878 68328 155774 173837 173483 174293 170834 91809

Table II and Fig. 2 summarize the Cu/Ta system after the deposition of the film. In Figs. 2(c)-2(f) the four most interesting Cu planes, Cu 2 to Cu 5, are shown with the atoms colored according to local symmetry type. In Table II, NTa, NCu, and Ntot are the observed number of atoms in the plane, and nCu is the hypothetical number of Cu atoms in the plane if the area outside the terrace area would be representative for the whole plane. We will call the area outside the terrace area the outside area of a plane. These numbers nCu reflect, better than the observed numbers NCu, the filling factor of the plane (i.e. the outside region) and therefore the defect content of the plane. Plane Cu 1 is fully epitaxial on the bcc-Ta (100) plane Ta 0. This follows not only from visual

94

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– inspection but also from the fact that nCu in plane Cu 1 is equal to NTa in plane Ta 0. The planes Cu 2 to Cu 5 all have numbers nCu in the range 1.70 × 105 to 1.74 × 105. To put these numbers in perspective, we note that in stress-free and defect-free conditions fcc-Cu (111) planes of this size contain 1.77 × 105 atoms. This is a clear indication that plane Cu 2 (outside area) has the most defects, followed by planes Cu 3 to Cu 5 which have smaller packing deficits. Below we will discuss the defects that are observed in these planes. The rapid decrease of nCu for planes Cu 6 to Cu 9 has another reason: these planes were not yet fully covered when the deposition stopped. There is more information in Table II. First we note that in stress-free equilibrium conditions, fcc-Cu (111) planes are separated by a distance of 2.08 Å and bcc-Ta (100) planes by 1.65 Å. Comparison with Table II shows that the surface relaxation in the Cu film has a considerable range, much more than in the Ta substrate. The first Cu layer has a distance of only 0.92 ± 0.01 Å to the underlying Ta plane. This is the fully epitaxial Cu layer on bcc-Ta (100). This shows that the initial Cu layer penetrates far into the Ta plane. The Cu-Ta atomic neighbor distance between these two planes is 2.51 Å, which is surprisingly small. Finally, the RMS thickness of each plane is listed in Table II for completeness. Note that this thickness does not contain a contribution from thermal atomic vibrations, because these are absent. Therefore, the value 0.02 Å for the bottom Ta plane indicates a small curvature, possibly due to stresses. 6.4.1 Plane by plane description In this section we describe the local crystallographic symmetries, the microstructures and the defects observed in the atomic planes of the deposited Cu film. Fig. 2 will serve as the main illustration for reference. On the scale of Fig. 2, individual atoms are not visible. In later sections we will therefore discuss some of the issues in more detail, showing atomic-scale patterns and arrangements, which justify the conclusions on the microstructure presented in this summary. For brevity we mention here only the properties of the film area outside the terrace region in each plane. For the film area inside the terrace region of a plane one should refer to the underlying plane, i.e. the plane with the next lower index; this can also be seen in Fig. 2. Plane Cu 1. Being fully epitaxial on bcc-Ta (100), plane Cu 1 contains no defects. As mentioned above, it sinks deep into the top Ta layer. Because of this, and because the Cu 2 layer above it is not a bcc (100) plane, the first deposited Cu layer is not recognized by the local symmetry algorithm as having bcc symmetry type but as being “unclassified”. This can be seen in the terrace area in Fig. 2(c), which is the part of plane Cu 2 that is in fact the first deposited Cu plane. Plane Cu 2. In Fig. 2(c), plane Cu 2 shows up as blue, but it has no true 3D bcc symmetry. As we will show later, this plane has a misfit superstructure with a (7√2 × √2)R45º supercell and is in a sense intermediate between a bcc (100) and an fcc (111) plane. The magenta curved lines in Cu 2 are grain boundaries (GBs), or rather the beginning of GBs which stretch out upwards. There are two types of GBs: • GBs between areas of different in-plane orientations (two such orientations occur, which will be designated as I and II)

95


Fig. 2. (a) Schematic side and top views as in Fig. 1(a). (b) Small section of the Cu/Ta system viewed at atomic scale (blue=Ta, orange=Cu). Note how deep the epitaxial plane Cu 1 has sunk into the top Ta layer. (c)-(f) Top views of the planes Cu 2 to Cu 5 after deposition. The atoms are colored according to local symmetry type (green=fcc, blue=bcc, orange=hcp, magenta=unclassified). Boxes denote regions discussed in the main text.

96

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– GBs between areas of identical in-plane orientation but shifted by in-plane vectors of the type (1/6), i.e. areas with three possible “anchor positions” A, B, and C, as they are usually indicated in stacking sequences of close packed planes. This terminology is used as if this plane is a genuine fcc (111) plane. Also the relatively low number of atoms in this plane (nCu = 170834) compared to the next higher planes (Table II) shows that there is a packing deficit in this plane. •

Plane Cu 3. Cu 3 is the first fcc (111) plane but one with many defects visible in Fig. 2(d). Since from this plane upwards all planes and their neighboring planes above and below are essentially fcc (111) planes, green is the main color for these planes. GBs show up mainly as blue. At this point it should be illustrated why the local symmetry type should be interpreted with caution, and why the true defect content of plane Cu 3 is much less than Fig. 2(d) suggests. The reason is as follows. An orange area (hcp) denotes a stacking fault plane, one for which the next lower and higher planes have the same anchor positions (such as for example in the vertical stacking sequence A,B,A in planes 2,3,4) rather than different anchor positions (such as A,B,C in planes 2,3,4). In the first of these sequences the B plane shows up as hcp (orange), in the second the B plane shows up as fcc (green). Therefore, neighboring orange and green regions in the same plane do not necessarily have different anchor positions. Instead, the different colors may be the result of different anchor positions in the next lower or upper plane. The A,B,A, and A,B,C example just given illustrates this: the B region may show up as orange or as green. The “boundary” between these two B regions is then artificial (nonexistent). The true GB is one plane up or down. Note that a pair of orange planes, at least if they are adjacent planes, denotes an intrinsic stacking fault, and these have indeed been found, see plane Cu 4. Another notable phenomenon are the blue stripes in plane Cu 3. We will show that they are associated with the presence of misfit supercells in the underlying plane Cu 2. Stripes run in the direction normal to the orientation (I or II) of the grain in plane Cu 2. Later we will show that the occurrence of these stripes and the fact that the orange areas all have approximately the same width are intimately connected with misfit supercells formed in plane Cu 2. Plane Cu 4. As is seen in Fig. 2(e), plane Cu 4 is also an imperfect fcc (111) plane, but less imperfect than the underlying plane. There are no more stripes, which indicates that the supercell modulations in plane Cu 2 do not propagate upwards upon growth. GBs are still present in two modifications: between different orientations (I and II) and between different anchor positions (A, B and C). An example of the second one is the boundary between the two regions in box R. Close inspection shows that the regions have the same orientation, but that, although they both are green, they are (1/6) vector shifted. The blue boundaries in the plane below, in box O in Fig. 2(e), are actually artificial according to the explanation given for plane Cu 3. To fully understand the nature of the boundary in box R, one should turn to Table III and follow the vertically stacked regions in the boxes L, O, R and T, which appear in Figs. 2(c)-(e)-(d)-(f) respectively. The orange lines bounded by two blue dots, such as seen in e.g. box Q in Fig. 2(e), are intrinsic stacking faults, bounded by two Shockley partial dislocations, of which the stacking fault strip runs approximately vertically.

97

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Table III. Schematic illustration of boundaries between grains having the same in-plane orientation but having different “anchor positions” A, B, or C. The formation and propagation in the growth direction (down-up in the table) are indicated by the white boxes L, O, R and T in Figs. 2(c)-(f). Plane Selected area in Figs. 2(c)-(f) Explanation schematically shown. Boundaries are no longer present. The whole Cu 6 area is B, just as the area below in box O. To maintain the fcc structure a split of the regions is no longer required. Plane Cu 6 is not shown in Fig. 2. The blue line is the boundary between Cu 5 Box T different “anchor position” areas C and A. The Fig. 2(f) split of the regions is necessary to maintain the fcc structure within both of them.

Cu 4

Box R Fig. 2(e)

Cu 3

Box O Fig. 2(d)

Cu 2

Box L Fig. 2(c)

The blue line is the boundary between different “anchor position” areas A and C. Formed freely, no influence of the Ta substrate. All blue boundaries are artificial, indicating region shifts from A to C in plane Cu 2 (horizontal lines) and in plane Cu 4 (vertical line). The whole area is close packed B (111) fcc or hcp. The magenta lines are GBs boundaries between different “anchor positions” areas A and C. Formed under the influence of the Ta bcc substrate below (see the main text for an explanation).

Plane Cu 5. This is the first plane that looks homogeneous, in the sense that the terrace and outside areas look very similar, see Fig. 2(f). There are just a few “imprints” of the original step edge visible in this plane, but they are not very obvious. The GBs run vertically. Compared to plane 4, more Shockley partials are visible here. Some 65 intrinsic stacking faults were identified in the whole system, eight of them very near the original step edge locations on the Ta substrate and the rest distributed within the grains. Also one new growth stacking fault is observed (box S in Fig. 2(f)). This stacking fault is the usual one, in the sense that the Ta lattice did not influence its formation and shape. Note that it is not a “long orange patch” like those seen in plane Cu 3. Plane Cu 6 and higher. Upward from plane 5, the planes are no longer fully occupied, see also Fig. 1(b). The microstructure of the underlying planes is continued, however. As the coverage decreases, i.e. in increasingly higher planes, the formation of islands is clear. To give a numerical illustration: plane 7 (outside) and plane 8 (terrace area) are approximately 40% occupied and consist of irregular islands with a mean size of 60 Å.

98

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Before moving on to the atomic-scale description of the defects, we should return once more to plane Cu 4 (Fig. 2(e)) and compare the microstructure of its outside area with that shown in Fig. 3. Also in Fig. 3 a Cu 4 plane is depicted, but here the result was taken from the simulation on the much smaller system mentioned earlier (18 nm × 18 nm) [1]. In the smaller system several intrinsic stacking faults are visible but not any of the other phenomena seen in Fig. 2(e), like growth stacking faults or grain boundaries. This comparison clearly shows how simulations on too small systems may hide essential features of the physics under study. It is one of the key motivations of presenting these results.

Fig. 3. Plane Cu 4 from a Cu deposition on an 18 nm × 18 nm flat bcc-Ta (100) substrate [1]. Note the differences with the outside area of plane Cu 4 of Fig. 2(e), which was produced under the same deposition conditions but on a 100 nm × 100 nm substrate. Clearly the boundary conditions of the small substrate have suppressed defects such as grain boundaries.

6.4.2 Atomic-scale descriptions In this section we illustrate in atomic detail some of the features that were observed in the film: grain orientations and grain boundaries, misfit supercells, and the origin of the narrow orange strips in plane Cu 3, Fig. 2(d). As far as point defects are concerned, almost no vacancies were found: only several tens in a system of over a million deposited atoms. Grain orientations and grain boundaries. Fig. 4 shows an atomic view of box K in Fig. 2(c) and is therefore a close up of plane Cu 2. This figure summarizes the different Cu grains that are found in the film. The coloring of the atoms is by local symmetry type, just as in Fig. 2(c). On the left of Fig. 4 is the bcc (100) part of this Cu plane, which is epitaxial on the Ta terrace just below it. Across the GB indicated by the blue dashed line, on the right of it, one sees the typical atomic arrangement of the outside area of plane Cu 2. As said earlier, it is intermediate between a bcc (100) and an fcc (111) plane, and we will describe it in the terminology of an fcc (111) plane. 99

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– [010]

AI

[100]

CII

CII

AII

Fig. 4. Atomic-level view of box K in Fig. 2(c). The atoms are all in plane Cu 2 and are colored by local symmetry type. On the left is the bcc (100) part of this Cu plane, which is epitaxial on the Ta terrace just below it. Across the blue dashed grain boundary, on the right of it, is the typical atomic arrangement of the outside area of plane Cu 2. Described in the terminology of an fcc (111) plane, the two groups of three parallel lines run in the two different close packed directions; the associated atomic hexagons are also shown. These two fcc directions make angles of +45º or –45º with the bcc [100] direction of the Ta substrate, and they are designated as orientations I and II, respectively. In the figure, four grains are visible: two having anchor position A and two having anchor position C. The AII and CII grains are translated by (1/6) vectors, and so are –although not shown here– AI and CI grains. For A/B and B/C grain combinations the same translation vectors apply. The red dashed lines in the figure indicate grain boundaries of various types: boundaries between orientation differences (AI and AII), between anchor position differences (AII and CII), and between both of these (AI and CII).

The two groups of three parallel lines shown in Fig. 4 run in the two different close packed fcc directions that are found throughout the copper film; the associated atomic hexagons are also shown in the figure. These two fcc directions make angles of +45º or –45º with the bcc [100] direction of the Ta substrate, and they are designated as orientations I and II, respectively. In the figure, four grains are visible: two having anchor position A with different grain orientations (I and II) and two having anchor position C with the same grain orientation (II). The AII and CII grains have the same grain orientation but they are translated by (1/6) vectors. For A/B and B/C grain combinations the same translation vectors apply. The red dashed lines in the figure indicate

100

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– grain boundaries of various types: boundaries between orientation differences (AI and AII), between anchor position differences (AII and CII), and between both of these (AI and CII). It is seen that in this, plane Cu 2, the local symmetry of many atoms in these GBs is colored magenta. Following these types of GBs through Figs. 2(c)-(f) one notices that many of the GBs between grains with different orientations run upwards (and show up as blue in planes Cu 3 and higher). These are 30° [111] fcc tilt grain boundaries. However, as explained earlier, the GBs between grains with different anchor points in plane Cu 2 give rise to orange patches in plane Cu 3, see for example box N in Fig. 2(d) and compare with Fig. 4 (magnified box K, Fig. 2 (c)). Above plane 3, Figs. 2(e)-(f) show that these GBs between grains with different “anchor points” have largely disappeared. However, they are still present, and Table III explains their propagation. Only in plane 3 these GBs are completely absent. The orange patches and their blue “boundaries” say nothing about the anchor positions of the grains in plane Cu 3 but only reflect the difference in anchor positions of the underlying grains in plane Cu 2. The only real boundaries in plane Cu 3 are those between grains of different orientations. In addition to the orange patches, plane Cu 3 exhibits one more special feature: blue stripes, see Fig. 2(d). We will discuss the orange patches and the blue stripes next, because they are both related to the misfit supercells in plane Cu 2. Misfit supercells. One of the most interesting results of the local symmetry analysis is the observation of blue stripes in the third deposited Cu layer. They are not tiny strips of bcc phase, as the color would suggest, but they are the result of the misfit supercells formed in the second deposited Cu layer, which happen to generate arrays of blue atoms in the symmetry analysis of the next higher layer. This could be concluded from an atomic-level analysis of the Cu 2 plane. Fig. 5 shows in detail (magnified box K, Fig. 2(c)) how plane Cu 2 is intermediate between plane Cu 1, a bcc (100) plane with the Ta lattice constant, and plane Cu 3 (magnified box N, Fig. 2(d)), an fcc (111) plane with the Cu lattice constant. The figure displays part of grain CII shown in Fig. 4. Fig. 5(a) shows the atoms in plane Cu 2 (grey) together with the atoms in the next lower plane Cu 1 (magenta), while Fig. 5(b) shows the atoms in plane Cu 2 (grey) together with the atoms in the next higher plane Cu 3 (green and blue). It is interesting to see how this intermediate plane Cu 2 is built up. In both Fig. 5(a) and 5(b) the lower strip of atoms connected by black triangles is the misfit supercell in plane Cu 2. The supercell is a compromise between the rectangular atomic pattern in the plane below and the perfect triangular pattern of the plane above. In terms of the underlying bcc (100) Cu 1 layer, it is a (7√2 × √2)R45º supercell, containing 26 Cu atoms. Fig. 5(a) shows that in the [1 10] (horizontal) direction, 13 Cu-Cu nearest neighbor bonds match the combined length of 7 diagonals of the underlying Cu squares, or equivalently, the combined length of 7√2 Ta lattice constants. As 13 unstrained Cu-Cu bonds would measure 13 × 2.555 Å = 33.22 Å for the potential used, and 7√2 Ta lattice constants add up to 7√2 × 3.303 Å = 32.70 Å, the supercell is under a 1.6 % horizontal compressive strain with respect to the underlying plane, and this strain is accommodated by the slight zigzag in the horizontal lines connecting the Cu atoms in Fig. 5. In the [110] (vertical) direction, 2 Cu triangle heights, or equivalently √3 Cu-Cu nearest neighbor bonds, match one diagonal of the underlying Cu squares, or equivalently, of √2 Ta lattice constants. In this

101

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– direction the size ratio is therefore (2 × (√3/2) × 2.555) / (√2 × 3.303) = 0.947, and hence the supercell is under 5.3 % vertical tensile strain with respect to the underlying plane.

[010]

[100]

(a)

[010]

[100]

(b) Fig. 5. Illustration of the 26-atom misfit supercell in plane Cu 2 (grey atoms), shown connected by black triangles. The area shown is a closeup of the CII grain in Fig. 4, but rotated over 45°. In (a), plane Cu 2 is shown together with the underlying plane Cu 1 (magenta atoms, connected by bcc (001) squares). In (b), plane Cu 2 is shown together with the plane above it, Cu 3 (green and blue atoms, connected by fcc (111) triangles). The blue lines are shown to emphasize the atoms in the “blue stripes”. Colors, except grey, represent local symmetry.

Fig. 5(b) shows why the supercell in plane Cu 2 assumes this particular (7√2 × √2)R45º shape. At the cost of relatively small strains with respect to plane Cu 1, which can be well accommodated by a zigzag distortion, at least in the horizontal direction, the fit of plane Cu 2 with the fcc (111) Cu triangles in plane Cu 3 above it is in fact very good. Apparently then, this is how the Cu plane

102

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– manages to interpolate between two different Cu planes. We will come back to the vertical direction shortly. If such supercells would fit fully periodically in the present simulation volume (they do not), and if the plane consisting of these supercells would contain no defects, the plane would contain 170502 atoms. Table II shows that the actual value of nCu is 170834 for plane Cu 2. The fact that these two occupancy numbers are in excellent agreement is less trivial than it seems, because it is due to a superposition of several influences. First, the value of nCu was derived from a scaling relation that includes some uncertainty. Also, plane Cu 2 does contain defects, notably grain boundaries as we have seen, and therefore a smaller occupation number than 170502 should be expected. But on the other hand, we will show that relaxation of the supercell structure has taken place, which has opened up more positions for atoms to occupy. Together then, these three effects happen to have caused an almost exact match of the two occupation numbers. The blue stripes in plane Cu 3 are the blue atoms in Fig. 5(b), some of which are shown connected by blue lines in the top of the figure. It can be seen that they form a regular pattern (although not perfectly everywhere) perpendicular to, and in phase with, the unit cells of the supercell. There is another observation to make. In the vertical direction, a stack of 6 supercells (12 Cu triangles), each √2 × 3.303 Å high in order to fit the underlying plane, measures 28.03 Å. However, the total height of 12 undeformed fcc (111) Cu triangles, as they are present in the plane above, is 12 × (√3/2) × 2.555 Å = 26.55 Å. The strain energy of the 6 supercells with respect to the plane above is therefore considerable. Interestingly, the total height of 12 + 2/3 undeformed fcc (111) Cu triangles is also 28.03 Å. It is very well possible, therefore, that the presence of the plane above, once it is rigid enough by its firm connection to the next higher planes, makes it energetically favorable for the vertical stack of supercells in plane Cu 2 to break into pieces. The reason is the following. If each of these pieces would be 6 supercells (12 Cu triangles) high, it could contract from 28.03 Å to 26.55 Å in order to fit the plane above perfectly, while leaving the Cu atom that starts the next piece also in its perfect fitting position, only this perfect fitting position is shifted with respect to those of the previous piece by the 2/3 fcc (111) Cu triangle height mentioned, or which is the same, by a (1/6) vector. In other words: the energetic profit that plane Cu 2 can attain to fit plane Cu 3, once plane Cu 3 is present and rigid enough, stimulates plane Cu 2 to break up into pieces that are mutually shifted by (1/6) vectors, or –in our earlier terminology– into pieces that have different anchor positions. Fig. 6 (enlarged area of boxes M and P, Figs. 2(c)-(d)) confirms that this is indeed what has happened. In reality the actual number of supercells in the vertical direction is not always precisely equal to the ideal value 6. Nevertheless, deviations from this number are never large. This can be concluded from the fact that the widths of most orange strips in Fig. 2(d) are close to 30 Å. In the next section we will show that this breaking up of plane Cu 2, or “relaxation” of plane Cu 2 as we have called it earlier, indeed had to wait longer than the deposition of just plane Cu 3. The fitting together of two very different crystal structures through a single distorted atomic monolayer represents a mechanism for enabling epitaxy that is very different from the textbook mechanism of misfit dislocations.

103


[010]

[100 ]

Fig. 6. Misfit supercell, as in Fig. 5 (b), only here the atomic-level view of boxes M (grey atoms) and P (green, blue and orange atoms) in Fig. 2(c)-(d) is given. The central part of the supercell is shown, repeated in the vertical direction (black triangles connecting grey atoms). The red triangles also connect grey atoms, analogous to the black triangles but shifted in the vertical direction by (2/3) triangle height. This shift is due to an energetically favorable relaxation, see the main text for a discussion. Red arrows indicate the first grey atoms out of registry, near the (blue) edges of the green area. Six perfect (111) fcc hexagons fit within the green area in the vertical direction.

6.5 Changes in the film during deposition Fig. 7 gives an impression of the planes Cu 2, 3, 4 in three successive stages of deposition (the fourth and final stage was already depicted in Fig. 2(c)-(f)). The number of Cu monolayers deposited is, respectively, 2.96, 3.59, and 4.56, and the progression of the plane occupancies is shown schematically at the top of Fig. 7. Close inspection of these results shows that in plane Cu 2, a small grain with orientation I is consumed by the surrounding grain which has orientation II (white arrow). Also, as one follows the successive stages of plane Cu 2 all the way to Fig. 2(c), one notices how the grain boundaries become increasingly sharper defined. One should combine this information with the changes in plane Cu 3, all the way to Fig. 2(d), where it is seen that the orange patches appear only in the last stage. Together this shows that the breaking up of plane Cu 2 in regions with different anchor points, as was discussed in the previous section, is completed not before approximately 6 ML deposition. Finally, also plane Cu 4 in Fig. 7 is far from its final state. All this is clear evidence of the changes that occur in deposited atomic planes while the real arrival of new atoms at the surface takes place several planes higher in the film. 104


Ta

Ta

(a)

(b)

Ta

(c)

Fig. 7. Successive stages of planes Cu 2 to Cu 4 during deposition, (a) after 432091 Cu atoms deposited or 2.96 ML, (b) after 541967 Cu atoms deposited or 3.59 ML, (c) after 709866 Cu deposited or 4.56 ML. Atoms are colored according to local symmetry. Plane occupancy is shown schematically at the top of the figure. The white arrows in plane Cu 2 indicate a grain which is consumed by its surrounding grain having a different orientation. For reference we mention that the last stage of the deposition, after 6.31 ML, is shown in Fig. 2.

105

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– 6.6 Annealing Experimentally [11] it is known that Cu de-wets bcc-Ta (100) and (110) surfaces at elevated temperatures. To be more specific, Fig. 8 shows how a 50 Å thick copper film, deposited on a bccTa (100) surface by electron gun evaporation, de-wets the substrate after a 10 s anneal at 995 K. The film breaks up and forms rectangular 3D islands, aligned along the crystallographic directions of the substrate. It could be concluded from the experiments that the process of island formation takes place from approximately 400 to 700 K when the Cu/Ta system is subjected to a heating ramp of 40 K/s. The areas between the islands are not the bare, exposed Ta substrate but a Cu film of one or two monolayers thickness remaining on top of the substrate. Hence, the experiments showed that the de-wetting is actually a Cu-on-Cu phenomenon, where of course the remaining Cu film is strained because of the underlying Ta substrate.

Fig. 8. Secondary electron image of a 50 Å Cu film deposited on Ta (100) and annealed at 995 K for 10 s [11]. The white square has the same size as the system used in the current simulations (100 nm × 100 nm).

In order to investigate whether the Cu film created in the current simulations would also dewet the Ta substrate at elevated temperatures, the system was annealed after deposition of the film. For this purpose the unmodified Cu potentials were used, and not the Cu potentials modified for deposition as described earlier (Sec. 6.2). In this way, truly elevated temperatures were simulated. It was not expected that a full island formation as shown in Fig. 8 would be found in the simulations. For this, the feasible simulation times are too short, the simulation area is too small to accommodate even a single island of the size actually observed, and the Cu film is thinner than in the experiments.

106

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Nevertheless, it would be interesting to find out if at least a tendency towards de-wetting could be identified in the simulations. The results are shown in Fig. 9 and Table IV. First, the system was annealed at 550 K for 1.76 ns (Fig. 9(a)). Thereafter the temperature was increased to 1000 K and annealing was continued for another 3.39 ns, leading to a total annealing time of 5.16 ns (Figs. 9(b)-(d)). The figures clearly show that some in-plane agglomeration and coarsening of the island structure occurs. In addition, Table IV shows that the occupancy of plane Cu 7 (the yellow atoms in Figs. 9(a)-(d)) increases significantly, and that of the neighboring planes Cu 6 and 8 also, but to a lesser extent. Conversely, the other Cu planes, from plane Cu 2 onwards, have lost atoms at the end of the annealing period. In summary then, the simulations indicate that during annealing there is horizontal as well as vertical transport of Cu atoms, both supporting the tendency towards agglomeration and island formation. These observations agree with what could be reasonably expected, as explained above. The much more pronounced de-wetting found in the simulations by Hashibon et al [3] was the result of different simulation conditions (see Sec. 6.1) and a higher simulation temperature (1400 K). This makes it almost impossible to compare the two simulation results.

(a)

(b)

(d)

(c)

Fig. 9. Top views of the system started from Fig. 1(b) and annealed as follows: during the first 1.77 ns at 550 K, thereafter at 1000 K. Annealing times: (a) 1.77 ns, (b) 2.21 ns, (c) 4.00 ns, (d) 5.16 ns. The atoms are colored according to height, using a different scale than Fig. 1(b). The green atoms are in plane Cu 9. A coarsening of the islands is seen. For atom counts in the planes, see Table III.

107

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Table IV. Cumulative changes in the number of atoms per plane during annealing, see also Figs. 9(a)-9(d). The annealing temperature during the first 1.77 ns was 550 K, thereafter 1000 K. After 1.77 ns After 2.21 ns After 4.00 ns After 5.16 ns Plane ΔNTa ΔNCu ΔNTa ΔNCu ΔNTa ΔNCu ΔNTa ΔNCu Cu 10 Cu 9 Cu 8 Cu 7 Cu 6 Cu 5 Cu 4 Cu 3 Ta t + Cu 2 Ta 0 + Cu 1 Ta –1 Ta –2

0 0 0 2 14 9 2 17 274 -316 -17 -98

-15 -2182 -547 5920 4687 -1795 -1967 -2383 -1746 -11 15 2

0 0 0 1 21 9 6 16 -4 -47 8 -123

-18 -2255 1976 9572 3716 -3248 -3436 -3952 -2418 18 16 7

0 0 0 1 20 14 6 34 35 -109 16 -130

-20 -2564 1741 11176 3801 -3578 -3785 -4408 -2549 122 28 14

0 0 0 3 22 19 5 44 41 -138 7 -116

-18 -2651 1442 11504 3833 -3579 -3805 -4381 -2480 73 23 17

6.7 Conclusions We have performed large-scale Molecular Dynamics simulations using Johnson-Oh EAM potentials to study the growth and subsequent annealing of a Cu film (6.3 monolayer thickness) on a bcc-Ta (100) substrate (area 100 nm × 100 nm). The main conclusions are as follows. Compared to an earlier simulation on a considerably smaller substrate area (18 nm × 18 nm), a much richer microstructure is found in the current simulations. Newly found features include the presence of grain boundaries throughout the simulation, misfit supercells that enable epitaxy without misfit dislocations, the splitting up of the misfit supercells as the result of continued deposition and the coarsening and agglomeration of the surface morphology. The present results therefore clearly warn against the use of too small simulation domains, especially in cases where complex strain fields are expected, which may be too strongly limited by the boundary conditions. The first deposited Cu plane grows heteroepitaxially and essentially defect-free on the bccTa (100) surface. The second Cu plane is the most interesting of all planes. It grows in the form of a superstructure of 26-atom misfit supercells which after prolonged film deposition relax energetically, by breaking up in groups and forming 30 Å wide, in-plane island strips separated by fcc (1/6) vectors. This prolonged deposition is needed to give the planes above the second plane sufficient mechanical rigidity to cause the relaxation to happen. The distorted second Cu monolayer represents an alternative way of enabling epitaxy between very different crystal structures without introducing misfit dislocations. The boundaries between the shifted island strips are grain boundaries of one or two monolayers height. Compared to the higher planes, the second Cu plane contains approximately 2 % fewer atoms. The third and higher Cu planes are fcc (111) planes, with grain boundaries as main defects. The fully deposited planes have 1.9 % fewer atoms than an ideally close-packed fcc (111) plane. The Cu film is polycrystalline, starting from the

108

Chapter 6 Copper on Tantalum –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– second plane up, with two different in-plane crystal orientations. Some 65 intrinsic stacking faults bounded by Shockley partials were found. Practically no vacancies were detected. The influence of a one-monolayer Ta terrace step on the substrate is minimal. It turns out that it is not the source of interesting breaks in the film morphology. The most important (and trivial) effect is that –in the same plane– the microstructure of the part of the film that has grown on the terrace lags one monolayer behind the microstructure of the part that has grown outside the terrace area. Upon annealing the film at elevated temperatures, a distinct tendency towards island coarsening and agglomeration is observed, but a full 3D island formation such as found experimentally is not observed.

Acknowledgements Author would like to thank Prof. dr. Barend J. Thijsse (TU Delft, The Netherlands), Dipl. Ing. Carmen Schäfer (IMM, RWTH Aachen, Germany) for discussions and critical review of the manuscript. Author would also like to thank dr. Peter T. C. Klaver (TU Delft, The Netherlands) for establishing the original simulation set-up. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organisation for Scientific Research (NWO). This research is also supported by the Netherlands Materials Innovation Institute (M2i) formerly the Netherlands Institute for Metal Research (NIMR) within the project 02EMM31.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

T. P. C. Klaver, B. J. Thijsse, J. Comput-Aided Mater. Des. 10 (2003) 61. T. P. C. Klaver, B. J. Thijsse, Mat. Res. Soc. Symp. Proc. 721 (2002) 37. A. Hashibon, A.Y. Lozovoi, Y. Mishin, C. Elsässer, P.Gumbsch, Phys. Rev. B 77 (2008) 094131. D. J. Oh, R. A. Johnson, J. Mater. Res. 3 no. 3 (1988) 471. R. A. Johnson, D. J. Oh, Embedded atom method for close-packed metals, in ‘Atomistic Simulations of Materials: Beyond Pair Potentials’ eds. V. Vitek, D. J. Srolovitz, (1989) p. 233. R. A. Johnson, D. J. Oh, J. Mater. Res. 4 (1989) 1195. R. A. Johnson, Phys. Rev. B 41 (1990) 9717. F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema, and A. K. Niessen, Cohesion in metals, ed. F. R. de Boer and D. Pettifor (North Holland Physics Publishing 1989) p. 544. T. P. C. Klaver, B. J. Thijsse, Thin Solid Films 413 (2002) 110. P. T. Moseley, C. J. Seabrook, Acta Cryst. B 29 (1973) 1170. V. Venugopal, B. J. Thijsse, Thin Solid Films 517 (2009) 5482. P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B 28 (1983) 784.

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110

Chapter 7 Summary and Conclusions ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

7. Summary and Conclusions Improvement of wear resistant and tribological properties of materials is of great technological importance and is the main relevance of the current oxide coatings study. Aluminum oxide is especially interesting because of its very rapid self-repair capacity. As a second subject of this thesis, Cu film growth on Ta, a diffusion barrier material, is of great interest for further miniaturization of IC devices to overcome the interdiffusion problem of Cu in submicron applications. The aim of the present work was to improve our understanding of oxide and metal film growth on metal substrates by performing simulations at the atomic scale. For this type of simulations, molecular dynamics (MD) is a powerful method, because of its atomic resolution and full evolution monitoring. To operate on the required length scale for sufficiently large systems, the combination of classical MD with the help of density functional theory (DFT) overcomes the difficulties that appear if just pure first principles or experimental techniques are applied. In this work a powerful MD model for metal-oxide film growth has been developed, and MD simulations of Al-O/Al growth (on a relatively small surface area) and Cu/Ta growth (on a large surface area) have been performed. In both oxide and metal growth, atom-by-atom deposition was simulated, as opposed to the often applied method of letting two completed pieces of material approach each other and relax. Our method is much closer to reality. In the oxidation setup, an Al (111) surface was exposed to oxygen atoms with thermal energies to form the oxide. For a proper physical description of this system the consideration of variable ionic, long range interactions is required. In the metal film growth, a Ta (001) surface was exposed to impinging high-temperature Cu atoms with a well-defined deposition rate. In this pure metal-metal system ionic bonds are not present, so that the interactions can be treated locally. Aluminum oxidation The bigger part of the thesis covers Al oxidation and the self-repair phenomenon. For this the best possible MD model for metal/metal-oxide systems was developed and implemented into an appropriate MD code. The MD code camelion developed in our group at Delft University of 111

Chapter 7 Summary and Conclusions –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Technology had already the Modified Embedded Atom Model (MEAM) formalism built in, including angular dependences and sophisticated angular screening (AS) interactions. It was extended in the present work by the Charge Transfer Ionic Potential (CTIP). Adding ionic interactions to camelion enables the combination of CTIP with any type of non-ionic interaction model that is based on pair potentials, EAM, or MEAM. It was shown that for the successful treatment of the metal/metal-oxide systems, in particular to model the very beginning of the oxidation process, a combination of the so-called Reference Free MEAM (RF-MEAM) with CTIP was required. The RF-MEAM is a new version of the MEAM. The implemented code was tested in three important ways. First, the very high accuracies of the electrostatic part of the model (CTIP) as well as of the long-range Coulomb interaction solver Particle-Particle-Particle-Mesh (PPPM) were confirmed. Analytically determined Madelung constants for properly constructed structures agreed extremely well with values obtained from PPPM. Excellent numerical results followed for the CTIP charge transfer process. Second, it was demonstrated that the angular force terms of RF-MEAM can be conveniently used to control the relative stability of certain crystal phases, which was a particular requirement for the current work. This was shown for a hypothetical III-V compound. Third, in a preliminary simulation, and using a parameter set from the literature, one O atom approached an Al surface without showing any unrealistic behavior except for the too low value of its final charge. The Al oxide growth performed with the existing EAM+CTIP model from the literature showed some serious shortcomings. The two most important ones were the not fully negatively charged O atoms when surrounded by Al atoms, as mentioned above, and the existence of binary Al-O crystal structures with lower minimum energies than those calculated by ab initio. For some of the binary crystal structures the energies were even lower than that of corundum Al2O3, the most stable Al oxide crystal structure, clearly an unphysical situation. These and other shortcomings were successfully overcome by a much better optimization of the parameters, from the ground up, and by simultaneously upgrading the non-electrostatic part of the potential from EAM to RF-MEAM. The development of the new RF-MEAM+CTIP potential for the Al-O system required a new fitting process, which was much more complicated than in the case of regular pure metal or metal-alloy systems. Optimization of the electrostatic and nonelectrostatic parameters demanded two different fitting approaches, both of them requiring extensive DFT calculations to obtain large enough fitting target sets. The electrostatic part was fitted to the ab initio electrostatic potential field in between atoms for a number of binary Al-O structures. This led to chemically correct atomic charges without actually including them as fitting targets, which is particularly relevant for the structures representing the beginning of the oxidation process, such as one O atom in pure Al bulk. To complete the potential development, the RF-MEAM energy equations have been fitted to ab initio energies from which the energies calculated by the just determined CTIP model have been subtracted. The reason for this subtraction is that ab initio energy calculations are not aware of separate “electrostatic” and “non-electrostatic” contributions. The structures involved in this nonelectrostatic fitting process were several Al-O crystal structures, each having their energies 112

Chapter 7 Summary and Conclusions –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– evaluated for a collection of different lattice constants, and strained corundum and fcc-Al crystals. These strained crystals were used to fit the experimental values of the elastic constants. Both the potential energy data collections and the elastic constants had a crucial role in the final determination of the Al-O potential: the first to eliminate the occurrence of unnatural low energy structures, the second to impose structure stability. It turned out that for a proper fit it was absolutely necessary to have upgraded from EAM to RF-MEAM. A few tests and several basic MD runs have clearly indicated that our new model performs well. Although still more tests are needed in the future, the structure, properties, and thermal stability of an oxygen molecule, an oxygen atom in fcc aluminum, and the α-Al2O3 phase (corundum) proved to be excellent. Also, the corundum surface relaxed as expected. It is no exaggeration to say that our new CTIP|+RF-MEAM potential is one of the most advanced and intricate potentials developed thus far in the materials modeling field. The two-step fitting method and the charge transfer model in combination with a powerful angular-dependent potential have opened new roads for modeling metal/non-metal or fully non-metal binary systems.

Copper film growth on tantalum Metal on metal film growth was investigated in the system Cu/Ta. More than a million Cu atoms, organizing themselves into up to ten fully and partially occupied atomic planes, were deposited on an (100) oriented bcc Ta substrate. For investigation of step edge effects a special Ta-terrace was introduced. Besides deposition also the wetting/dewetting behavior of the Cu film during annealing was studied. This enabled atomic scale understanding of Cu on Ta film formation and microstructure evolution during growth. In this work the Cu/Ta simulation was performed using an EAM model for metals from the literature. It turned out that model worked quite well, taking into account the simulation speed, which was of great importance for simulating the large system (100 nm × 100 nm) that we wanted to study. The principal aim was to find out whether a large system would give rise to a richer microstructure than the small system studied earlier in our group. To analyze local effects during growth and during later annealing, a local angular environment type was determined for each atom, facilitating visualization, analysis and explanation of the microstructural evolution of the system in a very effective way. Spectacular, and rather surprising results were obtained from the study of the large-scale system. The peculiar development of the Cu-fcc film on a bcc-Ta substrate involved different grain orientations, grain boundary types, supercells, and lattice defects –all to be interpreted as the particular way in which these two materials build up their connection region–, but no misfit dislocations. During the simulation their evolution and propagation during different stages of the deposition and the subsequent annealing was monitored. Compared to the earlier work on the much smaller system (18 nm × 18 nm) the present simulations indeed showed much more microstructural richness, which is a testimony of the dangers involved in performing simulations in too small systems with periodic boundaries.

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Chapter 7 Summary and Conclusions –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– We briefly summarize a number of important results. Out of plane texture – On top of a bcc (100) Ta surface, the texture of the grown fcc Cu film is almost perfectly (111) after three Cu monolayers. The first Cu monolayer is epitaxial, the second Cu monolayer is the most interesting, “connective” layer. In plane texture – Two well-defined grain orientations are formed. Also two types of grain boundaries occur: between two grains of different orientations and between shifted regions (grains) within the same oriented area. Supercells – A modulation of the positions of the atoms in the 2nd Cu plane, which try to adjust to the underlying epitaxial bcc (100) layer but also to the fcc (111) layer above, forms misfit supercells (7√2 × √2)R45º. Grains and grain boundary evolution – The growth itself influences the local crystallinity and the grains in the lower layers. Grain growth during deposition was also detected. Relaxations – The appearance of Shockley partial Burgers shifts forming the second type of grain boundaries in the 2nd Cu plane magnifies drastically after 4 extra Cu layers deposited on top. Annealing – Formation and agglomeration of small islands during annealing occurs, but not so intensively as in the experiment.

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Samenvatting en Conclusies ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Samenvatting en Conclusies Verbetering van de beschermende en tribologische eigenschappen van materialen is van technologisch belang en is een belangrijke drijfveer voor de hier gepresenteerde studie van oxidecoatings. Met name vanwege zijn snelle zelfhelend vermogen is aluminiumoxide bijzonder interessant. De vorming van dit oxide vormt het eerste onderwerp van dit proefschrift. Het tweede onderwerp, de groei van koperfilms op tantaal, is van belang voor de verdere miniaturisatie van ICcomponenten. Dit onderzoek beoogt bij te dragen aan het beter onder controle krijgen van het interdiffusieprobleem van koper in submicronapplicaties (tantaal wordt gebruikt als diffusiebarrière), in combinatie met de vorming van een gewenste microstructuur in de koperkanalen. Het werk beschreven in dit proefschrift heeft tot doel het begrip van oxide- en metaalfilmgroei op metalen substraten te vergroten door simulaties uit te voeren op atomaire schaal. Voor dit type simulaties is moleculaire dynamica (MD) een krachtige methode, want deze methode bezit atomaire resolutie en de tijdsevolutie van een proces of systeem kan gedetailleerd zichtbaar worden gemaakt. Om op de vereiste lengteschaal te kunnen werken met voldoende grote systemen is de combinatie van klassieke MD met ondersteuning van Density Functional Theory (DFT) geschikt. In dit werk is een effectief en hoogwaardig MD-model voor metaaloxide filmgroei ontwikkeld en zijn MDsimulaties van Al-O/Al groei (op een relatief klein oppervlak) en Cu/Ta groei (op een groot oppervlak) uitgevoerd. Voor zowel de oxide- als de metaalgroei zijn deposities gesimuleerd door atomen één-vooréén op een substraat te laten aankomen, in tegenstelling tot de vaak toegepaste simulatiemethode waarbij twee stukken materiaal tegen elkaar worden geplaatst en daarna relaxatie mogen ondergaan. De hier gebruikte methode staat dichterbij de realiteit. In de oxidatiesimulatie werd een Al (111) oppervlak blootgesteld aan zuurstofatomen met thermische energieën, die vervolgens de oxidelaag vormen. Voor een goede fysische beschrijving van dit systeem is het nodig dat men ionische langeafstandsinteracties en omgevingsafhankelijke ladingen in rekening brengt. In het geval van metaalfilmgroei is een Ta (001) oppervlak blootgesteld aan hoge-temperatuur Cu-atomen, die een precies gedefinieerde depositiesnelheid hadden. In dit pure metaal-metaalsysteem zijn geen ionische bindingen aanwezig, zodat de atomaire interacties slechts lokaal hoefden te worden beschouwd.

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Samenvatting en Conclusies –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Aluminiumoxidatie Het grootste deel van dit proefschrift behandelt Al-oxidatie en het verschijnsel zelfheling. Hiertoe is een niew, hoogwaardig en krachtig MD-model voor metaal/metaal-oxide systemen ontwikkeld en geïmplementeerd in de MD-code camelion. Deze code ontwikkeld in onze groep aan de TU Delft, had reeds het Modified Embedded Atom Model (MEAM) formalisme ingebouwd. Het MEAMformalisme heeft, in vergelijking met het EAM formalisme, geavanceerde hoekafhankelijkheden in de atomaire interacties ingebouwd. In dit werk is het MEAM-formalisme verder uitgebreid met de Charge Transfer Ionic Potential (CTIP). Door op deze manier ionische interacties toe te voegen aan camelion is het mogelijk om CTIP te combineren met elk type niet-elektrostatische potentiaal die is gebaseerd op paar- of tripletinteracties. In dit werk wordt aangetoond dat voor een succesvolle beschrijving van de metaal/metaal-oxide systemen, in het bijzonder voor het modelleren van het vroege begin van het oxidatieproces, een combinatie van de zogenaamde Reference Free MEAM (RF-MEAM) met CTIP nodig was. De RF-MEAM is een nieuwe versie van de MEAM. De geïmplementeerde code is getest op drie belangrijke manieren. Om te beginnen is de nauwkeurigheid van het elektrostatisch deel van het model (CTIP) en van de berekeningsmethode van de lange-afstands Coulombinteractie (de Particle-Particle-Particle-Mesh (PPPM) methode) onderzocht. Deze nauwkeurigheid bleek zeer hoog. Analytisch bepaalde Madelungconstanten voor specifiek geconstrueerde structuren stemden precies overeen met de waarden verkregen met PPPM. Ook de CTIP-waarden voor het ladingstransport (ionisatie) bleken correct. Ten tweede is aangetoond dat de hoekafhankelijke termen van de RF-MEAM potentiaal effectief kunnen worden gebruikt om de relatieve stabiliteit van bepaalde kristalstructuren te regelen, een vereiste voor dit werk. Dit werd aangetoond voor een hypothetisch III-V halfgeleiderkristal. Ten derde werd aangetoond in een simulatie die gebruik maakte van een bestaand EAM+CTIP model en parameterset uit de literatuur, dat een O-atoom op een Al-oppervlak wel invalt zonder duidelijk onrealistisch gedrag te vertonen, maar toch uiteindelijk, eenmaal in de Al-laag opgenomen, een te lage waarde voor zijn lading krijgt. Gaandeweg werd ook in meer algemene zin duidelijk dat Al-oxide groei gemodelleerd met het bestaande EAM+CTIP model een paar grote problemen heeft. Er zijn twee belangrijke tekortkomingen in deze potentiaal. De eerste, al hierboven genoemd, is dat de O-atomen niet volledig negatief geladen blijken te zijn wanneer ze omringd worden door Al-atomen. De tweede is dat van sommige binaire Al-O kristalstructuren de minimale energie beduidend lager wordt voorspeld dan de energie die volgt uit ab initio-berekeningen. Voor sommige van deze binaire kristalstructuren zijn is de energie zelfs lager dan die van α-Al2O3 (korund), de meest stabiele Aloxide kristalstructuur. Dit is duidelijk een onfysische situatie. Deze en andere tekortkomingen van het bestaande EAM+CTIP model zijn met succes aangepakt door een betere en geheel nieuwe optimalisatie van de parameterwaarden uit te voeren en door gelijktijdig het niet-elektrostatische deel van de potentiaal uit te breiden van EAM naar RFMEAM. De ontwikkeling van deze nieuwe MEAM+CTIP potentiaal voor het Al-O systeem vereiste een nieuw fitproces, dat veel gecompliceerder is dan dat voor pure metalen of 116

Samenvatting en Conclusies –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– metaallegeringen. Voor de optimalisatie van de elektrostatische en de niet-elektrostatische parameters waren twee verschillende, complementaire fitmethodes nodig. Voor beide was een uitgebreide set targetwaarden nodig om de fit te verwezenlijken. Deze waarden zijn verkregen uit een grote hoeveelheid DFT-berekeningen. Het elektrostatische deel van de potentiaal, het CTIP-deel, werd gefit aan het ab initioberekende elektrostatische potentiaalveld tussen de atomen voor een groot aantal binaire Al-O structuren. Dit leidde tot chemisch correcte atomaire ladingen zonder dat deze daadwerkelijk zelf tot de gefitte waarden behoorden. Dit resultaat is et name relevant voor de structuren die voorkomen in het begin van het oxidatieproces, zoals een O-atoom in puur Al. Om de ontwikkeling van de potentiaal te voltooien met het niet-elektrostatische deel, werden de RF-MEAM energievergelijkingen gefit aan de ab initio-berekende niet-elektrostatische energieën. Hiertoe werden de eigenlijke ab initio energieën verminderd met de energieën die volgen uit het CTIP model zoals dat in de eerste fase bepaald was. Dit was nodig omdat ab initioenergieberekeningen geen gescheiden “elektrostatische” en “niet-elektrostatische” bijdragen kennen. Het fitten van dit niet-elektrostatische deel van de potentiaal is uitgevoerd voor verschillende Al-O kristalstructuren, waarbij voor elke structuur verschillende roosterconstantes gebruikt werden, en voor elastisch vervormde korund-Al2O3 en fcc-Al kristallen. Deze vervormde kristallen zijn gebruikt om de experimentele waarden van de elastische constanten te fitten. Het gebruiken van een groot aantal potentiële-energie data en van de elastische constanten bleek een cruciale rol voor de uiteindelijke Al-O potentiaal te spelen: de potentiële-energie data voorkwamen het ontstaan van onrealistisch lage energieën, de elastische constanten garandeerden de stabiliteit van structuren. Het bleek eveneens noodzakelijk geweest te zijn om voor een correcte fit de EAM potentiaal uit te breiden tot RF-MEAM. MD-tests hebben aangetoond dat het nieuwe model goed werkt en binnen het testgebied superieur is aan de eerdere Al-O potentiaal. Alhoewel nog in een beginstadium van verificatie, bleken nu al de structuureigenschappen, energieën, ladingen, en de thermische stabiliteit van een zuurstofmolecuul, een zuurstofatoom in fcc-aluminium en de α-Al2O3 fase (korund) uitstekend. Voor een meer uitgebreide evaluatie van de potentiaal zullen in de toekomst meer tests nodig zijn. Zonder overdrijving kan gesteld worden dat de nieuwe CTIP+MEAM potentiaal één van de meest geavanceerde en complexe potentialen is die tot nog toe voor het atomair modelleren van materialen en materiaalgedrag is ontwikkeld. De tweestapsmethode voor de fit en het ladingstransportmodel, gecombineerd met een effectieve hoekafhankelijke potentiaal heeft nieuwe wegen geopend voor het modelleren van metaal/niet-metaal systemen en van volledig niet-metaal systemen. Groei van een koperfilm op tantaal De groei van een metaalfilm op een niet-compatibel ander metaal, interessant niet alleen vanuit toepassingsoogpunt maar ook fundamenteel gezien, is onderzocht met MD-simulaties aan het Cu/Ta systeem. Een van de unieke eigenschappen van deze atomaire depositiestudie is de grootte van het systeem. Meer dan een miljoen Cu-atomen zijn gedeponeerd op een (100)-georiënteerd oppervlak van een bcc-Ta substraat. De koperfilm organisserde zich in vijf volledig bezette en daarboven vijf 117

Samenvatting en Conclusies –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– gedeeltelijk bezette atoomvlakken. Om stap- en randeffecten te onderzoeken werd voorafgaand aan de depositie tevens een terras op het Ta-substraat aangebracht. Doel van de simulaties was het onderzoeken van de microstructuur van de film tijdens de depositie en van het bevochtigen/ontvochtigen van het substraat door de Cu-film tijdens verhitting. Hiermee werd beoogd om op atomaire schaal de vorming van een Cu-film op Ta te begrijpen. In dit proefschrift is de Cu/Ta-simulatie uitgevoerd met een EAM-potentiaal uit de literatuur. Het was eerder gebleken dat dit EAM-model behoorlijk goed werkt en vanuit het oogpunt van computerefficiency snel genoeg was. Die snelheid was van groot belang voor het simuleren van het grote systeem (100 nm × 100 nm oppervlak) dat we voor ogen hadden, en waarvoor acht processoren nodig bleken. Een belangrijk oogmerk van het onderzoek was om er achter te komen of een dusdanig groot systeem, door de geringere beperkende invloed van randvoorwaarden, een rijkere microstructuur zou opleveren dan een klein systeem, zoals het systeem dat al eerder in de groep was bestudeerd. Om lokale effecten tijdens het groeien en het verhitten beter te kunnen bestuderen, werd aan ieder atoom een typering toegekend die gebaseerd is op de geometrie van de lokale omringing. Hiermee werden visualisatie, analyse en verklaring van de ontwikkeling van de microstructuur van het systeem op een zeer efficiënte manier mogelijk. De bestudering van het grote systeem leverde opvallende en nogal onverwachte resultaten op. Bij de groei van de fcc-Cu film op het bcc-Ta substraat ontstonden verschillende korreloriëntaties, korrelgrenstypes, supercellen en roosterfouten, die alle verklaard kunnen worden door de typische manier waarop deze twee materialen hun grensvlakgebied opbouwen. Tijdens de simulatie werd hun evolutie en ruimtelijke groei in de verschillende stadia van de depositie en de daaropvolgende verhitting gemonitord en geanalyseerd; een samenvatting volgt hieronder. Dislocaties kwamen slechts in kleine hoeveelheden voor. In vergelijking met het eerdere werk aan het kleine systeem (18 nm × 18 nm) hebben deze simulaties inderdaad meer rijkdom in de microstructuur laten opgeleverd. Dit onderstreept nog eens de gevaren van het doen van simulaties in te kleine systemen met periodieke randvoorwaarden. De belangrijkste resultaten van de Cu/Ta simulaties kunnen als volgt worden samengevat. Textuur loodrecht op het kristalvlak. Op een bcc (100) Ta oppervlak is de textuur van de gegroeide fcc-Cu film na drie Cu-monolagen bijna perfect (111). De eerste Cu-monolaag is epitaxiaal op bcc (100) Ta, de tweede Cu-monolaag is de “verbindingslaag”, de laag met de interessantste microstructuur. Textuur in het kristalvlak. Er ontstaan twee goed gedefinieerde korreloriëntaties. Ook twee types korrelgrenzen verschijnen: een type grens tussen twee korrels van de twee verschillende oriëntaties, en een ander type grens tussen verschoven gebieden in dezelfde korrel. Deze verschuiving wordt beschreven door de Burgersvector van een partiële Shockly-dislocatie in fcc-Cu. Supercellen. In het tweede Cu vlak leidt een modulatie van de atoomposities, die zich zowel aan de onderliggende epitaxiale bcc (100) laag als aan de bovenliggende fcc (111) laag proberen te conformeren, tot de vorming van supercellen van het type (7√2 × √2)R45º. Korrel- en korrelgrensgroei. De groei beïnvloedt de locale kristallisatiegraad en de korrels in de onderliggende lagen. Tijdens depositie wordt ook korrelgroei waargenomen. Relaxatie. Het aantal verschuivingen van het type dat het tweede type korrelgrens definieert, neemt drastisch toe nadat 4 Cu lagen zijn gedeponeerd. Verhitting. Tijdens verhitting vormen en agglomeren kleine eilanden, maar niet in dezelfde mate als eerder gevonden in het experiment. 118

Ivan Lazić Acknowledgements –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Acknowledgements In general having optimistic character, writing this very last chapter is harder than one might think. I remember catching myself often during night wondering about the final opportunity to say a few words about my PhD years. These thoughts varied from ultimate anger against the never-ending nightmare up to finding that everything I passed through was just the best set of experiences that could ever happen to me. The truth is of course somewhere in the middle. The PhD time was, I had to realize with some disappointment, not just about science. Actually, one would be surprised (as I was myself) how little time was actually dedicated to productive scientific work. Often it was an undefined and unconnected set of activities leading not always towards the expected amount of results. I was trying to explain to myself why that was the case, finally concluding that the main task of the PhD student is to learn how to deal with the situation. And somehow, fortunately or unfortunately, I managed it. I did it. You are holding something in your hands. That is the fact. If you are enthusiastic enough you might even read it. It might be not the greatest accomplishment ever, but it required quite some effort and beside me involved a lot of other people. Me, being what I am and doing what I do, is not just like that. Many are to be thanked for this and probably there is no way to include them all properly within these few pages. I believe that no one would feel less appreciated if I put my parents in the very first place. My mother, Ljubinka Lazić, without whose infinite love my personality would never form the way it did and my father, Dušan Lazić, who had always nothing but the greatest support and encouragement for me. My father has always been, beside a great parent, my first and the best life and science teacher. The discussions we had on various mathematical, physical and scientific topics in general were the deepest and the most honest I have ever had. My sister and brother, Milica and Mladen, were whom I missed the most during these years. No money or honor or diploma will ever pay me back the time that I did not spend with them. Their support was what kept me going when it was the hardest. The rest of my family, grandparents, cousins, which I am very proud of, are to follow. My closest friends are also those whom I owe a great acknowledgement. They all have influenced my personality, surprisingly for me to discover their strong imprint in a variety of everyday situations I was passing through. Next is education. Getting a PhD degree at Delft University of Technology does not make you entirely a TU Delft PhD. It was the knowledge I brought from home, the Faculty of Electrical

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Ivan Lazić Acknowledgements ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– Engineering, University of Belgrade, and my earlier education in Serbia, that enabled me to handle the highly demanding PhD tasks successfully. I must thank all my teachers starting from my primary school on, for discovering and supporting all of my potentials from the very beginning. My first physics teacher Olivera Žikić is to be mentioned explicitly. She had the necessary knowledge and approach to push me in the direction, which was, I have never doubted that, the best for me. Further on, the following people during my education are to be especially acknowledged. My gymnasium professors: Svetlana Jelenković (Mathematics), Tihomir Stojanović (Physics), Tihomir Zafirović (Informatics) and Miljana Golubović (Philosophy and Sociology). My university professors: Prof. dr. Vitomir Milanović (Quantum Mechanics), Prof. dr. Jovan Radunović (Statistical Physics, Optoelectronics and Laser Systems) and Prof. dr. Branimir Reljin (Electrical Circuit Theory, Digital Image Processing). For my first international scientific experience I must thank Prof. dr. Hanry Baltes and my supervisor dr. Petra Kurzawski for giving me the opportunity to do the internship at their group at ETH Zürich. Finally, my special acknowledgement goes to my graduation thesis supervisor Prof. dr. Zoran Ikonić (Optoelectronics and Quantum Nano Structures) from the School of Electrical Engineering, University of Leeds. The time I spend working with him showed me how fulfilling and inspiring scientific work can be. Names of the several people from Delft University of Technology, together with some others, are already mentioned in the acknowledgements of the particular chapters. Here again, and first of all, I must thank my promoter Prof. dr. Barend J. Thijsse with whom I shared most of the troubles and who was the only one always reliable and available for me. I would like to thank him for the opportunity to work in his Virtual Materials Laboratory group and for all the patience he showed to support me. I have almost never seen him in a bad mood or loosing his temper. Just with his presence, he would always improve my heavy thoughts making things look much, much less difficult and problematic. I would like to thank him for deep understanding of all troubles a foreign employee passes through and for delightful discussions we had on all kind of topics. Second, I would like to thank my dear college and old friend Darko Simonović who happened to get the position and join the same group at the right moment to pull me out of the impression that nothing really made sense. Also to be thanked are dr. Ir. Wim Sloof and dr. Ir. Marcel Sluiter for critical reviews of the manuscripts, discussions and for directing my steps at the important moments. One person outside the TU Delft I owe an acknowledgement is dr. Pierre Anglade from UCL Leuven, Belgium (at that time), whom I met only once. Nevertheless we met shortly, the suggestion he gave me in half a minute discussion completely turned things towards the right direction. I already mentioned that it was not always only about science. Bureaucracy and troubles with documents, visas, residence-permit extensions, taxes etc. were part of the life too. The one at our group who was the true guru, always reliable and knowing precisely what should be done, being always endlessly and unselfishly willing to help, aside from being a terrific college, pleasant person and good friend was dr. Ir. Diana E. Nanu. Thank you Diana, you and Marius for all pleasant moments we had and thank you for your patience to help me and others with the most annoying things each foreigner needs to deal with. During the PhD time one must live somewhere and usually with someone. I was extremely lucky to share my apartment with three absolutely terrific guys, my colleges and also my good friends: dr. Filip Miletić, then Aleksandar Borisavljević and finally Marko Mihailović. I would like 120

Ivan Lazić Acknowledgements ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– to thank them for all the tolerance and understanding, for the support through the tough moments and for all the unforgettable times we shared. I should certainly not skip here to mention the “Material Boys”: Riccardo (drums), Frederick (bass), Andy (solo guitar), Nick (voice) and me (rhythm guitar). Our noise making during the latehour rehearsals at the TU Delft Cultural Centre would completely clean up my head like nothing else could do. The “number-one” band of the fourth floor of Material Science and Engineering Department performed during the Christmas Borel 2008. Big thanks to Marcello and other people from our group for our specially designed T-shirts outfit and Vadym for the announcement. That was a great night. Finally the last person is the one I could hardly ever thank enough. She was like the sun after fuzzy days of heavy rain and grey clouds. There is an acknowledgement in one of the chapters and the scientific support is unquestionable, but that is simply nothing comparing to what I would like to express. I am being so lucky and happy to have her by my side. Since I met her everything else was forgiven, forgotten and less important... thank you Carmen. At the end, my appreciations and thanks go to all the people I anyhow got in contact with during this time. All of them made considerable impressions and imprints on me, some influencing my behavior, way of thinking, habits and wishes strongly. Especially I would like to thank my dear friend dr. Sergey Grachev for his honest opinions and views, all the happy occasions we shared and the enormous support he showed at the moments that were very, very difficult for me. I also must thank my friend and delightful companion Maksym Chernishov for the great times, lessons about tea, working out in the gym, and my Russian language mastering. Finally, with all the respect to everybody, certainly not being able to finish the following list, many thanks for the happy moments to people being daily present, currently or in the past: Maria, Aleksey, Oksana, Nail, Andrey, Denis, Demian, Otto, Raymond, Veronika, Jasmina, Arkady, David, Andy, Tobias, Frederick, Stefanie, Chaitanya, Vadym, Santiago, Emre, Vlada M., Agata, Vlada J., Stevan, Miloš, Mihajlo, Milan, Nick, Ricardo, Marcello, Deniz, Vanya, Onaz, Fidel, Sebastian, Ryoji Sahara, Marius … cheers guys! One might say that there are different ways to handle the finances of the PhDs. However, if there is no unselfish financial, educational and technical support available, PhD student’s life certainly includes much more difficulties and much less dignity. In my case I was very lucky to have this support and I can thank for it, only with the best regards, to my employer, Foundation for Fundamental Research on Matter (FOM).

Delft, November 2009. Ivan Lazić

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Ivan Lazić Acknowledgements –––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

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Ivan Lazić Curriculum Vitae ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Curriculum Vitae Ivan Lazić was born in Knjaževac, Serbia on April 3, 1979. Primary school and gymnasium he finished in Bor, Serbia. From 1998 until 2003 he accomplished his studies at the Faculty of Electrical Engineering, University of Belgrade, Department of Physical Electronics, Section for Optoelectronics and Laser Engineering. He graduated in November 2003 obtaining a Dipl.-Ing. degree (equivalent to Master’s degree). His diploma work on Si/Ge quantum cascade lasers was done in cooperation with dr. Zoran Ikonić from School of Electrical Engineering, University of Leeds, UK. During his undergraduate studies he carried out an internship at Swiss Federal Institute of Technology, ETH, Zurich, Department of Physical Electronics, group of Prof. dr. Henry Baltes, working on a smart single-chip, chemical micro-sensor. After graduation he worked shortly as a post-graduate student researcher at the Institute of Microelectronic Technologies and Single Crystals (IHTM), University of Belgrade, Serbia. In the period November 2004 until April 2009 Ivan Lazić did his PhD research at Delft University of Technology (TU Delft) developing a variable charge molecular dynamics model for studying self-healing oxide coatings and performing atomic scale simulations on metal and oxide film growth. The work was carried out at the Department of Materials Science and Engineering, Structure and Change in Materials, group of Prof. dr. Barend J. Thijsse. Ivan Lazić is author of several international journal and proceeding papers and has presented his work on several international conferences within the USA and Europe, including being invited speaker on seminars in Leeds (2004) and Oxford (2008), UK. Since April 2009 he is working as a post-doc at the Faculty of Applied Sciences, Kavli Institute of Nanoscience and Charged Particle Optics Group at Delft University of Technology. His current research area is electron and ion beam interaction with material aiming at absolute (atomic) resolution of electron and ion beam induced deposition and lithography (EBID, IBID and EBL).

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Ivan Lazić Publications ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Publications

1.

P. Kurzawski, I. Lazić, C. Hagleitner, A. Hierlemann, and H. Baltes, “Multi-Transducer Recordings from a Single-Chip Gas Sensor System Coated with Different Polymers”, 12th International Conference on Solid-State Sensors, Actuators and Microsystems, Boston, Massachusetts, Jun 2003, Digest of Technical Papers, Volume 2, IEEE, New York, USA, 1359-1362 (2003).

2.

I. Lazić, V. Miletić, D. Dujković, “Analiza gresaka pri rotaciji binarnih slika primenom razlicitih metoda”, “Error analysis of binary images rotated using different methods” (only in Serbian language), ETF Journal of Electrical Engineering, Podgorica (2004).

3.

Z. Ikonić, I. Lazić, V. Milanović, R. W. Kelsall, D. Indjin, P. Harrison, “n-Si/SiGe quantum cascade structures for THz emission”, Journal of luminescence 121, 311-314 (2006).

4.

I. Lazić, Z. Ikonić, V. Milanović, R. W. Kelsall, D. Indjin, and P. Harrison, “Electron transport in n-doped Si/SiGe quantum cascade structures”, Journal of Applied Physics 101, 093703 (2007).

5.

I. Lazić, B. J. Thijsse, “Exploring simulation methods for self-healing oxide films”, Mater. Res. Soc. Symp. Proc., 978E, Warrendale, PA, 0978-GG08-05.

6.

I. Lazić, M. A. Ernst, B. J. Thijsse, “Atomistic simulation methods for studying self healing oxide films”, Proc. 1st International Conference in Self Healing Materials, CD, Editors A. J. M. Schmets and S. van der Zwaag, Springer, Dordrecht, The Netherlands (2007).

7.

I. Lazić, B. J. Thijsse, “Self-healing of aluminum oxide films: construction of a high precision Al-O potential for molecular dynamics”, Proc. 4th International Conference on Multiscale Materials Modeling MMM, Ed. A. El-Azab, Department of Scientific Computing, Florida State University, Tallahassee, Florida, USA, 454-458 (2008).

8.

I. Lazić, T. P. C. Klaver, B. J. Thijsse, “Microstructure of a Cu film grown on bcc Ta (100) by large-scale Molecular Dynamics simulations”, resubmitted to Physical Review B (2009).

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Ivan Lazić Conferences and Talks ––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Conferences and Talks

1.

XLVII conference of ETRAN, Herceg Novi, 8-13 June 2003, Montenegro. Proceedings paper and talk: Analysis of quality of the scanned binary images rotated using different Methods.

2.

M2i (Materials innovation institute, earlier NIMR) annual conference “Building Bridges in Metallurgy”, Noordwijkerhout, The Netherlands (2007 talk, 2004-2006 and 2008 poster).

3.

FOM (Foundation for Fundamental Research on Matter) annual conference Physics@FOM, Veldhoven, The Netherlands (2005-2008, poster).

4.

MRS Fall Meeting Nov. 27 – Dec. 1, 2006, Boston, Massachusetts, USA. Proceedings paper and talk: Exploring simulation methods for self-healing oxide coatings.

5.

First International Conference on Self-Healing Materials, 18-20 April 2007, Noordwijk, The Netherlands. Proceedings paper and talk: Atomistic simulation methods for studying selfhealing mechanisms in Al/Al2O3.

6.

Material Modeling Laboratory (MML) seminar, 14 June 2008, Department of Materials, University of Oxford, UK. Invited speaker: Variable charge models in molecular dynamics.

7.

MMM-2008, 4th International Conference on Multiscale Materials Modeling, 27-31 October, 2008, Tallahassee, Florida, USA. Proceedings paper and poster: Self-healing of aluminum oxide films: construction of a high precision Al-O potential for molecular dynamics.

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