DEPARTMENT OF PHYSICS HAZARA UNIVERSITY MANSEHRA …
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STUDY OF ANHARMONIC EFFECTS IN THE PRESENCE OF ADPARTICLES AT THE SURFACES OF CRYSTALS OF NOBLE METALS By ZULFIQAR ALI SHAH Ph.D SCHOLAR DEPARTMENT OF PHYSICS HAZARA UNIVERSITY MANSEHRA 2015
Transcript of DEPARTMENT OF PHYSICS HAZARA UNIVERSITY MANSEHRA …
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STUDY OF ANHARMONIC EFFECTS IN THE PRESENCE OF
ADPARTICLES AT THE SURFACES OF CRYSTALS OF
NOBLE METALS
By
ZULFIQAR ALI SHAH
Ph.D SCHOLAR
DEPARTMENT OF PHYSICS HAZARA UNIVERSITY MANSEHRA
2015
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HAZARA UNIVERSITY MANSEHRA
Department of Physics
STUDY OF ANHARMONIC EFFECTS IN THE PRESENCE OF ADPARTICLES AT THE
SURFACES OF CRYSTALS OF NOBLE METALS
By
Mr. Zulfiqar Ali Shah
This research study has been conducted and reported as partial fulfillment of the requirements of PhD degree in Physics awarded by Hazara University Mansehra, Pakistan
DEPARTMENT OF PHYSICS HAZARA UNIVERSITY MANSEHRA
2015
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Declaration
The work presented in this thesis was carried out by myself under the
supervision of Dr. S. Sikandar Hayat Assistant Professor and Co-Supervision
of Dr. Najmul Hassan Assistant Professor. The conclusions are my own
research after numerous discussion with my supervisor and co-supervisor. I
have not presented any part of this work for any other degree.
Zulfiqar Ali Shah
We certify that to the best of our knowledge the above statement is
correct.
Research Supervisor
Dr. S. Sikandar Hayat
Co-Supervisor
Dr. Najmul Hassan
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It is stated that Mr. Zulfiqar Ali Shah defended his PhD dissertation under
title "Study of Anharmonic effects in the presence of adparticles at the
surfaces of crystals of Noble Metals" in the Senate Hall of Hazara University,
Mansehra on February, 27 2015. He successfully defended his research thesis.
The examiners are satisfied with his performance and recommended him for
the award of the degree of Doctor of philosophy in Physics.
External Examiner-I ---------------------------------- Prof. Dr. Syed Zafar Ilyas Chair Department of Physics
Allama Iqbal Open University, Islamabad
External Examiner-II ---------------------------------- Dr. Shahzad Shifa Assistant Professor
Department of Physics
GC University, Faisalabad
Internal Examiner/Supervisor ---------------------------------- Dr. S. Sikandar Hayat Assistant Professor Department of Physics Hazara University, Mansehra
Head of Department ------------------------------------ Dr. Saleh Muhammad Department of Physics Hazara University, Mansehra
DEPARTMENT OF PHYSICS HAZARA UNIVERSITY MANSEHRA
2015
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HAZARA UNIVERSITY MANSEHRA
APPROVAL SHEET OF THE MANUSCRIPT
PHD THESIS SUBMITED BY
Name Mr. Zulfiqar Ali Shah
Father's name Faqir Shah
Date of Birth January 01, 1986 Place of Birth Haripur
Postal Address Mohallah Gul Masjid Village & Post Office Gudwalian
District & Tehsil Haripur
Telephone Business +92-343-3039970
Email: [email protected]
PhD Thesis Title Study of Anharmonic Effects in the Presence of Adparticles at
the Surfaces of Crystals of Noble Metals
Language in which the thesis has been written English
APPROVED BY Signatures
1. Dr. Sardar Sikandar Hayat ..............................................
(Supervisor)
2. Dr. Najmul Hassan ..............................................
(Co-Supervisor)
RECOMENDED BY
1. Prof. Dr. Habib Ahmad ..............................................
Dean Faculty of Science
2. Dr. Saleh Muhammad ..............................................
Head Department of Physics
3. Dr. Muhammad Farooq ..............................................
Assistant Professor (Physics)
4. Dr. Muhammad Tauseef ..............................................
Lecturer in Physics
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STUDY OF ANHARMONIC EFFECTS IN THE PRESENCE
OF ADPARTICLES AT THE SURFACES OF CRYSTALS OF
NOBLE METALS
Submitted by ZULFIQAR ALI SHAH
PhD Scholar
Research Supervisor DR. SARDAR SIKANDAR HAYAT
Assistant Professor Department of Physics Hazara University Mansehra
Co-Supervisor DR. NAJMUL HASSAN
Assistant Professor Department of Physics Hazara University Mansehra
DEPARTMENT OF PHYSICS HAZARA UNIVERSITY MANSEHRA
2015
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TABLE OF CONTENTS
S. No. TOPICS PAGE #
DEDICATION I
ACKNOWLEDGEMENT II
LIST OF TABLES IV
LIST OF FIGURES V
LIST OF PUBLICATIONS IX
LIST OF ABBREVIATIONS XI
ABSTRACT XII
Chapter I INTRODUCTION 1
Chapter 2 THE ANHORMONIC EFFECTS IN METALS 9
2.1 The Anharmonic Effect 9
2.2 Fissure and Dislocation 12
2.3 Creation of Vacancy and its Migration 13
2.4 Surface Diffusion 15
2.5 Diffusion Mechanisms 16
2.5.1 Vacancy Diffusion 17
2.5.2 Interstitial Diffusion 17
2.5.3 Substitutional Diffusion 17
2.5.4 Self-Diffusion or Ring Mechanism 18
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2.5.5 Self-Interstitial 18
2.6 Steady-State Diffusion 18
2.7 Non-Steady-State Diffusion 19
2.8 Factors that Influence Diffusion 20
2.8.1 Diffusing Species 20
2.8.2 Temperature 20
2.8.3 Lattice Structure 21
2.8.4 Presence of Defects 21
Chapter 3 COMPUTATIONAL TECHNIQUES 25
3.1 Density Functional Theory 25
3.2 Monte Carlo Method 27
3.3 Molecular Dynamics (MD) 28
3.4 Born-Oppenheimer Approximation 28
3.5 Potentials 30
3.5.1 Born-Oppenheimer MD (BOMD) 30
3.5.2 Car-Parrinello MD (CPMD) 30
3.6 Equations of Motion 32
3.7 Procedure 35
3.7.1 Initialization 36
A. Interaction Potential 36
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B. Crystal Preparation 38
C. Velocity Initialization 39
D. Integration of Classical Equations of Motion 39
E. Periodic Boundary Conditions 41
F. Energy Minimization 42
3.7.2 Equilibration 46
A. Constant Volume Canonical Ensemble (NVT) 47
B. Isothermal-Isobaric Ensemble (NPT) 47
C. Micro-Canonical Ensemble (NVE) 49
3.7.3 Simple Statistical Quantities to Measure 49
A. Average Potential Energy 49
B. Average Kinetic Energy 50
C. Average Total Energy 51
D. Temperature 51
E. Mean Square Displacement 51
F. Pressure 52
G. Melting Temperature 53
3.8 More on MD 53
Chapter 4 MULTILAYER RELAXATIONS NEAR/AT THE
SURFACES OF COPPER 55
4.1 Introduction 55
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4.2 Computational Procedure and Modeling 57
4.3 Results and Discussion 59
4.3.1 Cu(311) Surface 60
4.3.2 Cu(210) Surface 61
4.4 Conclusions 64
Chapter 5 DIFFUSION OF COPPER PENTAMER ON AG(111)
SURFACE 75
5.1 Introduction 75
5.2 Modeling and Computational Details 77
5.3 Result and Discussions 79
5.4 Conclusions 85
Chapter 6 ANHARMONIC EFFECTS IN THE PRESENCE OF
Cu- AND Ag TRIMER ISLAND ON Cu(111) AND
Ag(111) SURFACES 93
6.1 Introduction 93
6.2 Computational Details 95
6.3 Results and Discussion 97
6.3.1 Ag3/Ag(111) 98
6.3.2 Ag3/Cu(111) 99
6.3.3 Cu3/Ag(111) 100
6.3.4 Cu3/Cu(111) 101
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6.4 Conclusions 106
Chapter 7 THE VACANCY GENERATION AND ADSORPTION OF
COPPER ATOM AT Ag(111) SURFACE 119
7.1 Introduction 119
7.2 Computational Details 121
7.3 Results and Discussion 122
7.4 Conclusions 128
Chapter 8 SUMMARY AND CONCLUSION 138
RECOMMENDATIONS 142
ACKNOWLEDGEMENT 142
REFERENCES 143
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Dedication
To my great father Faqir Shah and nice mother Hajra Khatoon
without whose love, support and prayers I could not have become
what I am now.
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ACKNOWLEDGEMENTS
I have no words to express my deepest sense of gratitude to The
Gracious, The Greatest “Almighty ALLAH”, whose innumerable blessings
enabled me to complete this difficult task. I also pay my sincere gratitude to
The Holy Prophet Muhammad (S.A.W); the most perfect and the most exalted
among all the human being ever born on the surface of the earth.
I would like to extend my deepest thanks to Dr. S. Sikandar Hayat for
his kind supervision during this dissertation and being an outstanding
academic advisor during my graduate studies. He is definitely a great source
of motivation and encouragement for me. His patience, scholarly supervision
and technical guidance have been sole asset for completing this important
task.
I wish to express my warm appreciation to Dr. Saleh Muhammad,
Chairman, Department of Physics for providing facilities, his advice and
encouragement during my research work. I am thankful to my Co-Supervisor
Dr. Najmul Hassan, Dr. Muhammad Farooq Dr. Muhammad Tauseef and all
faculty members of the Physics Departments. I am also thankful to Dr.
Muhammad Ayaz (Chairman Department of Political Science).
I would like to convey my heart fill thanks to Lab Co-Researcher Mr.
Zakir-ur-Rehman and my friends Mr. Bakhtiar ul haq (PhD Scholar at UTM)
to help and provide research materials during my research work. In addition
of these, special thanks are extended to Mr. Quaid Ali, Mr. Sajid Hussain, Mr.
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Anwar Ali, Mr. Baber Shahzad and Mr. Sadique Rahman. I am very grateful
of my wife. Her love and understanding encourage me to work hard and to
pursue PhD studies, her firm support has encouraged me to remain steadfast
in difficult situations. She has always let me know that she is proud of me,
which motivated me to work with satisfaction. I express my gratitude for my
daughter Eshal Zulfiqar, brothers and sisters whose affection, prayers and
wishes have been a great source of comfort during my research work. I also
express my gratitude to acknowledge the generous financial support of
Higher Education Commission (HEC), Government of Pakistan, for
Completion of this research endeavor.
ZULFIQAR ALI SHAH
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LIST OF TABLES
4.1 The unrelaxed and relaxed inter-layer separation for Cu (311)
and (210) surfaces.
65
4.2 Comprising Between percentage change in the plane registry
relaxation '' 1, iir and percentage change in interlayer relaxation
'' 1, iid near Cu(311) surface.
66
4.3 Comprising Between percentage change in the plane registry
relaxation '' 1, iir and percentage change in interlayer relaxation
'' 1, iid near Cu(210) surface.
67
5.1 Diffusion coefficient of Cu pentamer island on Ag(111) surface
at three temperatures and its effective diffusion barrier and
diffusion prefactor. Here square brackets represent the values of
molecular dynamics results while the figures in curved brackets
represent the Self Learning Kinetic Monte Carlo (SLKMC)
results.
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LIST OF FIGURES
S. No Topics Name Page No
2.1 The hard-sphere and square-well potentials. 22
2.2 Diffusion mechanisms. 23
2.3 Steady-state and Non-steady-state diffusion processes. 24
4.1 MD simulated values of lattice parameter at various
temperatures for Cu.
68
4.2 Interplaner spacing for Cu (311) surface. 69
4.3 Interplaner spacing for Cu (210) surface. 70
4.4 Percentage interlayer relaxation for Cu(311) surface. 71
4.5 Percentage plane registry relaxation for Cu(311) surface. 72
4.6 Comparison of percentage interlayer relaxation for Cu(210)
surface.
73
4.7 Comparison of percentage plane registry relaxation for
Cu(210) surface.
74
5.1 Comparison between simulated and experimental values of
lattice parameter at various temperatures for Ag.
88
5.2 Mechanism of Cu five-atom island diffusion observed during
molecular dynamics run.
89
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5.3 Trace of center of mass of Cu pentamer on Ag (111) surface at
(a) 300 K, (b) 500 K and (c) 700 K.
90
5.4 Center of mass of Cu pentamer on Ag (111) as a function to
time at (a) 300 K, (b) 500 K and (c) 700 K.
91
5.5 Arrhenius plot of the diffusion coefficient for Cu Pentamer on
Ag(111). This plot gives the energy effective barrier value as
0.20525 eV and the diffusion prefactor value as 5.549×1012
Å2/s. The inset represents the mean square displacement
(MSD) for Cu pentamer on Ag (111), as a function of time at
300, 500 and 700 K, respectively.
92
6.1 Structure of Cu(111) surface 109
6.2 Structure of Ag(111) surface 110
6.3 (a) Mechanism of trimer Ag/Ag(111) adatom diffusion
observed during molecular dynamics simulations. (b) From
left to right, snapshots representing anhormonic effects at the
Ag(111) surface in the presence of trimer Ag island at 300, 500
and 700 K, respectively.
111
6.4 Trace of center of mass of Ag trimer on Ag(111) surface at 500
K for 2000 ps.
112
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6.5 (a) From left to right, snapshots representing the breaking
effects of Ag(111) surface in the presence of the trimer Cu
island at 300, 500 and 700 K, respectively. (b) From left to
right, snapshots representing anhormonic effects at Cu(111)
surface in the presence of the trimer Ag island at 300, 500 and
700 K, respectively.
113
6.6 Trace of center of mass of Ag trimer on Cu(111) at 500 K for
2000 ps.
114
6.7 (a) Mechanism of Cu3/Ag(111) adatom diffusion observed
during molecular dynamics simulation. (b) From left to right,
snapshots representing anhormonic effects at the Ag(111)
surface in the presence of the trimer Cu island at 300, 500 and
700 K, respectively.
115
6.8 Trace of center of mass of Cu trimer on Ag(111) surface at (a)
500 K for 2000 ps.
116
6.9 (a) Mechanism of opening of island for Cu3/Cu(111) surface.
(b) From left to right, snapshots representing anhormonic
effects on Cu(111) surface in the presence of the trimer Cu
island at 300, 500 and 700 K, respectively.
117
6.10 Trace of center of mass of Cu trimer on Cu(111) surface at 500
K for 2000 ps.
118
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7.1 The circle represents fissure, the rectangle represents
dislocation, and the square represents vacancy. The pink-
colored atom is a surface atom, popped-up from the Ag(111)
surface. (a) At 300 K only a fissure seen (left hand snapshot).
(b) Both fissures and dislocations are observed in the middle
snapshot at 500 K. (c) Snapshot representing the vacancy at
700 K. (d) At 700 K, Green-colored atom is the exchanged
atom of Cu with the vacancy at the Ag(111) surface.
130
7.2 Snapshots (a) show the instantaneous migration of the
vacancy. The pink-colored atom represents the popped-up
atom on the Ag surface. Snapshot (b) shows the shifted
vacancy.
132
7.3 Trace of center of mass of Cu-trimer on Ag (111) surface
during 798 ps.
133
7.4 XY-plot of Ag monomer on Ag(111) surface during 798 ps. 134
7.5 Mixed-tetramer (comprising of three Cu- and one Ag-atom)
trace of center of mass of on Ag(111) surface during 765 ps.
135
7.6 XY-plot of the mixed-trimer island (comprising of two Cu-
and one Ag-atom) represents the position of the island over
Cu substituted atom in Ag(111) surface about 437 ps.
136
7.7 Trace of center of mass of mixed-trimer (Cu2-Ag island) on
Ag(111) surface during 750 ps.
137
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LIST OF PUBLICATIONS
This thesis consist of the following published and under-review papers
1. Zulfiqar Ali Shah, S. S. Haya, Z. Rehmana and f. Bouafia (2014) Phys. Lett. A
378 (1727–1732).
2. Zulfiqar Ali Shah, S. S. Hayat, Z. Rehman, S. S. Rahman and F. Bouafia
(2014) Surf. Rev. and Lett. 21, 5 (1450072-1450079).
3. F. Hussaina, Zulfiqar Ali Shah, S. S. Hayata, N. Hassanb and S. A. Ahmad
(2013) Chin. Phys. B 22, 9 (096102-096111).
4. F. Hussain, S. S. Hayat, Zulfiqar Ali Shah and S. A. Ahmad (2013) Chin. J. of
Phys, 51, 2 (356-367).
5. Zulfiqar Ali Shah, Z. Rehman and S.S. Hayat, "Study of Anharmonic effects
in the presence of Cu and Ag island on (111), A Molecular Dynamics
approach" March 7,8 (2014), AIOU Islamabad Pakistan.
6. Z. Rehman, Zulfiqar Ali Shah and S. S. Hayat "Computational Study of
Thermal Diffusive Properties of Cu/Ag(111) surface" March 7,8 (2014), AIOU
Islamabad Pakistan.
7. Zulfiqar Ali Shah, N. Hassan and S.S. Hayat, "Anhormonicity at Ag(111)
Surface in the Presence of Cu and Ag Trimer Island" Second Internal
Workshop on Materials Modeling and Simulations (IWWMMS)" (2012),
University of Malakand Pakistan.
8. Zulfiqar Ali Shah, S. S. Hayat and Z. Rehman, "Temperature role for
calculating Diffusion coefficient and Energy barrier in case of pentamer on
Ag(111)" (Under review in Computational Material Science).
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9. Zulfiqar Ali Shah and S. S. Hayat, "Multilayer Surface Relaxation of Cu (311)
and (210) Surfaces". (Under review in Surface Review and Letters).
10. Z. Rehman, Zulfiqar Ali Shah and S. S. Hayat, "Diffusion of Small Two-
Dimensional Cu Islands on Ag(111) Surface: A Molecular Dynamics
Approach" (Submitted to Computational Material Science).
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LIST OF ABBREVIATIONS
BOMD Born-Oppenheimer Molecular Dynamics
CEM Corrected Effective Medium
CPMD Car-Parrinello Molecular Dynamics
DFT Density Functional Theory
EAM Embedded Atom Methods
EMT Effective Medium Theory
EV Electron-Volt
FCC Face Center Cubic
FIM Field Ion Microscopy
HCP Hexagonal Close Packed
HK Hohenberg Kohn Theorems
ISS Ion Scattering Spectroscopy
KMC Kinetic Monte Carlo
LEED Low Energy Electron Diffraction
LEIS Low-Energy Ion Scattering
MC Monte Carlo
MD Molecular Dynamics
MSD Mean Square Displacement
NN Nearest Neighbor
PBC Periodic Boundary Condition
SLKMC Self Learning Kinetic Monte Carlo
STM Scanning Tunneling Microscopy
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ABSTRACT
The anharmonic effects at/near the surfaces of noble metals in the
presence of adparticles have been presented at 300, 500 and 700 K
temperatures. The Molecular Dynamics (MD) simulations are carried out for
Cu(111), Cu(311), Cu(210) and Ag(111) surfaces using realistic many-body
interatomic potentials obtained from the Embedded Atom Methods (EAM). In
a comparative study of the structure and the dynamics of the close packed as
well as open surfaces, the calculated shifts in the interlayer spacing, mean
square displacement, surface diffusion and surface defects (fissure,
dislocation and vacancy) show the onset of anharmonic effects at the
characteristic temperatures. Fir the Cu(311) surface interlayer relaxations
show a uniform damping in magnitude of oscillatory order (-, +, - , +, . . . ),
while for the Cu(210) surface they reveal a non-uniform damping in
magnitude with moving away from the surface. The plane registry relaxation
at both (311) and (210) surfaces exhibits no order for damping. The diffusion
of adatoms on the (111) surface of Ag and Cu has been studied to examine the
surface anharmonicity. The diffusion coefficient, effective energy barrier and
diffusion prefactor of Cu pentamer on Ag(111) are calculated. The MD
simulations at 300, 500 and 700 K show that the diffusivity obeys the
Arrhenius Law. An Arrhenius plot of the diffusion coefficients provides an
effective energy barrier of 205.25 ±10 meV and a diffusion prefactor 5.549×
1012 Å2/s. A striking feature of a pop-up of single-atom at 700 K among 5-
atom island is observed. The effective energy barrier obtained from the
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Arrhenius plot is in excellent agreement with those extracted from scanning
tunneling microscopy experiments. During the diffusion of Cu- and Ag-
trimer on Cu- and Ag(111) surfaces at 300, 500 and 700 K temperatures, the
constant energy MD simulation are consistent with anharmonic effects at the
surface such as fissures, dislocations and vacancy generation in the presence
of a island. The fissures and dislocations formed are in the range of 1.5–4 Å
and 1–7 Å respectively from the island position. The Cu and Ag islands both
diffuse easily on the Cu(111) surface, and indicate that diffusion is faster on
the Cu surface as compared to the Ag surface. The process of breaking and
opening of the island has been also observed. A surface atom is popped-up at
700 K by generating a vacancy near the Cu island at the Ag surface. The
energy of vacancy generation at the Ag surface is found to be Ev = 1.078 eV,
in the presence of a Cu trimer. The popped-up Ag adatom combines with Cu
the trimer, making a mixed-tetramer (Cu3–Ag island). The adsorption energy
of a copper atom from the mixed tetramer island into the substitutional site is
Es = ∼– 0.813 eV, leaving a mixed-trimer (Cu2–Ag island) at Ag(111). The rate
of diffusion increases with increase in temperature, both for homo and hetero
cases. Furthermore, we comment on the effect of adparticles diffusion on the
harmonic behavior of the surfaces in the anharmonic regime.
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Chapter 1
INTRODUCTION
The main objective of this dissertation is the detailed theoretical study
of the anharmonic effects at surface of noble metals by molecular dynamics
(MD) simulation techniques. The simulation is carried out for different
temperatures (300, 500 and 700 K) to study the different parameters of
anharmonicity on the surfaces. The access to the objective is achieved by
calculating mean square displacement (MSD), diffusion coefficient, diffusion
prefactor, and effective energy barrier. The diffusion of the islands on the
surfaces of the noble metals yields the anharmonic effects by generation of
vacancies on surface, breaking of islands, formation of fissures and
dislocations at the surfaces, pop-up of atom and adsorption of islands' atom.
The diffusion dynamics and their resulting anharmonic features are studied
both for homo and hetero cases. The multilayer relaxations of Copper (Cu) at
(311) and (210) surfaces with a focus on interlayer/interplanar spacing,
interlayer relaxation and plane registry relaxation are also estimated.
The noble metals are a group of metals that resist oxidation and
corrosion in moist air. The most common noble metals are ruthenium
(Ru), rhodium (Rh), palladium (Pd), silver (Ag), osmium (Os) , copper (Cu),
iridium (Ir), platinum (pt), and gold (Au). In the industry, alloys formed from
these metals are used to make various corrosion-resistant apparatus, electric
heaters, equipments for producing optical glass and glass fiber,
thermocouples and resistance standards.
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Chemical reactors and their parts may be made entirely of noble metals
or merely covered with a noble-metal foil. In medicine, noble metals are used
in instruments, parts, devices, artificial limbs, and various preparations,
chiefly based on silver. Noble metals are used in radiation therapy and in
compounds that increase the defensive abilities of the organism. In the
photographic and cinematographic industries, noble metals in the form of
salts are used to make light-sensitive materials. Alloys of noble metals are
used in jewelery-making and applied decorative arts. Noble metals are used
in the manufacture of mirrors. The reflectors covered by these metals are used
in infrared drying equipment, electronic contacts and components for
conductors, radio apparatus and equipment for X-ray therapy and
radiotherapy. They are used as an anticorrosion covering for specialized
tubes, fans and containers.
To study the anharmonic effects, molecular dynamics (MD)
simulations are carried out at three different temperatures i-e; 300, 500 and
700 K. These calculations are based on many-body interaction potentials
based on the Embedded-Atom Method (EAM). This potential is proposed by
Daw and Baskes [1,2] and is based on quasi—atom concept [3] and density—
functional theory. This method is equally applicable to simple as well as to the
transition metals. It has wide application to point defects [4], surface and
thermal expansion [5,6]. Foiles applied this potential [7] to liquid transition
metals and provided a realistic description of the energetic and structural
properties of the liquid phase. Mei [8] obtained the structural and dynamical
3
results of Cu, Ni, Au and Ag by performing molecular dynamics (MD)
simulations. The EAM smartly addresses the surface phenomena due to
thermal dynamics at the surfaces in the presence of ad-islands, the creation of
a vacancy at the surface and the responsible anharmonic features.
The details of the anharmonic effects for lattice dynamics and
thermodynamics have been extensively discussed in the literature [9-13].
However, the quantitative information on anharmonic effects, both
experimental and theoretical, is rather poor so far. The experimental
difficulties are due to the fact that at high temperatures (T ≈ Tm, where Tm is
the melting temperature) for which the anharmonic effects in the
thermodynamic properties become noticeable, it is usually difficult to identify
their contribution to the heat capacity and thermal coefficient of expansion
from the contributions of vacancies and other thermally excited lattice defects
[9,13-16].
The anharmonic features related to the diffusion of adatoms on metal
surfaces are very important for understanding many physical phenomena
related to surfaces, such as epitaxial thin film, crystal growth, heterogeneous
catalysis and morphological changes. The experimental techniques used to
investigate the surface diffusion parameters and their related phenomena are
field ion microscopy (FIM), scanning tunneling microscopy (STM), low-
energy ion scattering (LEIS) and helium-beam scattering. The diffusion of Ag
on Ag(100) has been studied by using the LEIS method [17] with a calculated
migration energy value of 0.40 ± 0.05 eV. Similarly, the STM study for the
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system of Ag/Ag(111) gave a corresponding migration energy of (97 ± 10)
meV [18]. From the theoretical point of view, the Ag/Ag(100) system has been
studied by several groups using semi-empirical models, such as the EAM [19,
20], by using the corrected effective medium (CEM) theory [21-23], the
effective medium theory (EMT) [24], and first-principle techniques [25,26].
Furthermore, theoretical investigations have also been performed for the
Ag/Ag(l11) system using the EAM [19, 20], CEM [27], EMT [18, 24-28] and ab-
initio calculations [25, 29]. Papanicolaou et al. used the molecular dynamics
simulations of the self-diffusion processes of Ag/Ag(l00) and Ag/Ag(l11),
using a many-body potential scheme within the second-moment
approximation of the tight-binding (TB) theory. He deduced the mean square
displacements (MSD) and relaxations of both the surface atoms and adatoms
in the normal to the surface direction as a function of temperature, as well as
the adatom-vacancy formation energy [30]. In the present work, the EAM is
employed for self diffusion of Ag/Ag(111) and Cu/Cu(111) surfaces and for
hetero diffusion of Ag/Cu(111) and Cu/Ag(111) surfaces. The effects of ad-
islands on the surfaces and the related anharmonicity produced by the
dynamics of the systems during the time and temperature evaluation, are
monitored.
Surface diffusion of atoms and clusters has been studied for decades,
due to their important technological role in thin film and crystal growth
processes. The advent of field ion microscopy and, later, scanning tunneling
microscopy, allowed the imaging of individual atoms on surfaces, enabling
5
diffusion of atoms and clusters to be observed directly. Direct observation of
atom and cluster diffusion has revealed a diversity of surface diffusion
phenomena and has lead to the discovery of many interesting and unexpected
mechanisms [31-33]. This is especially true for cluster diffusion, because of the
numerous ways a collection of atoms can move on a surface. Despite
predictions of interesting diffusion phenomena resulting from lattice
mismatch [34,35], relatively little work has been done in heteroepitaxial
systems with a large lattice mismatch.
Advances in atomic scale experimental techniques, supplemented by
enhanced computational power, have led to a surge in the studies of
structural properties of high Miller-index surfaces. Although, there exists a
number of experimental methods [36], yet the brunt of experimental data
have come from the low energy electron diffraction (LEED) technique. For
example in one of the LEED experimental study, Ismail et al., [37] studied the
structure of Cu(210) and found the oscillation of the interlayer relaxations to
be (-,-,+,+,-) with uniformly damping magnitude away from the surface into
the bulk. The LEED observation with a uniform damping in magnitude of
multilayer relaxation contradicts the experimental work on Al(210), in which
a significant interlayer relaxation has been found for the neighboring layer
away from the surface into the bulk [38]. In the experimental work for
Al(210), the magnitude of the interlayer relaxation for d34 is found to be larger
than d23 which is interesting due to the fact that d23 is near to the surface as
compared to d34 [39]. The recent theoretical work on Pd(320), based on first
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principle electronic structure methods, Makkonen et al., have predicted that
the magnitude of interlayer separation relative to the bulk, leads the following
observation |d45|>|d23|>|d34|[40].
The diffusion of particles on surfaces is an important step in many
surface phenomena, such as layer growth, heterogeneous catalysis, phase
transitions, segregation and sintering [41]. A large amount of experimental
information is available for single-atom diffusion on different surfaces [42,43],
while little is known about the details of cluster diffusion on surfaces. As a
result, many growth models still assume that once a cluster is nucleated it
remains immobile or, the mobility of the cluster simply decreases as the
cluster grows. However, it has been shown experimentally [44,45] and
theoretically [46–48] that the mobility of small clusters may exhibit a non
monotonic dependence on increasing cluster size. For instance, Wang and
Ehrlich [45] have shown that, for self-diffusion of Ir clusters on the Ir(111)
surface, the energy barrier increases sharply from adatom, dimer to trimer,
but drops for tetramer, and then rises again for pentamer and larger clusters.
Similar behavior was also obtained by embedded atom method [2]
calculations for Irn/Ir(111) [48] and Nin/Ni(111) [6].
The existence of vacancies in metals and intermetallic compounds
plays an important role for the kinetic and thermodynamic properties of
materials. In this connection, vacancy formation energy is a key concept in
understanding the processes that occur in different compounds and their
alloys during mechanical deformation or heat treatment. Over the last decade
7
many successful studies determined the energies of vacancy formation from
positron annihilation experiments [49-51] as well as from ab-inito full-potential
calculations for metals and their compounds [52-54] including for some
transition and noble metals [54-58]. The most recent work even includes
studies of vacancy-vacancy and vacancy-solute interactions in Cu, Ni, Ag and
Pd [59-62] and in Al [63] by means of the full-potential Korringa-Kohn-
Rostoker and Green’s-function method. On the other side, there are many
experimental observations and theoretical expectations which reveal that the
vacancy formation energy also effects other physical properties such as
cohesive and surface energies [64-66], melting and Debye temperatures [67,68]
and elastic constants [69]. Therefore, one of the goals of the present work is to
perform a systematic MD study of vacancy formation energies in noble
metals.
The diffusion of small islands on surfaces plays a key role in the
formation of epitaxial nanostructures and in thin film growth and deposition.
It provides a conclusive understanding about surface dynamical processes,
such as chemical reactions and the growth of islands and epitaxial layers. To
understand the migration mechanism and energetic of small atomic clusters
on crystal terraces, is of fundamental technological importance. To get a deep
insight into the thin film growth kinetics, it is of crucial importance to
visualize the adatom-surface interaction and adatom-adatom interaction on
surfaces [70]. The diffusion kinetics of the hetero-epitaxial metallic systems is
8
less well understood than that of homo-epitaxial metallic systems, based on
experimental [71,72] and theoretical [73,74] work done so far.
The rest of the chapters of this dissertation are organized as follows:
Chapter 2 contains the literature review of the anharmonic effects and the
factors which are responsible for these effects on the surfaces. In Chapter 3,
the computational details and simulation method employed in our
calculations are presented. The Chapter 4 contains the results of multilayer
relaxations for Cu (311) and (210) surfaces. The diffusion of Cu pentamer on
the Ag(111) surface is explained, in Chapter 5. The results of a molecular
dynamics study of anharmonic effects at Cu(111) and Ag(111) surfaces in the
presence of Cu- and Ag-trimer islands are described in Chapter 6. The
generation of vacancy and adsorption of Cu atom at Ag(111) surface during
diffusion of Cu trimer, is the part of the Chapter 7. The manuscript is
summarized in Chapter 8.
9
Chapter 2
THE ANHORMONIC EFFECTS IN METALS
The anharmonicity due to atomic vibrations in solid materials is well
recognized, and is required to account for many macroscopic properties,
including thermal expansion, volume dependence of elastic constants, mean
square displacement, radial distribution function, finite thermal conductivity
and the asymptotic value of the specific heat at high temperatures.
Anharmonicity can be observed directly by using phonon spectroscopy, and
is responsible for the finite lifetime of phonons and the temperature
dependence of phonon energies.
2.1. The Anharmonic Effects
The inter-atomic/inter-ionic potential energy does not end at the
quadratic term, but has a cubic term as well as a quadratic term. The
importance of these terms increases with increasing the temperature. The
effects produced by these additional terms are called anharmonic effects, as
the potential energy cannot be reduced to a simple harmonic potential.
In order to get a better understanding of the anharmonic potential, we
need a simple model to use as our starting point [75,76]. The simplest model
of all is a linear chain of atoms, each of mass m and separated by the unit
cell length a, as illustrated in Fig. 2.1.
For the moment consider that each atom only feels the force due to
its immediate neighbor. It is named as the nearest-neighbor interaction.
10
If the energy between two neighbors at a distance of a is φ(a), the total energy
of a chain of N atoms when each atom is at rest is:
(2.1)
Now, we assume that each atom can move about a little, and we represent the
displacement of an atom along the chain by the symbol u. If the
displacements are small in comparison with a, then we can calculate the
energy of this flexible chain using a Taylor series, summing over all the atoms
given by:
!
11
1
s
n
nns
s
s
uuus
NE
(2.2)
If un is the displacements of the nth atom from its equilibrium position, the
distance between two atoms n and 1n is )( 1 nn uuar . Thus the
derivatives of ϕ with respect to u are equivalent to the derivatives with
respect to r. Since a is the equilibrium unit cell length, then the first derivative
of ϕ is zero, so that the linear term in the expansion (s = 1) can be dropped. All
the other differentials correspond to the point u = 0. As u is small in
comparison to a, this series is convergent, and we expect that the dominant
contribution will be the term that is quadratic in u. Therefore, we start with a
model that includes only this term (s = 2), neglecting all the higher order
terms. The energy of this lattice is the same as the energy of a set of harmonic
oscillators, and so we call this approximation, the harmonic approximation.
The higher-order terms that we have neglected, are called the anharmonic
terms . By considering these terms, we can more closely approach the real
aNE
11
system. For example, if heat is added to a harmonic lattice and divided
unequally between the normal modes, there is no way for the system to reach
equilibrium because the normal modes do not interact. However, if slight
anharmonicity exists in the lattice, we expect that equal partitioning of energy
can occur and that the system could thus reach equilibrium. An intermediate
approach is the quasiharmonic approximation, which is both simpler then
dealing with all the anharmonic terms in the potential and more versatile than
the harmonic approximation. One example of quasi harmonic approximation
is adding thermal expansion to an otherwise harmonic system by changing
the lattice constant of the system corresponding to the desired temperature.
As we are aware, anharmonic effects manifest themselves in thermal
expansion, in non-linear variations of mean square vibrational amplitudes
with temperature, and in the temperature dependence of phonon frequencies
and line-widths. With further enhancements in anharmonicity, surfaces may
disorder or roughen [76]. While these effects are well defined, the
experimental observation of the temperature variation of these quantities,
particularly at higher temperatures, is not simple. It is thus not surprising that
evidence of anharmonic effects at surfaces as accumulated through
observations of surface structural phase transitions, roughening, premelting,
disordering, diffusion of adatoms, vacancies and clusters, and other thermally
activated phenomena have been at times controversial. Theoretical treatment
of anharmonic interactions is also non-trivial. Instead of exact analytical
approaches, theoretical methods consist of either perturbative schemes which
12
are valid in regions of low anharmonicity, or the use of the quasiharmonic
approximation, or the resort to molecular dynamics simulations using
anharmonic interaction potentials [75,76]. In any event, deeper understanding
of the characteristic temperatures at which these inherently anharmonic
phenomena are initiated or accentuated is necessary, not only for
fundamental reasons but also for technological development in the synthesis
of new materials and thin films. The factors responsible for the generation of
anharmonicity at the metallic surfaces are briefly described in the following
sections of this chapter.
2.2. Fissures and Dislocations
Dislocations are abrupt changes in the regular ordering of atoms, along
a line (dislocation line) in the solid surface whereas fissures are gaps that
appear at the surfaces, due to a small displacement of the atoms. They occur
in high density and are very important in mechanical properties of materials.
They are characterized by the Burgers vectors, found by doing a loop around
the dislocation line and noticing the extra interatomic spacing needed to close
the loop. The Burgers vector in metals points in a close packed direction [77].
Mostly two types of dislocations may occur on the solid surfaces. These are
edge and screw dislocations.
Dislocations play a key role in the mechanical properties of solid
materials and are also responsible for the anharmonicity by enabling
crystalline materials to deform plastically when subjected to stress orders of
magnitude lower than their theoretical critical shear stress [77]. They have
13
also been shown to relieve stress in strained films, greatly affecting the
growth mode [78-80], and have been predicted [81,82] but never
experimentally implicated in adatom island diffusion. The majority of
experimental studies of island diffusion have been limited to homo-epitaxial
systems where motion is usually a result of diffusion at steps, particularly for
large islands (102 atoms) [83-88]. In these cases, the barrier is insensitive to
size, but diffusivity scales with size depending on the rate-limiting process
[89-91]. In contrast, there have been numerous theoretical predictions [92-99]
and a few experimental demonstrations [100,101] of non-trivial size
dependencies of the diffusion barrier and/or prefactor for smaller homo-
epitaxial clusters (2-20 atoms).
2.3. Generation of Vacancy and it's Migration
Vacancy and its migration, is the dominant mechanism behind atomic
transport and is of fundamental importance in processes like solid phase
transformations, nucleation and defect migration. Vacancies also play an
important role for surface morphology [102]. Vacancies, created at the surface,
exist at a certain concentration in all materials depending exponentially on the
formation energy of the vacancy and the temperature. The importance of
vacancies and calculation of formation energies have been a natural subject of
theoretical studies. A theoretical accuracy in energy of the order 0.025 eV is
desired if rate predictions are to be made within a factor of three for processes
occurring at room temperature.
14
When calculating energy differences, it is helpful to remember the rule
of thumb: ‘‘always try to compare similar systems.’’ The bulk and the vacancy
systems differ by the introduction of a void in the system [103]. There is an
exposed surface area in the case of a vacancy and not in the bulk case,
resulting in a qualitative difference between the perfect bulk and vacancy
systems, in turn increasing the demands of applicability on any theoretical
method used.
The formation energy for the vacancy is calculated from:
bulkvac En
nEE
1 (2.5)
where Evac is the total energy of the cell containing a vacancy, n is the number
of atoms in the bulk cell, and Ebulk denotes the total energy for a bulk
calculation using the same cell and parameters. For the relaxed geometries,
we calculated the total energy for several volumes and found the minimum
by fitting. Then the unrelaxed vacancy geometries are calculated at different
volumes [103].
Two kinds of relaxation are involved in a vacancy calculation, volume
and structural relaxations. First looking at volume relaxation, a void causes
electrons to partly fill it, to lower their kinetic energy. This rearrangement of
electrons, in turn, reduces the density from the optimal bulk value in other
parts of the system [104]. Reducing the volume of the solid compensates for
this effect. For calculations, this effect is large for small unit cells, since the
15
reduction in bulk volume caused by the vacancy is a significant fraction of the
cell volume. However, for large cells we find that this volume relaxation is
irrelevant.
2.4. Surface Diffusion
Surface diffusion is a general process involving the motion
of adatoms, molecules, and atomic clusters (adparticles) at solid material
surfaces. The process can generally be thought of in terms of particles
jumping between adjacent adsorption sites on a surface. Just as in
bulk diffusion, this motion is typically a thermally promoted process with
rates increasing with increasing temperature. Many systems display diffusion
behavior that deviates from the conventional model of nearest-neighbor
jumps [105]. Various analytical tools may be used to elucidate surface
diffusion mechanisms and rates, the most important of which are field ion
microscopy and scanning tunneling microscopy. While in principle the
process can occur on a variety of materials, most experiments are performed
on crystalline metal surfaces. Due to experimental constraints most studies of
surface diffusion are limited to well below the melting point of the substrate,
and much has yet to be discovered regarding how these processes take place
at higher temperatures [106].
Diffusion is the process by which atoms move in a material. Many
reactions in solids and liquids are diffusion dependent. Structural control in a
16
solid to achieve the optimum properties is also dependent on the rate of
diffusion [107].
Atoms are able to move throughout solids because they are not
stationary but execute rapid, small-amplitude vibrations about their
equilibrium positions. Such vibrations increase with temperature and at any
temperature a very small fraction of atoms has sufficient amplitude to move
from one atomic position to an adjacent one. The fraction of atoms possessing
this amplitude increases markedly with rising temperature. In jumping from
one equilibrium position to another, an atom passes through a higher energy
state since atomic bonds are distorted and broken, and the increase in energy
is supplied by thermal vibrations. As might be expected defects, especially
vacancies, are quite instrumental in affecting the diffusion process on the type
and number of defects that are present, as well as the thermal vibrations of
atoms[108].
Diffusion can be defined as the mass flow process in which atoms
change their positions relative to neighbors in a given phase under the
influence of thermal treatment and a gradient [107]. The gradient can be a
compositional gradient, an electric or magnetic gradient, or stress gradient.
2.5. Diffusion Mechanism
In pure metals, self-diffusion occurs where there is no net mass
transport, but atoms migrate in a random manner throughout the crystal. In
alloys inter-diffusion takes place where the mass transport almost always
17
occurs so as to minimize compositional differences. Various atomic
mechanisms for self-diffusion and inter-diffusion have been proposed.
2.5.1. Vacancy Diffusion
Vacancy diffusion can occur as the predominant method of surface
diffusion at high coverage levels approaching complete coverage. This
process analagous to the manner in which pieces slide around in a sliding
puzzle. It is very difficult to directly observe vacancy diffusion due to the
typically high diffusion rates and low vacancy concentration [105].
2.5.2. Interstitial Diffusion
Solute atoms which are small enough to occupy interstitial sites diffuse
by jumping from one interstitial site to another. The unit step here involves
jump of the diffusing atom from one interstitial site to a neighboring site.
Hydrogen, Carbon, Nitrogen and Oxygen diffuse interstitially in most metals,
and the activation energy for diffusion is only that associated with motion
since the number of occupied, adjacent interstitial sites usually is large [105].
2.5.3. Substitutional Diffusion
Substitutional diffusion generally proceeds by vacancy mechanism.
Thus interstitial diffusion is faster than substitutional diffusion.
18
2.5.4. Self-Diffusion or Ring Mechanism
Three or four atoms in the form of a ring move simultaneously round
the ring, thereby interchanging their positions. This mechanism is untenable
because exceptionally high activation energy would be required.
2.5.5. Self-Interstitial
Self-Interstitial diffusion is more mobile than for a vacancy as only
small activation energy is required for a self-interstitial atom to move to an
equilibrium atomic position and simultaneously displace the neighboring
atom into an interstitial site. However, the equilibrium number of self-
interstitial atoms present at any temperature is negligible in comparison to the
number of vacancies. This is because the energy to form a self-interstitial is
extremely large [108].
2.6. Steady-State Diffusion
Diffusion processes can be either steady-state or non-steady-state.
These two types of diffusion processes are distinguished by use of a
parameter called flux. It is defined as the net number of atoms crossing a unit
area perpendicular to a given direction per unit time. For steady-state
diffusion, flux is constant with time, whereas for non-steady-state diffusion,
flux varies with time [105,106]. A schematic view of concentration gradient
with distance for both steady-state and non-steady-state diffusion processes is
shown in Figure 2.3.
19
Steady-state diffusion is described by Fick’s first law which states that
flux, J, is proportional to the concentration gradient. The constant of
proportionality is called diffusion coefficient (diffusivity), D (cm2/sec).
Diffusivity is characteristic of the system and depends on the nature of the
diffusing species, the matter in which it is diffusing, and the temperature at
which diffusion occurs. Thus under steady-state flow, the flux is independent
of time and remains the same at any cross-sectional plane along the diffusion
direction. For the one-dimensional case, Fick’s first law is given by:
dt
dn
Adx
daDJ x
1 (2.6)
and
txfJ x , (2.7)
where D is the diffusion constant, da/dx is the gradient of the concentration a,
dn/dt is the number atoms crossing per unit time a cross-sectional plane of
area A. The minus sign in the equation means that diffusion occurs down the
concentration gradient. Although, the concentration gradient is often called
the driving force for diffusion (but it is not a force in the mechanistic sense), it
is more correct to consider the reduction in total free energy as the driving
force [105].
2.7. Non-Steady-State Diffusion
The most interesting cases of diffusion are non-steady-state processes
since the concentration at a given position changes with time, and thus the
20
flux changes with time. This is the case when the diffusion flux depends on
time, which means that a type of atom accumulates in a region or is depleted
from a region (which may cause them to accumulate in another region). Fick’s
second law characterizes these processes and is expressed as:
dx
daD
dx
d
dx
dj
dt
da (2.8)
where da/dt is the rate of change of concentration at a particular position.
2.8. Factors that Influence Diffusion
Ease of a diffusion process is characterized by the parameter D,
diffusivity. The value of diffusivity for a particular system depends on many
factors as many mechanisms could be operative.
2.8.1. Diffusing Species
If the diffusing species is able to occupy interstitial sites, then it can
easily diffuse through the parent matter. On the other side, if the size of
substitutional species is almost equal to that of parent atomic size,
substitutional diffusion would be easier. Thus size of diffusing species will
have great influence on diffusivity of the system.
2.8.2. Temperature
Temperature has a most profound influence on the diffusivity and
diffusion rates. It is known that there is a barrier to diffusion created by
neighboring atoms, as these need to move to let the diffusing atom pass. Thus,
atomic vibrations created by temperature assist diffusion. Empirical analysis
21
of the system resulted in an Arrhenius type of relationship between
diffusivity and temperature.
RT
QMD exp (2.9)
where M0
is a pre-exponential constant, Q is the activation energy for
diffusion, R Boltzmann’s constant and T is absolute temperature. From the
above equation, it can be inferred that a large activation energy means a
relatively small diffusion coefficient. It is observed that there exists a linear
proportional relation between lnD and 1/T. Thus by plotting and considering
the intercepts, values of Q and M0 can be found experimentally [105].
2.8.3. Lattice Structure
Diffusion rate is faster in open directions than in closed directions.
2.8.4. Presence of Defects
As mentioned in earlier sections, defects like dislocations, grain
boundaries act as short-circuit paths for diffusing species, where the
activation energy of diffusion is less. Thus the presence of defects enhances
the diffusivity of diffusing species [108].
22
2+n U 1 n U Un 1- Un 2- Un
Fig. 2.1: The hard-sphere and square-well potentials.
23
Fig: 2.2: Diffusion mechanisms
24
Fig. 2.3: Steady-state and Non-steady-state diffusion processes.
25
Chapter 3
COMPUTATIONAL TECHNIQUES
Computational science plays an important role in all disciplines of
science on atomic and subatomic level of molecules and complex compounds
in Physics, Chemistry and Bio-sciences, it has a wide range of applications.
The Noble prizes of 1998 and 2013 show the significance of computational
tools. We can apply different simulation techniques on atoms or molecules in
material science to predict all the physical properties of a material before
synthesis. Ultimately through computational materials science we can predict
the properties of known materials even at extreme conditions like high
temperature, pressure or radiation and can design new materials with desired
properties under these conditions [109-114].
To reduce the hurdles in computational work of solid state physics,
theoretical studies play a key role to decide a best approximation, while
preserving as much information as possible about the objects of study.
Sudden breakthroughs were made possible by the development in the power
of computers, so simulations often need breakthroughs in theory. We have
different simulation techniques like molecular dynamics, Density functional
theory (DFT), ab-initio methods and Monte Carlo (MC) etc.
3.1. Density Functional Theory
Density Functional Theory is a quantum mechanical theory applied in
physics and chemistry to study the electronic structure (principally the
ground state) of many-body systems, in special atoms, molecules and the
26
condensed phases. From DFT the properties of a many-electron systems can
be calculated by using functionals, i.e. function of another function. DFT is
very useful and accepted method accessible in condensed-matter physics,
computational physics and computational chemistry. Using DFT, the basic
properties of solids both in the bulk form and at the interfaces can be
explained successfully. Kohn and Sham proved that the Hamiltonian equation
derived from this variation approach has in a very simple form. The so-called
Kohn-Sham equation is related in form to the time-independent Schrodinger
equation with the exception that the potential skill by the electrons is properly
expressed as a functional of the electron density. DFT was given a compact
theoretical grip by the Hohenberg-Kohn theorems (H-K) [115]. The basic H-K
theorems held only for non-degenerate ground states in the absence of a
magnetic field, although they have since been generalized to include these
[116, 117].
Despite further advances, there are difficulties in using the density
functional theory to suitably illustrate the intermolecular interactions, mainly
in Van der Walls forces (dispersion), charge transfer excitations, transition
states, global potential energy surfaces, some other strongly correlated
systems, and in calculations of the band gap in semiconductors. Its shortened
treatment of spreading can negatively affect the validity of DFT (at least when
used alone and uncorrected) in the behavior of systems which are dominated
by spreading (e.g. interacting noble gas atoms) or where spreading competes
significantly with other effects.
27
3.2: Monte Carlo Method
The word "Monte Carlo (MC) method" was coined in the 1940s by
physicists working on nuclear weapon projects in the Los Alamos National
Laboratory [118]. MC methods are a category of computational algorithms
that rely on repeated random case to compute their outcome. Monte Carlo
methods are frequently used in simulating physical and mathematical
systems. Since these rely on repeated computation of random or pseudo-
random numbers these methods are mostly run using a computer and tend to
be used when it is not possible or feasible to compute an accurate result with
a deterministic algorithm [119]. MC methods are techniques which are based
on the use of random numbers and probability statistics to study the
problems. Using MC methods we examine the model physical problems and
solve their equations which consist of hundreds or thousands of atoms [120].
MC simulation methods are mainly used for modeling phenomena with
trivial uncertainty in inputs and in studying systems with a large number of
combined degrees of freedom. In particular, MC methods play an important
role in computational physics, physical chemistry and related applied fields,
and have different applications from complex quantum chromo dynamics
calculations to scheming heat shields and aerodynamic forms. The MC
method is broadly used in statistical physics, particularly Monte Carlo
molecular modeling as an alternative for computational molecular dynamics
as well as to calculate the statistical field theories of simple particle and
polymer models. In experimental particle physics, these methods are applied
28
for designing detectors, understanding their actions and comparing
experimental data to theory or on much larger scales of galaxy modeling [121,
122].
The major problem which DFT and MC techniques face is the variation
in temperature. Temperature is an essential variable for the change in the
physical properties of solid materials. The majority of our concern lies in
calculating temperature dependent quantities like energy barriers, thermal
expansion, rate of diffusion and pre-melting, etc. To calculate these quantities
MD is superior on DFT and MC techniques. Molecular dynamics can
successfully predict deviation in the physical properties of solid compounds.
3.3: Molecular Dynamics
The molecular dynamics technique used at atomic level is totally based
on classical mechanics, including a classical approach for quantum mechanics.
Molecular dynamics treats atomic movements and predicts static as well as
dynamic properties of solid materials. The motion of atoms on atomic scales
can be explained within the limit of classical mechanics, described by
fundamental Newton's equations. Although quantum mechanical approaches
are used to calculate inter-atomic forces in case of First-principles calculations
the motions of the nuclei can still be safely considered as classical.
3.4: Born-Oppenheimer Approximation
The Born-Oppenheimer approximation explains the re-arrangement of
electrons that occurs instantaneously in response to the motions of nuclei.
29
Newton’s equations of the system have nothing to do with the electrons. The
future and past trajectories of atoms can be calculated accurately if we have
the exact knowledge about positions, velocities and inter atomic forces inside
the system. While in real quantum mechanics the same process cannot be
true as the time-reversal symmetry is broken.
In the molecular dynamics method, the classical equations of motion
are solved according to time evolution:
The forces on each atom can be calculated by assuming the initial
positions, velocities and inter atomic potentials of every atom in the
system.
Under these forces, the positions and velocities of the atoms are
updated by a time step.
These equations depend on the choices of initial conditions,
approximations of using finite time steps, adding up of truncation and
rounding errors, obviously there are many problems. These problems are
briefly discussed in the following sections and the detailed explanation can be
found in many textbooks and review articles [123-131].
The information about a system obtained directly are the positions and
the velocities (which can be calculated from the former) of all atoms in the
system. Other parameters, such as energies and forces, may be saved during
the calculation and are easy to calculate from the positions.
30
3.5: Potentials
Molecular dynamics is commonly classified on the basis of potentials
used or how the inter atomic forces are calculated; using a good potential
results a good Molecular dynamics simulations. Inter atomic potentials can be
attractive or repulsive. Classical MD uses simple pair or multiple particle
potentials in analytical forms. The inter-atomic forces are calculated through
DFT in First-principles MD [123].
There are two categories of First-principles MD; i) Born-Oppenheimer
MD (BOMD), ii) Car-Parrinello MD (CPMD).
In the first case the inter atomic forces can be calculated with approximate
electronic structure, such as insulators and semiconductors. The second is
applicable to metals.
3.5.1: Born-Oppenheimer MD (BOMD)
The minimization method is used in BOMD with a standard DFT
calculations. The inter atomic forces are calculated from the relaxation of
electronic states to the ground states for every time step.
3.5.2: Car-Parrinello MD (CPMD)
In CPMD method the electronic degrees of freedom are related to the
classical coordinate system. Also a set of the plane wave basis set is
considered classically as an additional set of coordinates. This can be done by
assigning a fictitious mass to these orbitals. The CPMD method is beneficial in
31
respect that it reduces the computational expenses and there is no need to
relax electronic states to ground states. This type of approach may work well
for some systems but may work less for others.
In CPMD a short time step is often required, therefore the convergence
of CPMD is poorer as compared to BOMD, and this partially undermines the
cost benefit. Firstly we assign certain functions to the structural and/or
thermodynamical properties of the system for the generation of classical
potentials. These properties are lattice parameters, bulk modulus, elastic
modulus, thermal expansion, and vibrational spectrum among many others.
There are some good packages to calculate Classical potentials through first
principles methods. These potentials are transferable: how well the potentials
can be used for other systems or systems under other conditions is discussed
below.
Classical potentials tend not to be easily transferable, and it is
necessary to find or generate the right potentials for each system of interest.
Suppose that there are N particles in a system. The potential energy can be
expanded into two-particle, three-particle, ........ n-particle terms:
N
KJI
N
JI
rkrjriUrjrirnrrU ,,,,....2,1 (3.1)
Where near-range interactions are dominant in systems, then all higher-order
potentials are ignored and only consider pair potential ., ji rrU need to be
considered.
32
There are tens of well known pair potentials which are widely used. For
example, the Lennard-Jones (LJ) potential is:
62
1
, 4
rU ji
(3.2)
where r is the inter atomic distance and + and ‒ are parameters. LJ
potentials are commonly used for free noble gas atoms. The Buckingham
potential is oftenly used for solids and can be written as:
6exp
CBrArU Buck (3.3)
Where A, B, and C are the parameters of total energy. These two potential are
isotropic and the energy depends only on inter atomic distance. More general
empirical potentials can be develop that describe highly directional covalent
bonds in solid, such as diamond. One such potential is the Tersoff potential.
In metals and their alloys pair potentials usually cannot give a good
description of the system, because electrons are delocalized and for
delocalized electrons long-range interactions are more important.
3.6: Equations of Motion
It is quite difficult to solve many-body problem analytically. We apply a
numerical technique to solve many-body problem and the best choice is
molecular dynamics. Vesely [132] explain molecular dynamics on simple
fluids, Hoover [133, 134] and Evans and Morriss [135] described the non-
equilibrium simulations. Plasma simulations, galactic evolution, and electron
33
flow are emphasized by Hockney and Eastwood [136]. Collections of the
theory were edited by Ciccotti and Hoover [137], and Allen and Tildesley
[138], who explain the most comprehensive expositions of methods and
applications [139].
Molecular dynamics is a computer simulation technique in which time
evolution of a set of interacting atoms and molecules is followed by
integrating their equations of motion.
In molecular dynamics the Laws of Newtonian mechanics are used and
most notably Newton’s second law:
iii amF (3.4)
where i stands for ith atom, Fi is the force, mi is the mass and ai is the
acceleration of ith atom. Hence the force can be expressed as the gradient of
the potential energy
VFi (3.5)
Combining these two equations yields
2
2
dt
rdm
dr
dV ii
i
(3.6)
where V denotes the potential energy of the system. Newton’s equation
of motion can then relate the derivative of the potential energy to the changes
in position of atoms as a function of time [140].
A simple application of Newton's second law:
2
2
dt
xdm
dt
dvmamF (3.7)
34
If acceleration is constant then:
dt
dva (3.8)
Integrating above equation we get
0vatv (3.9)
whare 0v is the initial velocity
And since
dt
dxv (3.10)
By integrating again we obtain
0xtvx (3.11)
Combining this equation with the expression for the velocity, we obtain the
following relation which gives the value of x at time t as a function of the
acceleration a, the initial position x0, and the initial velocity v0 [140].
00
2 xtvtax (3.12)
The acceleration is given as the derivative of the potential energy with respect
to the position, r,
dr
dE
ma
1 (3.13)
35
The initial positions of the atoms are used to calculate a trajectory such
as: an initial distribution of velocities and the acceleration. The gradient of the
potential energy function determines these distributions. The positions and
velocities can be calculated at all other times, t by using the positions and the
velocities in equations of motions at time zero. The initial distribution of
velocities is determined from a random distribution with the magnitudes
conforming to the required temperature and corrected so there is no overall
momentum [140].
N
i
iivmP1
0 (3.14)
In 1957 Alder and Wainwright for the first time performed molecular
dynamics for a condensed phase system by using a hard sphere model [141].
Some early simulations are also used the square well potential. A. Rahman for
the first time used continuous potentials in simulation [142], and also
performed the first simulation for a molecular liquid [143]. He introduced
many methodologies in molecular dynamics [144].
3.7: Procedure
There are several steps to perform molecular dynamics for the solution
of many body problems. These steps are discussed below.
36
3.7.1: Initialization
A. Interaction Potential
The basic method used so far to calculate the interaction potentials was
Pair potential approximation. The superposition principle provides a basis for
this approximation. The superposition principle states that the net force on a
particle is the sum of the forces due to all of other particles that interact with
this particle. So the net force on a single atom in a many-body problem was
calculated by summing up the forces due to all other atoms, which was
calculated with the help of pair-potentials. But pair potential shows
inaccuracy for several properties of metal solids; for example, in the
calculations of elastic constants, vacancy formation energies, cohesive energy
and melting point. This inaccuracy is due to the fact that the interaction of
other atoms is missing in the interaction between a pair of atoms [144]. This
means that pair potential is not suitable for the Face Centered Cubic (fcc)
metal systems. Therefore a many body interaction potential needs to be
introduced to solve this problem. Different types of potentials are in practice.
Interatomic potentials
Pair potentials (for details see Reference [145])
Stillinger-Weber potentials (for details see Reference [146])
Tersoff potential (for details see Reference [147])
Embedded Atom model (for details Reference [4])
Tight-binding potentials (for details see Reference [148])
37
Out of these potentials we used thr Embedded-Atom Method (EAM) in
our calculations. The Embedded-atom method is a semi-empirical technique
used to compute the energy of randomly chosen arrangements of atoms.
Embedded-atom methods are widely used for computing the energy of FCC
metal systems including defects, surface relaxations, surface diffusion, surface
and bulk phonon properties and thermal properties. The embedded-atom
method is proposed by Daw, Foiles and Baskes [2]. This is based on two
terms.
1- The energy needed to “embed” an atom into the local electron density
provided by the remainder of the atoms.
2- An electrostatic interaction represented by a pair interaction.
)]([)( i
i
iij
ji
ij rFrU
(3.15)
Here )( ir
is local electron density at site ir
, iF is the energy to place atom i
into that electron density and ijij r is the pair interaction between atoms i
and j. The electron density at each site is calculated from superposition of
atomic electron densities.
ji
ij
ati rrr
)( (3.16)
rij is the pair interaction which is repulsive potential. The effective charge
is constrained to be positive and to decrease monotonically with increasing
separation.
38
r
rZrZr
ji
ij
(3.17)
The simple parameterized form for rZi is
rrZrZ vi
i exp10 (3.18)
The embedded-atom method requires more computation time as compared
to pair interaction models, even though it includes many-body interactions.
Using an embedded-atom method potential Foiles et al. [4] draw plots for the
effective charge Z(R) as a function R and embedding functions F(ρ) as a
function of background electron density for Cu, Ag, Au, Ni, Pd, and Pt
metals.
B. Crystal Preparation
A crystal with the desired surface is prepared such as a (111), (311), or
(210) surface. We can generate a crystal with any required surface having all
x, y and z mutually perpendicular directions. In our case, the (111) surface is
required for surface diffusion and vicinal surfaces such as (210) and (311) for
the calculation of surface relaxations. The (111) surface is the most compact
surface and (210) is the most open surface. Initial configuration of the crystal
is important for the desired direction and plane. After choosing the surface
type (compact or open) and unit cell, we can expand the unit cell by choosing
type of surface and unit cell. In this way we obtain the initial positions of all
the atoms that form our system.
39
C. Velocity Initialization
Depending on temperature we assign initial velocities to the atoms
randomly selecting from a Maxwell-Boltzmann distribution:
Tk
vm
Tk
mvp
B
ixi
B
iix
221
2
1exp
2 (3.19)
Maxwell-Boltzmann distribution provides the probability that an atom i
of mass mi has a velocity component vix in the x‒ direction at a temperature T.
Actually, the Maxwell-Boltzmann distribution in one direction is a
Gaussian distribution; we can obtain this using a random number generator.
The initial velocities are adjusted in such a way that momentum of the system
should be zero. First we calculate the total momentum of the system in each
direction, then this value is divided by the total mass of the system and the
result of this is subtracted from the atomic velocities to introduce an overall
momentum [144].
D. Integration of Classical Equations of Motion
When we assign initial positions and velocities to the interacting atoms
in the system then we need a reliable algorithm to integrate the equations of
motion. With the help of continuous potentials we generate molecular
dynamics trajectories. The main idea is to break down the integration into
many small time intervals δt. The total force on each particle in the
configuration at certain time t is the vector sum of its interactions with other
particles. Equations of motion are integrated with the help of different
40
algorithms. But predictor-corrector approach is a standard method for the
solution of ordinary differential equations. Sufficiently accurate atomic
positions, velocities, etc. are obtained. The time interval δt will be significantly
smaller and will be adjusted according to the method of solution.
For continuous classical trajectory, positions, velocities, etc. are
estimated by using Taylor expansions at time tt .
.....
.........
......2
1 2
tatta
ttatvttv
tatttvtrttr
P
P
P
(3.20)
Superscript P is used to indicate these “predicted” values; we shall be
“correcting” them shortly. A complete set of positions and velocities are taken
r, and v. We should store three “vectors” r, v, and a by truncating the
expansions, as describe above. In another way we use a slightly different
predictor equation and velocities: we take r, v and “old” values of velocities
ttv , ttv 2 or r, v, a and “old” accelerations tta . Since
we have not taken in to account the equations of motion with the passage of
time step, Equation 3.17 does therefore not generate correct trajectories. These
equations enter through correction the step. The forces at tt can be
determined with the help of new position pr and hence we can calculate the
“correct” acceleration ttac . To calculate the error in the predictor step
we can compare these with predictor acceleration Equation 3.20.
ttattatta pc (3.21)
41
This error as well as the results of the predictor step, are fed into the corrector
step, which reads typically,
ttcttatta
ttcttvttv
ttcttrttr
pc
pc
pc
2
1
0
(3.22)
Here we used the idea of ttrc , etc. for the true positions,
velocities, etc. the coefficients c0, c1, and c2 are discussed by Gear [135] (best
choice). By including more position derivatives in this scheme then different
values of coefficients are used. The coefficients also depend upon the order of
differential equation. To refine the position, velocities, etc. through the
equations above, the new “correct” accelerations are calculated by iterating
from the position cr and comparing values of
ca . The iterations are the key to
obtaining an accurate solution. The initial guess is provided by the predictor
then successive iterations converge rapidly to the correct answer [144].
E. Periodic Boundary Conditions
Defining the boundaries of a simulated system is not trivial. If we simply
terminate the boundaries, then we have fewer neighbors near the boundaries
as compared to atoms inside. In this condition a number of surfaces surround
the sample. It is not realistic if we want to simulate a cluster of atoms. In
reality a microscopic piece of matter contained number of atoms which is in
the order of 1023 and the ratio between the total number of atoms and surface
atoms is much larger. Therefore surface plays an important role in simulation.
42
Reliable calculations are obtained by applying Periodic Boundary Conditions
(PBC). In applying PBC, the particles are considered to be enclosed by a box.
The repetition of these boxes to infinity will completely fill the space by rigid
translation in all three Cartesian directions. If a particle is located at position r
in the box then this represents an infinite set of particles at positions
znymxlr , ,,, zyx (3.23)
where, x, y, and z are integers and l, m, and n are vectors which represent
translation directions of the box. We have to represent once in a
computational program while these “image” particles move together. Now
here a particle i is considered as interacting with other j particles in a box
along with images in near-by boxes. Interactions are allowed by box
boundaries. In this case virtual elimination of the surface effects on the system
takes place and position of the box boundaries has no effect [149].
F. Energy Minimization
Systems always exist in the minimum energy configuration. After
cutting a crystal along a specific direction we get a desired surface, in which
the upper most layer of atoms will be attracted to the atoms below in the
crystal. In the simulation, we have to minimize the energy of all the atoms.
A minimum point can be either global (lowest function value) or local
(lowest in a finite neighborhood and not on the boundary of that
43
neighborhood). A global minimum point cannot be obtained so easily. There
are two standard ways to find global minima:
1- Local minima are found from widely varying starting values of the
independent variables.
2- Taking a finite amplitude step away from it we have to see if our
routine returns to a better point or “always” to same one to perturb
local minima.
There are different methods for the calculations of energy minimization.
Conjugate-gradients
Newton’s method
Brent’s method
Steepest descent
Molecular dynamics cooling
We used the conjugate-gradients technique for energy minimization.
We consider a symmetric and positive–definite function with G the
gradient operator such as:
xGxxF 2
1 (3.24)
The minimization of xF in a certain direction (say 1d ) from some point 1x
will occur at 1112 dbxx where 1b satisfies after differentiation of this
equation with respect to 1b at 2x
44
01111 dGdbx (3.25 a)
A subsequent minimization along some direction 2d will then produce
2223 dbxx where 2b satisfies
0222111 dGdbdbx (3.25 b)
For the minimization of xF, we differentiate equation 3.24 with respect to
1b and 2b at 3x . This gives
0122111 dGdbdbx (3.26 a)
and
0222111 dGdbdbx (3.26 b)
In order to get consistency between Equation 3.21 and 3.22, and for the
independent minimization along 1d and 2d to be, we require that
0. 1221 dGddGd (3.27)
The directions 1d and 2d must be conjugate to each other [150]. Generalizing
to
,0 mn dGd For mn (3.28)
The conjugate-gradients technique is a simple and effective procedure
to find energy minimization. The initial direction is taken to be the negative of
the gradient. Linear combination of the new gradient and the previous
direction that minimized xF is used for the construction of a subsequent
conjugate direction. In two-dimensional problem we need only two conjugate
45
directions. These directions are enough to span the space. The minimum is
obtained in just two steps while the Steepest-descents method allows
convergence in many steps. The previous direction vector and the current
gradient maintain the necessary information about multidimensional space.
Thus we get an important result that in this conjugate-gradient manner, the
directions generated are indeed conjugate. The algorithm used to obtain the
precise search direction id generated by the conjugate-gradients method, is
given below [144].
1 mmmm dgd (3.29)
where
11
mm
mmm
gg
gg (3.30)
and 01 .
The dimensionality of the vector space in each iteration is reduced by
one because minimization along the conjugate direction is independent. Hare
when the dimensionality of the function has been reduced to zero then the
trial vector must be at the position of minimum. Conservatively speaking, to
locate exactly the minimum of a quadratic function takes in a number of
iterations equal to the dimensionality of the vector space. However, a
minimum can be located in practice by performing the calculations so that far
less iteration is required to locate the minimum [144].
As we increase the numbers of iterations then the value of energy of the
crystal decreases gradually. The energy cannot be decreased after the
46
perpendicular cut line, and a constant value (minimum) of the energy is
obtained.
3.7.2 Equilibration
Every time there is a change in the state of a system, the system will
change in a way that maintains thermal equilibrium. This one is
thermodynamics equilibrium. It means that the indicators of systems state are
changing continuously and relaxing towards a new value. We can change the
state of the system by changing a parameter of the simulation. The system can
be changed spontaneously when it undergoes a phase transition. In other
words the system is moving from one equilibrium state to another. In each
case to start measurements on the system, we require equilibrium. Generally a
physical quantity A approaches its equilibrium value exponentially with time,
accordingly to:
tCAtA exp0 (3.31)
where A(t) represents average of physical quantities so that instantaneous
fluctuations may be avoided over a short time step. Here relaxation time is
relevant variable. If is of the order of hundreds of time steps then A(t)
converges to 0A and this makes possible a direct measurement of the
equilibrium value. If is much larger than the overall simulation time (of the
order of one second) then we do not see any relaxation during the run. In
47
intermediate situations for , we may get the drift but cannot wait long
enough to see convergence of A(t) to 0A [149].
(A) Constant Volume Canonical Ensemble (NVT)
NVT is the ensemble in which number of particles (moles) N, volume V
and temperature T of the system are all constant. We have to keep constant
temperature during a simulation. The temperature is maintained by
multiplying the velocities at each time step by a factor actdes TT ,
where desT is the desired temperature and actT is the actual temperature.
Berendsen et al. [151] used another method to maintain the temperature. They
coupled the system to an external heat bath that is fixed at the desired
temperature [144]. We get that the rate of change of temperature proportional
to the difference in temperature between the bath and the system:
tTT
dt
tdTbath
1 (3.32)
Where represent that how tightly the bath and the system are coupled
together. The system tends to decay exponentially towards the desired
temperature. During successive time steps the change in the temperature is
tTTt
T bath
(3.33)
(B) Isothermal-Isobaric Ensemble (NPT)
In case of an Isothermal-Isobaric ensemble, the number of particles
(moles) N, pressure P and temperature T of the sample are all constant. A
48
constant pressure is maintained by changing the volume of the simulation
cell. The volume of the system is related to the isothermal compressibility, ;
TP
V
V
1 (3.34)
A substance is easily compressible if it has a large value of . In case of a
less incompressible substance large fluctuations in the volume occur at a
given pressure than for a more incompressible substance. On the other hand a
less compressible substance shows larger fluctuations in the pressure in a
constant volume simulation [144].
In an isobaric simulation volume we can achieve a volume change by
changing length scale in all directions or just in one direction. For the
observation of volume changes of a “typical” system we might expect to
observe in a constant pressure simulation. The relation between isothermal
compressibility and mean square displacement is given by
2
22
1
V
VV
TkB
(3.35)
The value of isothermal compressibility for an ideal gas is
approximately 1 atm-1 and for relatively incompressible substance such as
water 4.75×10-6 atm-1. Many of the methods used for temperature control are
used for pressure control. Thus, by simply scaling the volume, pressure can
be maintained at a constant value. An alternative is to couple the system to a
“pressure bath” analogous to a temperature bath [151]. The rate of change of
pressure is
tPP
dt
tdPbath
P
1 (3.36)
49
where, P is the coupling constant, bathP is the pressure of the “bath” and
tP is actual pressure at time t. If we take as the volume coordinate factor
which is equivalent to scaling the atomic coordinates by a factor λ1/3, we have:
bath
P
PPt
1 (3.37)
(C) Micro-Canonical Ensemble (NVE)
The ensemble in which number of particles (moles) N, volume V and
energy E of the system are all constant is called micro-canonical ensemble and
form an isolated system [144].
3.7.3 Simple Statistical Quantities to Measure
During molecular dynamics calculations, we measure quantities by
taking time averages of physical quantities over the system trajectory. These
quantities are a function of the coordinates and velocities [149]. We can define
the instantaneous value of a generic physical property A at time t:
tvtvtrtrftA NN ,...,,,..., 11 (3.38)
And its average will be
TN
tT
tAN
A1
1 (3.39)
where t denotes time steps and varies from 1 to the total number of steps TN .
(A) Average Potential Energy
The average potential energy V is obtained by averaging instantaneous
potential energies. For two body interactions
50
i ij
ji trtrtV 2
1)( (3.40)
where V is required for the conservation of energy, which is an important
check for in molecular dynamics simulations [149].
(B) Average Kinetic Energy
The instantaneous kinetic energy is obtained by
i
ii tvmtK2
2
1)( (3.41)
Here we will call K its average on the run, and it is straight forwarded to
compute [149].
(C) Average Total Energy
According to the law of conservation of energy the total energy is
constant, and to check that the total energy is constant with time we have to
calculate it at each time step. During the run, the energy changes between
kinetic K(t) and potential V(t) but the total energy is constant [149].
)()()( tVtKtE (3.42)
(D) Temperature
The relation between temperature T and kinetic energy is given by the
equipartition formula. We take an average of 2TkB per degree of freedom:
TNkK B2
3 (3.43)
51
By using this equation the temperature is obtained from the kinetic energy
[149].
(E) Mean Square Displacement
the Mean Square Displacement of atoms in the system can be defined as
20rtrMSD (3.44)
The brackets .... denote averaging over all atoms present in the system.
When periodic boundary conditions are taken into account, we do not
consider “jumps” of particles to refold them into the box as contributing to
diffusion. The MSD provide the information about the atomic diffusivity. For
solid systems MSD saturates to a finite value and for liquid system it increases
linearly with time. In this situation the system behavior is characterized in
terms of the slope. In other words by the diffusion coefficient D:
20
6
1lim rtr
tD
t
(3.45)
In case of two-dimensional systems, 6 is replaced by 4 in the expression above
[149].
(F) Pressure
Clausius virial function is used for the measurement of the pressure in a
molecular dynamics simulation.
N
i
TOT
iiN FrrrW1
1..... (3.46)
52
Here TOT
iF is the total force on the ith atom. the statistical average of the work
done W will be obtained from
N
i
iii
t
trmrdt
tW
1
..
0
1lim (3.47)
This is given by Newton’s law. Integrating by parts we get
2
1
.
0
1lim
N
i
iii
t
trmrdt
tW (3.48)
This is twice the average kinetic energy. From the equipartition law in
statistical mechanics we can write,
TDNkW B (3.49)
where D represents dimensionality, N is the number of particles and Bk is
Boltzmann constant.
The sum of the forces acting on the particles will be
EXT
ii
TOT
i FFF (3.50)
iF is the internal force, and the walls of the container exert EXT
iF ( external
force). If we consider that the particles are enclosed in a parallelepipedic
container with sides ,,, zyx LLL and zyx LLLV , then EXTW can be
calculated as
DPVLPLLLPLLLPLLW yxzzxyzyx
EXT (3.51)
53
here zyLPL is the external force
EXT
xF applied by the yz plane along the x
direction to particles located at xLx , etc.
TDNkDPVFr B
N
i
ii 1
(3.52)
The value of PV is
N
i
iiB FrD
TNkPV1
1 (3.53)
This result is called the virial equation [143].
(G) Melting Temperature
The Mean square displacement is a function of time. This behavior
gives us information about the difference between solid and liquid. When we
increase the temperature of the simulated substance the caloric curve exhibits
a jump at the phase transition due to Latent heat. This needs to be included in
MD simulations [149].
3.8: More on Molecular Dynamics
Molecular dynamics is very helpful to explain the vibrational
properties of materials, especially, when first-principles DFT methods are
utilized. It can account for many of the effects that are neglected when using
other methods. For example, anharmonic effects in crystal lattices are
produced by intrinsic and extrinsic terms. The first term is induced from the
shapes of the interatomic potentials and the second term is induced from the
thermal distortion of lattice. On the other hand the use of DFT methods in
54
frozen phonon calculations, gives the intrinsic term correctly, but totally
misses the extrinsic contribution. That’s why we have to use classical MD
very cautiously. As we discussed above, these potentials are fitted to the
properties of materials.
Like measured phonon dispersion, these potentials are directly
generated from the vibrational properties. If it gives ideal results for some of
the properties and completely fails for others, it will not be a surprise. In
molecular dynamics or lattice dynamics it is not necessary that any calculation
is only as good as the potentials it uses. In case of nonequilibrium states a
system of particles can be simulated by using molecular dynamics. For
example as in the case of thermal transport and radiation damage. However
for these simulations the size of the system must be large enough, therefore in
most cases only classical MD is practical.
55
Chapter 4
MULTILAYER RELAXATIONS NEAR/AT SURFACES OF COPPER
The calculations presented here provide a detailed qualitative and
quantitative measure of surface relaxations. High index (or open, vicinal)
surfaces/interfaces relax more as compared to compact surfaces/interfaces. In
the area of heterogeneous catalysis these high index surfaces of metals are of
practical interest. That's why the present work calculates the multilayer
relaxations of Cu(311) and Cu(210) Surfaces.
4.1: Introduction
Knowledge of the geometrical arrangement of the atoms near the
surface is a basic ingredient for the study of structural and dynamic
properties of a metal surface. Therefore, these surface properties here received
much attention in the past two decades. In the surface geometry, the surface
relaxation, surface tension and stress play an important role in determining
whether and how surfaces reconstruct and relax [152].
During relaxation, the interlayer spacing near the metal surface relaxes
to values that are different to the bulk interlayer spacing. Using Low-Energy
Electron Diffraction LEED, which is a principal technique for surface
crystallography, first time structural results were obtained which show that
the interlayer spacing between the 1st and 2nd atomic layers is less than as it is
in the bulk [152,153]. In the relaxation process there are several layers
participating and from several early measurements it can be deduced that this
is a common phenomena [154-160]. Multilayer relaxation measurements are
56
mainly carried out by low-energy electron diffraction, Ion Scattering
Spectroscopy (ISS) and multilayer relaxation analyses various metal surfaces
such as silver, aluminum, copper and for many others [160-164].
Since the early days of surface science, high index surfaces have received
attention [165, 166]. Vicinal surfaces/interfaces have a lack of symmetry along
the direction normal to the surfaces/interfaces (neighbors less than 12). This is
therefore unlike bulk atoms which have symmetric electric and ionic charge
arrangement throughout the bulk. Therefore for high index
surfaces/interfaces some modifications in the charge configurations around
the steps in surfaces and near interfaces is required. As a result, the force field
is likely to change as compared to symmetric bulk atoms with normal
coordination. So the force field changes at vicinal surfaces as compared to flat
surfaces and is likely to change near interfaces. Due to the change of the force
fields some relaxations appear and the interplanar spacing is changed as
compared to the normal bulk spacing. Sometimes, it is the result of
contraction of two consecutive planes (-) and sometimes it is due to the
expansion of two consecutive planes (+) [167]. On the other hand, relaxations
parallel to the surface/interface also change the registry of the planes. It
means that when planes are moving perpendicular to the surface, the plane
registry also changes by the relaxation of the plane, parallel to the surface or
boundary. These multilayer relaxations facilitate proper charge redistribution
near the grain-boundaries and at the planes of vicinal surfaces [168].
57
A lot of work has been done in the area of surface science in recent years
[167,169-173]. The progress in experimental and theoretical techniques for
multilayer relaxation at the surfaces provides more avenues for surface
scientists to work on. In recent findings, most of the experimental data on the
surface relaxation has been emerged from low-energy electron diffraction
experiments [169-172]. On the theoretical side, some calculations are based on
first principle electronic structure calculations, which most other use many-
body potentials [167, 173]. Sun et al., [174, 175] use density functional theory
and correlate the sequence of the nearest neighbor number of atoms made-up
of the layers of clean vicinal surface with the characteristics of the multilayer
relaxations. The finding of ab-initio calculations for (910) and (111) surfaces of
Cu with tightly packed step-edges [176], vicinal (110) surface of Pd [100], and
the results of many-body potentials extracted from tight binding methods on (
_
19 11) surface of Cu [101], enhance our knowledge about the behavior of the
corner atoms of vicinal surfaces. The next section (4.2) describe the
computational and modeling procedures, while the results and discussion are
in 4.4. The summary of the chapter is in 4.4.
4.2: Computational Procedure and Modeling
Since details of molecular dynamics techniques can be found readily in the
literature [2, 4], we merely summarize the salient features of the procedure
applied in the present work. The atomic interaction is modeled through
many-body potentials [2, 4] as given by the embedded-atom method. The
58
potentials for Ag and Cu have cut-off distances of 5.55 × 10-1 and 4.95 × 10-1
nm, respectively, thus taking into account interactions up to the third and
fourth nearest neighbors. To investigate the surface relaxations of Cu(311) and
Cu(210) surfaces, a reasonable MD simulation involving several thousand
steps has been performed. The fluctuations in calculated results are
compensated by taking average of statistical quantities.
The potentials of Ag and Cu are constructed [2], so as to be minimized at
their lattice parameters of 4.09 × 10-1 and 3.62 × 10-1 nm at 0 K, respectively.
For finite-temperature calculations, the appropriate lattice parameter of the
Ag substrate has been obtained by simulating fcc bulk Ag with a periodic
cubic super-cell at constant number of atoms (256), pressure, and temperature
(NPT ensemble).
To attain the lattice parameters at the desired temperatures we used,
3/14a (4.1)
where Ω is the calculated average atomic volume at each temperature. The
computational cell was generated at a specific temperature while keeping the
pressure constant; the system was allowed to evolve till the cell edges and
volume became constant.
To calculate the percentage change in interlayer relaxation '' 1, iid for
different surfaces, we used:
59
b
biiii
d
dYYd
1
1, 100 (4.2)
where Y is the direction perpendicular to the surface and bd is the bulk
interlayer spacing.
To calculate the percentage change in plane registry relaxation ir of the ‘i’
plane near the considered surface, we used:
b
initialfinal
ir
XXr
100 (4.3)
where X is the direction in which whole plane relaxed and br is the bulk
interlayer spacing. The calculated Cu lattice parameter is plotted in Fig. 4.1.
Figs. 4.2 and 4.3 give the interplanar Cu(311) and Cu(210) spacing's
respectively.
4.3: Results and Discussion
In this section, a brief comparison of experimental and theoretical work
on the multilayer relaxation of Cu at (311) and (210) surfaces is presented. The
theoretical and experimental results for (311) and (210) surfaces of Cu for
multilayer relaxation of high-index surfaces are plotted in Figs. 4.4, 4.5, 4.6
and 4.7, Table 4.2 and 4.3 list the calculated values. The atoms near/at the
surface of low miller index surface undergo a high degree of surface
relaxation in parallel and perpendicular directions to the surface plane due to
steps at the surface. Perpendicular relaxation to the surface leads to a
60
characteristic interlayer separation, while parallel to the surface causes lateral
displacement of the surface atoms. Here, the calculated percentage interlayer
spacing (di, i+1) and registry interlayer relaxation (ri, i+1) for Cu(311) and
Cu(210) surfaces are considered. The unrelaxed and relaxed inter-layer
separation for Cu (311) and (210) surfaces is shown in Table 4.1.
4.3.1: Cu(311) Surface
First, let us talk about the relaxations at Cu(311) surface, since some of its
faces are ‘anomalous’. The surface interlayer and plane registry relaxations
are calculated for Cu(311) surface. The maximum interlayer relaxation is -9.17
percent for ‘d12’, which is a contraction and +3.94 percent for 'd23' which is an
expansion. The interlayer relaxation at Cu(311) shows a uniformly damped
magnitude away from the surface to the bulk with an alternating oscillatory
order of (-, +, -, +, . . . ). Here '+' denotes an expansion and '‒' denotes a
contraction. The interlayer relaxation near (311) surface is given in Fig. 4.4
which shows that relaxation decreases with increasing of plane number.
The percentage registry relaxation for 'r12' is -0.94%, while for 'r23' and
'r34' are 0.22% and -0.08%, respectively. The maximum percentage registry
relaxation is 2.26% for 'r45' which is an expansion and - 0.94% for 'r12', which
is a contraction. The percentage registry relaxation of Cu(311) shows an
alternating oscillatory order (-, +, -, +, . . ), shown in Fig. 4.5.
61
4.3.2: Cu(210) Surface
The interlayer relaxation near (210) exhibits random order, i.e. (-,-,+,-,
+, . . .) with a non-uniform damping. For the (210) surface, the first and second
interlayer spacing (d12, d23) are contracted (-) by 12.11% and 1.27%,
respectively. The magnitude of the next interlayer relaxation for d34 is more
than d23. The plane registry relaxation near the (311) interface is of random
nature; while near the (210) interface is a more or less ordered form. Interlayer
and plane registry relaxations are not found near the (111) surface due to high
atomic density of the (111) plane as compared to others. Comparison of
percentage change interlayer relaxation '' 1, iid for Cu(311) and Cu(210)
surfaces is given in Table 4.2.
The interlayer relaxation for Cu(210) is shown in Fig. 4.6, which is in
good agreement with LEED measurements [177]. The LEED study of
multilayer relaxation at the Cu(210) surface is an order of (-, -, +, +, - . . . .),
with uniform damping of the magnitude of the interlayer relaxations away
from the surface into the bulk. For Cu(210) the magnitude of interlayer
relaxation for d34 and d45 is found to be larger than d23. Interestingly, results
from an early LEED measurement to determine of multilayer relaxations of
Cu(210) by Guo et al. [178] are in line with our finding as damping of non-
uniform damping for the interlayer relaxations.
The maximum percentage interlayer relaxation near the (210) surface is
contraction of -12.13% and -3.38%, for 1st interlayer spacing ‘d12’ and of 4th
interlayer spacing 'd45', respectively. At the (311) surface, oscillatory damping
62
of interlayer relaxation is found to be (-,+,-,+, . .), while at the (210) surface
interlayer relaxation is random in nature. Plane registry relaxation at both
(311) and (210) surfaces are not ordered. Comparison of percentage change in
the plane registry relaxation '' 1, iir for Cu(311) and Cu(210) surface is given
in Table 4.3. The magnitude of both interlayer and plane registry relaxations
are damped away from the surface. While moving away from the surface, a
very small change in magnitude of registry relaxation is found. This change is
larger for the (210) surface as compared to the (311) surface.
In our EAM calculations and LEED calculation, the average error
deviation for interlayer relaxation for Cu(210) is nearly 1.94%, and for the
registry relaxation it is 0.74%. The possible reason for this discrepancy is the
temperature dependence of the relaxation, which has recently been attracting
attention. Both thermal expansion and contraction relaxation have been
observed on open surfaces of metals [189-181]. However, due to the limited
number of temperature dependent studies on the multilayer relaxation of
high index surfaces, the picture is not clear yet. Nevertheless, due to the low
temperature at which the LEED data set for Cu(210) is collected, the
temperature effect should not be very significant.
The varying trends in the interlayer and registry relaxation can be
better perceived by examining the atomic relaxations in the low coordination
region of the surface. Among different surfaces studied, in the present work
we choose the (311) and (210) surfaces of Cu with tight and loose packed step
63
edge on narrow and wide terraces and have calculated the relaxation of
individual atoms at/near the surface for both surfaces.
For the Cu (210) surface relaxation, previous experiments suggested an
inward relaxation ranging from −0.30 to −0.70% [182, 183], while the latest
low-energy electron diffraction [184] experimental results give a small
expansion. For the theoretical calculations, EAM [185] predicted an inward
relaxation while our EAM gives a small expansion of - 0.68%, which is
consistent with the latest experimental results [184]. Due to the surface
relaxation, the surface energy decreases. The change in surface energy due to
multilayer relaxation effects is, in general, of the order of 2.01% for (311)
surfaces. For the (210) surface, which is more open, the decrease in surface
energy is of the order of 4–9%. For a reconstructed surface, this is obviously
unrealistic.
For Cu surfaces with two-atom-wide terrace, the largest relaxation
occurs for the interlayer spacing of the first two layers exposed to the vacuum,
d12. This characteristic behavior in the first interlayer spacing has been
observed for several other metal surfaces as well [186-191]. On the other hand,
for Cu surfaces with a large terrace width there is no such a general trend in
the nature of multilayer relaxation, the most striking feature of which is the
stronger interlayer relaxation for the layer away from the top layer, as
compared to the first interlayer spacing.
64
4.4: Conclusions
The multilayer relaxations are calculated for the (311) and (210) copper
surfaces. The maximum interlayer relaxation for the Cu(311) surface is -9.17 %
for ‘d12’ which is a contraction and +3.94% for 'd23' which is an expansion. The
interlayer relaxation of Cu(311) shows uniform damping in magnitude away
from the surface to the bulk with an alternate oscillatory order of (-, +, -, +, . . .
). The interlayer relaxation near (311) shows that relaxation decreases with the
increase in plane number. The interlayer and atomic relaxations for Cu(210)
can be traced to the modifications in the local force fields around steps, we
have compared the calculated multilayer relaxations with recent low energy
electron diffraction data other surface relaxations. An interesting feature in
the calculated relaxations for Cu(210) is a pronounced expansion of the
interlayer distance between the 3rd and 4th layers, which leads to a non-
uniform damping in magnitude for the multilayer relaxations of Cu(210).The
lattice parameter and interlayer spacing's are increased with increase in
temperature which is 0.11% for Cu(311) surface and 0.08% for Cu(210)
surface per 100 K are formed. The increase is 0.04% larger for the Cu(311)
surface as compared to Cu(210).
65
Table 4.1. The unrelaxed and relaxed inter-layer separation for Cu (311) and
(210) surfaces.
Element Surface Unrelaxed d12 (Å) Relaxed d12 (Å)
Cu(311) 1.09 0.99
Cu(210) 0.81 0.71
66
Table 4.2. Comparison of percentage change interlayer relaxation '' 1, iid
for Cu(311) and Cu(210) surfaces.
'' 1, iid Cu(311) Cu(210)
d12 -9.17 -12.13
d23 3.94 -1.28
d34 -3.76 3.06
d45 2.28 -3.37
d56 -1.38 0.41
d67 1.01 0.81
d78 -0.47 -0.68
67
Table 4.3. Comparison of percentage change in the plane registry relaxation
'' 1, iir for Cu(311) and Cu(210) surface.
'' 1, iir Cu(311) Cu(210)
r12 -0.94 -0.78
r23 0.22 -1.44
r34 -0.08 0.75
r45 2.26 0.14
r56 -0.12 0.29
r67 0.09 0.08
r78 -0.05 0.04
68
Figure 4.1: MD simulated values of lattice parameter at various temperatures
for Cu.
69
Figure 4.2: Interplaner spacing for Cu(311) surface.
70
Figure 4.3: Interplaner spacing for Cu(210) surface.
71
Figure 4.4: Percentage interlayer relaxation for Cu(311) surface.
72
Figure 4.5: Percentage plane registry relaxation for Cu(311) surface.
73
Figure 4.6: Comparison of calculated percentage interlayer relaxation for
Cu(210) surface and experiment [177].
74
Figure 4.7: Comparison of percentage plane registry relaxation for Cu(210)
surface.
75
Chapter 5
DIFFUSION OF COPPER PENTAMER ON Ag(111) SURFACE
The purpose of this work is to obtain qualitative understanding of the
microscopic processes controlling the hetro-diffusion of two-dimensional Cu
clusters on Ag(111). We have calculated the diffusion coefficient, effective
energy barrier, and diffusion prefactor for 5-atom Cu islandw on the Ag(111)
surface, using molecular dynamics simulations based on the atomic
interactions yielded by embedded-atom method potentials. The Chapter is
organized as follows: The next Section carries a brief introduction of the
chapter. In Section 5.2 the computational details are given. Section 5.3
contains our results for the diffusion coefficient, energy barriers, and diffusion
prefactor as a function of the temperature. We also present and discuss briefly
the various mechanisms that take place for the cluster and their incidence at
temperatures 300, 500, and 700 K. Section 5.4 offers a summary and the
concluding remarks of the study.
5.1: Introduction
The diffusion of adatoms on metal surfaces is still the subject of very
active research [192, 193]. Indeed, it plays a crucial role in crystal growth
which is very important to master in view of the potential applications, for
instance in nanotechnologies. Detailed knowledge of surface diffusion is of
utmost importance for the understanding of a number of non-equilibrium
phenomena such as nucleation and growth [194]. On surfaces, for instance,
76
the rates at which particles diffuse determine the equilibrium shape of islands
and, on macroscopic time scales, the morphology of films. However, very
little is known of the fundamentals of diffusion, although a large amount of
experimental and theoretical research has been devoted to this subject [195].
The derive Understand how clusters move on a solid surface has
drawn an increasing research interest in surface science and materials
research. The incentive for this is not only its relevance to the development of
new thin-film structures [196] and one-dimensional and two-dimensional
nanostructures [197,198], but is also due to the fact that advances in
experimental atomic imaging of surface dynamics and theoretical tools for
simulating such surface dynamics have lead to the discovery of fascinating
modes of cluster motion. Most importantly, there have been quite a few
independent reports of fast cluster diffusion, which ensures the possibility of
cluster diffusion being a determining factor in the overall surface dynamics.
Diffusion barrier calculations for given atomic configurations are also
helpful to understand the dynamics, but we do not know the probability of
each atomic configuration emerging in realistic processes. Accordingly, real
dynamic simulations are highly desirable to thoroughly understand the island
behavior [199]. Diffusion barriers control mass transport during surface
deposition and determine the formation and stability of surface patterns,
which can be used for nanostructure fabrication. The diffusion of atoms on
flat surfaces has been exhaustively studied for many years using diverse
computational and experimental methods. In particular, Cu adatom diffusion
77
through hopping and exchange processes on Cu(111) and Cu(100) flat
surfaces has received great attention. Liu et al. [200] summarized the energy
barriers obtained by different methods for the diffusion of adatoms on diverse
fcc metal surfaces.
In the present work mean-square displacement is obtained by the trace of
an atom or cluster’s geometric center within the framework of MD relaxation,
which is sufficient for reaching a relaxation equilibrium. ˂R2(t)> is the mean-
square displacement (MSD) of the mass-center of the cluster and is given by:
222 00 ytyxtxtR (5.1)
The coordinate of adatoms on the surface are obtained by recording the
positions of atoms for each time step, when the system is in thermodynamic
equilibrium. The long-time behavior of the mean-square displacement at large
time gives us the diffusion coefficient. In experiments, the diffusion coefficient
for a cluster can generally be obtained by measuring the mean-square
displacement of the cluster’s mass-center (or center of mass) [43].
5.2. Modeling and Computational Details
Since details of molecular dynamics technique can be found in literature
[2,4], we just summarize the significant features of the technique applied in
the present work. The Nordsieck’s algorithm [201] with a time step of 10-15 s is
used to solve classical equations of motion for atoms interacting through
interatomic potentials [2,4] as given by the embedded-atom method. To
78
obtain the lattice constant appropriate to a specific temperature, we first
carried out a simulation of fcc bulk Cu and Ag using a periodic cubic supercell
containing 256 atoms.
In order to study the diffusion of Cu 5-atoms island on the Ag(111)
surface, a model crystallite is generated in the form of a rectangular block of
atoms, with (111) geometry. Here x and y axes lie in the surface plane, while z
is along the surface normal. All the atoms in the computational system are
allowed to move. In the surface diffusion simulations, periodic boundary
conditions are applied along the x and y directions only. Our system consists
of 6 layers with (20 × 20) atoms in a layer. The crystal is first allowed to relax
to minimize its energy by the conjugate gradient method [202]. The system is
then thermalized using NVT simulations with a 20 ps time step. Finally, the
system is subject to a long constant energy molecular dynamics run for 10 ns
with a 5-atom at island, so that we can monitor the path and trajectory of all
motion of the island.
For the model system, we consider an Ag(111) substrate with Cu adatom
island on top, as shown in Fig. 5.2. The blue-colored atoms are substrate
atoms, whereas the red-colored atoms are the island atoms, placed initially on
fcc sites, which are hollow sites having no atoms underneath them in the layer
below. The Molecular dynamics simulation begins by placing a 5-atom Cu
island in a randomly chosen configuration on the Ag substrate. Statistics are
recorded after every 0.05 ps. We thereby obtained 20000 sets of statistics for
each temperature. The diffusion coefficient of an island is calculated by:
79
dt
RtRD
CMCM
t2
)0()(lim
2
5.2
where D is the diffusion coefficient, tRCMis the position of the center of mass
of the island at time t, and d is the dimensionality of the system. The effective
energy barrier and diffusion prefactor values are deduced from an Arrhenius
plot.
5.3. RESULTS AND DISCUSSION
We calculated the lattice parameters for Ag at different temperatures and
compared these with the experimental values, as plotted in Fig. 5.1. Simulated
values of lattice parameter agree well with the experimental values in the
range of 300 to 1100 K for Ag. Our molecular dynamics results are closer to
experimental values as compared to the calculated values reported by
Kallinteris et al. [203], using tight binding potentials.
The results based on the MD simulation method are presented for
single and multiple atom processes involving jumps from one fcc site to an hcp
site, which are automatically accumulated and performed during the
simulation. As for the model system, the Ag(111) surface is considered as a
substrate with Cu island on top, as shown in Fig. 5.2. The blue-colored atoms
are substrate atoms, whereas the red-colored atoms are island atoms placed
initially on fcc sites. MD simulation begins by placing Cu 5-atom island, in a
randomly chosen configuration, on the Ag substrate. Statistics are recorded
after every 0.05 ps. An MD simulation for a simulation time of 2 ns was
80
performed at temperatures of 300 K, 500 K and 700 K. For each temperature
120,000 sets of statistics are recorded.
The diffusion process of a 5-atom Cu island on Ag(111) surface is shown
in Fig. 5.2. This shows mainly rotation. The strongly concerted motion
appears at 700 K as compared to 300 and 500 K. During this concerted motion,
the island atoms open along one side, almost forming a straight line, and then
closes towards the other side. Through the concerted motion, it jumps from an
fcc to another hcp site. During this concerted motion of the island at 700 K, one
of the Cu island atom is popped-up onto the island. During the diffusion, the
shape of the base of the island also changes, and the poped-up island atom
randomly moves on the island.
The calculated diffusion coefficients of the Cu island on the Ag(111)
surface at 300 K, 500 K, and 700 K are summarized in Table 5.I. The numerical
values for diffusion coefficient are 2.134× 109 Å2/s, 3.671×1010 Å2/s and
2.207× 1011 Å2/s for the pentamer at the said temperatures. The calculated
diffusion coefficient, effective energy barrier, and the prefactor sing MD
simulations are also summarized. These values are given in the square
brackets underneath the corresponding ones obtained when hcp-site assisted
processes are not included [204].
The calculated effective energy barrier for the Cu pentamer on the
Ag(111) surface is in reasonable agreement with the experimental value [204].
It can be noticed that the slight difference between the two values could be
81
related to the fact that the experimental barrier pertains mainly to fcc↔fcc
hoping whereas that from MD simulations stems largely from fcc↔hcp
hoping. Therefore, if the barrier for fcc↔hcp hopping is lower than that for
fcc↔fcc hopping [205,206], it is plausible to expect that fcc↔hcp hops slightly
bias our computed effective energy barrier toward lower values.
The MD simulation at 300 K shows that the Cu pentamer hops to and
stays at fcc sites approximately two times more often than on hcp sites. It can
be concluded that since the instability of the hcp site is derived from the
repulsive interaction of the atom directly below, the short and strong Cu–Ag
bonds, compared to the Ag–Ag bond [207], may help stabilize the hcp site in
the hetero-epitaxial case.
The movies generated from our MD simulations show that the intracell
processes (these can be zigzag motion, concerted rotations, and short
concerted translations) predominate in the kinetics of the Cu pentamer on
Ag(111). The rate of intercell mechanisms is, however, not much lower than
that of intracell mechanisms. These often occur via intercell zigzag or
concerted jumps. Translational concerted hops with rotation are considerably
less frequent. Sudden multiple translational or rotational concerted jumps
resemble a barrierless sliding motion. The long jumps are rarer still but
nevertheless occasionally present at 700 K. For intercell mechanism at 500 K,
zigzag or concerted jumps with rotation, multiple translational or rotational
and long jumps along with a kink, have all been observed. The pentamer
performs long jumps with a sharp kink even at 300 K, in the sense of
82
consecutive intercell mechanisms with residence times shorter than 0.2 ps.
The Center of mass of Cu pentamer on Ag (111) as a function of time at 300,
500 and 500 K for 2000 ps is given in Fig. 5.4. The rate of diffusion increases
with increasing temperature, as shown in Fig. 5.5. It was found that the
diffusion coefficient of the island nicely fits an Arrhenius curve, for
temperatures of 300, 500, and 700 K. The diffusion prefactor can hence safely
be considered temperature independent in the range 300–700 K.
It is useful to compare the kinetics of Cu pentamer on the Ag(111) surface,
is the contrast with the corresponding homoepitaxial case of Cu/Cu(111), for
which a considerable body of theoretical and experimental work is already
available [203,206,208-213]. The interpretation of our results outlined below is
reached as well in the light of a recent study on a Ag27 Cu7 core-shell nano-
particle[214], which suggests that there is a bond-strength hierarchy among
homo-bonds and hetero-bonds that mediates the minimum-energy structure
of any particular system. Such a hierarchy in Ag and Cu systems largely
favors the optimization of Cu-Cu bonds over that of Ag-Ag bonds, while Cu-
Ag bonds, if not constrained by the symmetry of the system may be almost as
short and strong as the Cu-Cu bonds. For this reason, the relatively weak and
loose Ag-Ag bonds (compared to Cu-Cu and Cu-Ag bonds) easily give way,
reducing the Cu-Cu and/or Cu-Ag bond lengths down to the bond length
inbulk Cu, often at the expense of expanding the Ag-Ag bonds [215] of those
Ag atoms which make bonds with Cu atoms.
83
For the pentamer, however, the relatively long bond length of the
substrate now works to empower diffusion routes (fcc-fcc, hcp-to-fcc) not
energetically favorable in the homoepitaxial case (Cu/Cu(111)). In this way,
the screening of the lattice-mismatch effect which hinders pentamer diffusion
is effectively screened, such processes account for the low effective diffusion
barrier of the Cu on Ag(111), relative to that of Cu on Cu(111). Our MD
simulations also suggest that at finite temperatures (300-700K) the close
similarity between Cu-Cu and Cu-Ag bonds in respect to bond strength and
bond length promotes off‒lattice sites and establishes a competition between
the optimization of these two types of bonds, resulting in an in-plane Cu-Cu
vibration that assists the kinetics of the pentamer (including dissociation at
700 K) and subjects the substrate to an alternate strain-release motion. Along
these lines, it is tempting to speculate that the Ag-Cu lattice mismatch and the
bond-optimization hierarchy [214] (among homo bonds and hetero bonds)
that minimizes the energy may establish the pentamer as the turning point of
a generalized enhanced mobility of Cu islets on Ag(111). Finally, we find that
adatom impurities seem to be localized perturbations of the periodic potential
that oscillates randomly and may generate a dynamic elastic displacement
field.
The mean-square displacements (MSD) are calculated as a function of
time. The slope of each MSD vs time curve gives one diffusion constant
measurement. The MSD for the Cu pentamer on Ag(111), as a function of time
at 300, 500 and 700 K, respectively is shown in the inset in Fig. 5.5. The results
84
indicate that the time duration of nearly 40 ns chosen for the simulation was
indeed long enough to ensure an acceptable statistics in the analysis of cluster
diffusion.
The diffusion coefficient D for the Cu pentamer on Ag(111) obtained from
our MD simulation at the three temperatures (300, 500 and 700 K) are given in
Table 5.1. From several sets of simulations we find the error in D to be less
than 4% for the pentamer. Extraction of the effective diffusion energy barrier
∆E and the diffusion prefactor D0 for the Cu pentamer is enabled by the
smooth Arrhenius behavior [205] of the diffusion coefficient D ( Fig. 5.5). The
negligible temperature dependence of the prefactors between ~ 300 and 600 K
is well understood since this temperature range is low enough that the
potential energy of the entire crystal can be considered harmonic [216,217]
and the atomic vibrations treated as small oscillations [216], while high
enough that quantum effects may be neglected [216]. The temperature
independence of the prefactor, however, cannot be extrapolated to low
temperatures, such as those at which the experiments of interest here are
performed, since the vibrational states—many of which are unoccupied at 25
K—must be described quantum mechanically [218] . It is noteworthy that MD
simulations by Ferrón et al. [219] have found that above 300 K, long and
recrossing jumps for the diffusion of a Cu adatom on Cu(111) lead to a
deviation of D from the Arrhenius behavior obtained from 100 to 250 K. The
calculated effective energy barriers in turn may be extrapolated down to zero
85
temperature because their temperature dependence arises only from the
expansion of the lattice, which is smaller from 0 to 300 K than from 300-700 K.
By allowing the system the possibility of evolving in time through all
types of possible processes of its choice, we are able to establish the relative
significance of various types of atomistic processes through considerations of
the kinetics and not just the energetic and/or the thermodynamics, as is often
done.
The comparison of the trace of the center of mass of the Cu pentamer on
the Ag(111) surface at the three study said temperatures is shown in Fig. 5.3.
This plot shows that the island did diffuse ∼ 2 Å along the x-axis and ∼ 6.5 Å
along the y-axis during 2 ns at 300 K. The same trend for 500 K shows that the
island diffuses ∼11 Å along the x-axis and ∼ 9.5 Å along y-axis. At 700 K, the
island diffuses along the x-axis ∼ 23 Å and ∼ 30 Å along the y-axis. This
comparison at three different temperatures shows that the diffusion rate
increases with an increase in temperature.
5.4: Conclusions
In summary, we performed a systematic study of the diffusion of a 5-
atom Cu island on the Ag(111) surface, using many-body interatomic
potentials developed by Foiles et al. [30]. We have studied the diffusion of the
Cu pentamer on the Ag(111) surface, and reported the effective energy
barriers Ea to be 0.20525 eV and diffusion prefactor D0 to be 5.549×1012 Å2/s ;
these are in excellent agreement with experiment [204]. The high barrier
86
expected for a Cu pentamer on the Ag(111) is due to the lattice mismatch,
since Cu the pentamer on Ag(111) diffuses through hops ∼ 10% longer than
those they exhibit for the homo-case (explained in next chapter), which makes
them detach significantly from other Ag nearest neighbors (NN) at the
transition state. It is found that the rate of diffusion increases with increase in
temperature. Our calculations infer that in the presence of the short fcc-hcp
configuration, with its relatively low-energy triggering processes it may act
together with those involving the long fcc-hcp site to establish an efficient
intercell zigzag diffusion thereby reducing the effective diffusion barrier . We
found the significant changes in the size dependent variations of diffusion
characteristics of the islands after including concerted motion. It is found that
small-sized islands (5-atoms) diffuse primarily through concerted motion
with a small contribution from single atom processes, even though for certain
cases the frequency of single atom processes is large because of lower
activation energies.
87
Table 5.1. Diffusion coefficient of Cu pentamer island on Ag(111) surface at
three temperatures and its effective diffusion barrier and diffusion prefactor.
Here square brackets represent the values of molecular dynamics results
while the figures covered brackets represent the Self Learning Kinetic Monte
Carlo (SLKMC) results [204].
Island
size
(atoms)
Diffusion coefficient
D(Å2/s)
300K 500K 700K
Effective
barrier
Ea (eV)
Diffusion
prefactor
D0(Å2/s)
5
[2.134× 109]
[3.671×1010]
[2.207×1011]
[0.20525]
[5.549×1012]
5 (7.87 × 107) (1.13 × 1010) (9.71 × 1010) (0.321)
88
Figure 5.1: Comparison between simulated and experimental values of
lattice parameter at various temperatures for Ag.
89
Fig. 5.2: Mechanism of Cu five-atom island diffusion observed during
molecular dynamics run. Complete repertoire of island shapes. Final
snapshot: pop-up of one of the Cu atoms on the other four atoms of the
island at 700 K.
90
Fig. 5.3: Trace of center of mass of Cu pentamer on Ag (111) surface at (a) 300
K, (b) 500 K and (c) 700 K
91
Fig. 5.4: Center of mass of Cu pentamer on Ag (111) as a function to time at (a)
300 K, (b) 500 K and (c) 700 K.
92
Fig. 5.5: Arrhenius plot of the diffusion coefficient for Cu Pentamer on
Ag(111). This plot gives the energy effective barrier value as 0.20525 eV and
diffusion prefactor value as 5.549×1012 Å2/s. The inset represent the mean
square displacement for Cu pentamer at Ag(111), as a function of time at 300,
500 and 700 K, respectively.
93
Chapter 6
ANHARMONIC EFFECTS IN THE PRESENCE OF Cu- AND Ag TRIMER ISLAND ON Cu(111) AND Ag(111) SURFACES
We have explored the structure and the dynamics of Ag(111) and
Cu(111) surfaces in the presence of trimer islands for the temperature range of
300-700 K, using molecular dynamics simulations. The interaction potential
employed is the embedded atom method potential. Calculations are carried
out for fissure, dislocation, mean square displacement, pop-up of atoms and
opening and breaking of islands with the variation in temperature. These
indicate that the observed anharmonic effects are small up to a temperature of
500 K, beyond which there is an enhancement.
6.1: Introduction
The anharmonicity in solids is well recognized and is required for
many macroscopic properties like thermal expansion, specific heat and
volume dependence of thermal conductivity. The dynamical diffusion
processes of adatoms/small clusters are affected by the harmonicity at the
surfaces of the solid materials. The increase in temperature causes the
anharmonic changes at the surfaces which can lead the pre-melting at the
surfaces of materials [220]. The addition of adatoms/small clusters on
surfaces gives an insight of the surface morphology and structure. The study
of anharmonic effects at (111) surfaces of Cu and Ag in the presence of trimer
islands, both for homo and hetero cases, is the aim of the present work.
94
The anharmonic effects of atomic clusters at surfaces along with their
dynamic diffusion behavior can be theoretically estimated by calculating
mean square displacement, structure factor, radial distribution function and
phonon properties [221-225]. A change in the structure of the Ag substrate
has been found during the diffusion of monomers and dimers which shows
an increase with the increase in temperature and is distinct for monomer and
dimer diffusion [226]. The anharmonic effects at the surfaces in the presence
of an island, and/or with the increase in temperature of the material can be
found such as fissures and dislocations. The creation of a vacancy on the
surfaces is also one of the causes of anharmonicity on surfaces.
The kinetics and thermal phenomena on surfaces responsible for
anharmonicity can be well understood by MD simulation techniques. This
technique provides sufficient computing power compared to an ab-initio
calculation, which is a limited approach in this regard [227-229]. In the case of
an MD simulation due to thermal vibration of atoms although more
computational time is required and it becomes difficult to capture each of the
microscopic processes accurately, it is a micro-canonical approach
compensated by a more efficient computational setup and parallel execution
of the appropriate package. If the statistics are recorded after every few
picoseconds, it becomes possible to record those dynamical processes
(anharmonic features at surfaces) which are not in direct reach of systematic
conventional approaches.
95
The reliance of results using MD simulations depends upon the
interatomic potentials, and this work is based on the semi-empirical
embedded atom method potentials. The EAM proposed by Daw and Baskes
[1, 2] is based on the quasi-atom concept [230] and density-functional theory.
It is applicable to the transition metals as well as to simple metals. It has been
used widely to calculate the point defects [231], surface [5] and thermal
expansion [6]. Foiles made the application of the EAM to liquid transition
metals and showed that the EAM also provides a realistic description of the
energetics and structural properties of the liquid phase [7]. This is the main
reason to choose the EAM as an interatomic potential for performing MD
simulations.
6.2: Computational Details
The interatomic potentials [1,2] for interacting atoms are calculated using
the formula of the form:
ciiijijji rfrU (6.1)
The first term is an electrostatic interaction represented by a pair
interaction and the second term is the energy needed to “embed” an atom into the
local electron density. The equations of motion for atoms are solved using
Nordsieck’s algorithm [4]. To obtain the appropriate lattice constant at a
specific temperature, preliminary simulations are performed. Initially, these
were carried out for fcc bulk Ag using a periodic cubic super cell containing
256 atoms. The interatomic potentials are used with a cut‒off distance of 5.55
96
× 10-1 nm for Ag, thus taking into account interactions among third and fourth
nearest-neighbors. The potentials are built so as to be minimized at a lattice
parameter of 4.09 × 10-1 nm for Ag. Then, the same procedure of simulation is
adopted for fcc bulk Cu with a lattice parameter of 3.65 × 10-1 nm, having the
same number of atoms. A reasonable MD simulation involving several
thousand steps has been performed to study the anharmonic effects in the
presence of Cu- and Ag trimer island on Cu and Ag(111) surfaces. The
fluctuations in calculated results are compensated by taking average of
statistical quantities. The simulation is carried out under the condition of a
constant number of atoms, pressure and temperature (NPT ensemble), when
calculating the lattice parameter at various temperatures.
In order to study the various diffusion processes of Cu and Ag trimers on
Ag- and Cu(111) surfaces, (the Ag timer on the Cu(111) surface and Cu trimer
on the Ag- and Cu(111) surfaces) model crystallites are generated in the form
of a rectangular block of atoms, with (111) the geometry of Cu and Ag as
shown in Figs. 6.1 and 6.2, respectualy. Here the x and y axes lie along the
surface plane, while z is along the surface normal. All the atoms of the
computational cell are allowed to move. Periodic boundary conditions are
applied along the x and y directions only. The crystals are first allowed to
attain minimum energy using the conjugate gradient method [203]. Then, the
systems are thermalized using NVT simulations with 20 ps. Finally, a long
constant energy MD run is executed for 10 ns with a trimer island on the
97
surface. Thus, the path and trajectory of all motions of the diffusing trimer is
monitored.
As a model system, an Ag(111) substrate with an Ag adatom island on
top is shown in Fig. 6.3. The substrates consist of 6 layers with 400 atoms in
each layer. The calculations are performed at three different temperatures
(300, 500 and 700 K). Statistics of the trajectories of an island are recorded after
every 0.05 ps. Thereby, 20000 sets of statistics are obtained for both islands at
the three temperatures.
6.3: Results and Discussion
During the diffusion of the trimer island on the (111) surface, the island
configuration randomly changes, even individual atoms of the island
experience a variation in position. Sometimes, atoms gain fcc positions and for
other times, they gain hexagonal close packed (hcp) positions. However,
snapshots show that the atoms are passing through a saddle point. For all
cases, red colored atoms representing the island atoms ofthe Cu and Ag,
while the blue colored atoms are showing the (111) surface atoms of Cu and
Ag metals. During the diffusion of the trimer, it is not necessary for all to
atoms to reside on the same fcc/hcp position types. Sometime one atom is at
the hcp site and other two atoms are at fcc sites and vice versa. Similarly, one
atom is at the saddle point and other atoms are at fcc/hcp sites and vice versa.
The homo and hetero diffusion of an Ag and Cu trimer island on Ag-
and Cu(111) surfaces offer the best examination of the effect of surface
98
anharmonicity as a function of temperature. The molecular dynamics
simulation method does make available all dynamics of the island like single
and multiple atom processes involving jumps from one fcc site to an hcp site
and to another fcc site and similarly, from an hcp site to an fcc site and other fcc
site.
6.3.1: Ag3/Ag(111)
An example of homo diffusion is one in which a Ag trimer is diffuses on
the Ag(111) surface (studied here by molecular dynamics simulation method).
Snapshots of the diffusion are shown in Figs. 6.3(a) and 6.3(b). The trimer
diffusion mechanism shows mainly rotation of the island. At 300 K, the
concerted motion of the island from one fcc site to another nearby fcc or hcp
site is less pronounced than the rotation, while concerted motion is observed
at 700 K. Sometimes, the island opens along one side, approximately forming
a straight line, and then closes towards the other side. During the concerted
motion, the Ag trimer does jump from one fcc site to another hcp site and then
back from the hcp to fcc site. One astonishing feature is the breaking of the
island at a high temperature of 700 K, forming a monomer and a dimer, which
then diffuse separately.
During the diffusion process of Ag on Ag(111), different anharmonic
effects are observed at the surface at different temperatures (300, 500 and 700
K) in the presence of the island. At 300 K, only fissure is found at ∼ 2.05 Å
from island's position {Fig. 6.3(b)}. At 500 K, the fissure's position is ∼ 2.5 Å
and a dislocation is observed nearly at a distance of about 2 Å from the island
99
{Fig. 6.3(b)}. At 700 K, both fissures (at ∼ 3 Å) and dislocation (at ∼ 1 Å) are
observed as shown in Fig. 6.3(b).
The XY-plot of the Ag 3-atom island on the Ag(111) surface at 500 K is
given in Fig. 6.4. This plot shows that the island did diffuse ∼ 20.3 Å along the
x-axis and ∼ 29 Å along the y-axis during 2 ns. The XY-plots at three different
temperatures show that the diffusion rate increases with increasing
temperature.
6.3.2: Ag3/Cu(111)
The schematic mechanism of diffusion of the Ag trimer on the Cu(111)
surface carried out at three different temperatures namely 300, 500 and 700 K,
is given in Figs. 6.5(a) and 6.5(b). In this case rotation is dominant at 300 K,
while the concerted motion of the island from one hcp site to another near fcc
or hcp site is less prominent than the rotation. However, the concerted motion
as well as the island breakage takes place at a high temperature of 700 K. The
island opens along one side, making a curved shape and then closes towards
the other side. Through concerted motion, the Ag trimer does jump from an
hcp site to an fcc site and then back from an fcc site to an hcp site.
The Cu island does not break during diffusion in the beginning, but with
the increase in temperature it breaks up into a single atom and a two atoms
pair. Further increase in temperature results into the separation of the island
into single atoms completely, as shown in Fig. 6.5(a). A fissure lies at about 3.5
Å (300 K), while at 500 and 700 K the fissures are observed at the distances of
100
∼ 2.05 Å and ∼ 4.1 Å, respectively and dislocations are found at ∼ 2.1 Å and ∼
3.1 Å, respectively from the island's position, as shown in Fig. 6.5(b). The XY-
plot of the trimer Ag island on the Cu(111) surface at 500 K is given in Fig. 6.6.
This plot shows that during 2 ns the island does diffuse ∼ 34.8 Å along the x-
axis and ∼ 61 Å along the y-axis.
6.3.3: Cu3/Ag(111)
The Cu trimer diffusion mechanism on the Ag(111) surface is shown in
Figs. 6.7(a) and 6.7(b). The island changes its shape and positions of the atoms
from the fcc site to hcp site and vice versa. The rotation of the island is also
observed during these changes of sites. Fig. 6.7(b) carries the anharmonic
features of the Cu island on a Ag surface. The fissure and dislocation are
observed at the distances of a few angstroms (1.7 and 2.5 Å, respectively)
from the Cu trimer island, at 500 K. While at 300 K, only a fissure is observed
at about 1.5 Å from the island. The process of opening and closing of the Cu
trimer island has also been observed, over the temperature range studied. On
the other hand, at 700 K, a vacancy is also created along with fissure and
dislocation.
Actually at high temperature, disorder appears in the layers of the
substrate which can be the cause of the creation of vacancies on the surfaces.
So, in case of Cu3/Ag(111) at 700 K, disorder in the first layer causes a
vacancy at the Ag surface in the form of an adatom. This adatom, first joins
with the trimer, forming a mixed island of four atoms (a tetramer) comprising
three Cu atoms and one Ag adatom. The Ag adatom separates from the Cu
101
trimer after 2 to 3 ps, which diffuses separately on the Ag surface. However,
they again combine to constitute a tetramer of mixed Ag and Cu atoms. Then,
the Cu trimer starts rotating about the Ag adatom, reflecting the difference
between Cu-Cu and Cu-Ag interactions. Sometimes, one of the Cu or Ag
adatoms is at an fcc site, and occasionally even two of the three atoms show
the various behaviors of the island atoms, resulting from the difference in
cohesive forces between Cu adatom pairs and adhesive forces between Cu
and Ag atoms.
On the other hand, the vacancy generated by the popping up of an Ag
atom may also migrate to a certain extent. Occasionally, all three Cu atoms
move around the Ag adatom. During the motion of this mixed tetramer, one
Cu atom drops (adsorbed) into the vacancy that was created by popping up of
Ag atom over the Ag surface. This is an exchange of Cu atom by an Ag atom,
following the rotation of an Ag atom and the remaining Cu atoms about the
substituted Cu atom. The details of vacancy generation have been explained
in chapter 7. The XY-plot of the 3-atom Cu island on the Ag(111) surface at
500 K is given in Fig. 6.8. This plot shows that during 2 ns the island did
diffuse ∼ 36.4 Å along the x-axis and ∼ 14 Å along the y-axis.
6.3.4: Cu3/Cu(111)
A typical diffusion process for the homo diffusion of the Cu trimer on
Cu(111) is shown in Fig. 6.9(a). The snapshots of the anharmonic behavior in
the presence of Cu3 island on Cu(111) surface are shown in Fig. 6.9(b). The
102
island opens a little bit at 300 K, a little more at 500 K making a curve but at
700 K it approximately forms a straight line. This dynamical behavior of the
island is absent in all the other three cases of diffusion, in the present work.
Another distinguishing feature of this case of homo diffusion is the formation
of fissures at 300 and 500 K at the distances of ∼ 3.1 and ∼ 2.05 Å respectively
{Fig. 6.9(a)}. At 700 K, fissures and dislocations are located at the distances of
3 and 3.5 Å respectively as shown in Fig. 6.9(c). However, the concerted
motion is observed at 700 K for Cu3/Cu(111).
At 500 K, the XY- plot of the Cu trimer island on the Cu(111) surface is
shown in Fig. 6.10. This plot shows that during 2 ns the Cu trimer island did
diffuse ∼ 61 Å along the x-axis and ∼28 Å along the y-axis on the Cu(111)
surface.
The XY-plots for all four cases of diffusion given in Figs. 6.4, 6.6, 6.8 and
6.10, show the rate of diffusion during a time of 2 ns. Fig. 6.4 shows that the
trimer Ag island moves ∼ 25 Å along x- and y-direction on average during
diffusion process, while the same island on Cu(111) surface travels ∼ 48 Å in
both directions. On the other hand, for the case of Cu on Ag(111) surface, the
trimer island covers almost 25 Å along x-/y-axis in the same duration, while
for the Cu surface it moves ∼ 44.5 Å along both directions. This comparison
shows that the Ag island diffuses more as compared to the Cu island during
the same duration. The analysis for the rates of diffusion also shows that the
diffusion is relatively prominent for Cu as a homo case {i.e. Cu3/Cu(111)}. The
hetero diffusion of the Ag island {i.e Ag3/Cu(111)} is relatively more
103
prominent. As a whole, we can say that the diffusion is faster on a Cu surface
as compared to a Ag surface for both homo- and hetero-diffusion.
Comparatively, the rate of diffusion of Ag island on Cu surface is high due to
bond length mismatch of Cu and Ag (13.5%), and the surface energy
difference of 0.15 eV [232]. Therefore, it is unfavorable for a Cu island to sit on
local minima of Ag substrate i.e., fcc and hcp sites at the (111) surface.
The perturbed structure of the Ag substrate as a function of instant
position and configuration due to the diffusion of Cu monomer and dimer
have been studied by Hayat et al. [226]. The perturbation is particularly
conspicuous for the dimer in which case the dislocations do not remain local.
While in our trimer case, all the snapshots apparently show that all the three
atoms of the island are in correlated positions to a high degree. In this case,
the perturbation remains local and the hollow sites are only observed near the
island’s position shown in the snapshots of Figs. 6.3(b), 6.5(b), 6.7(b) and
6.9(b) of our simulation. These hollow sites may result into the formation of
fissures and dislocations at the surfaces near the island’s positions. Notice that
the terms “fissures” and “dislocations” mean merely the rupture of the Ag-
Ag, Ag-Cu, Cu-Ag and Cu-Cu bonds along a line in the lattice rather the
crystallographic defect at the surfaces. These ruptures on the surfaces results
in the vibrations of the substrates, whose displacement patterns expand and
contract the Ag-Ag, Ag-Cu, Cu-Ag and Cu-Cu bonds along a line in the lattice
which are the cause of formation of fissures and dislocations at the substrate
surfaces [233].
104
The dislocation mechanism is more active for the diffusion of
Cu3/Ag(111) in the temperature range of 300 to 700 K which is likely to be
inaccessible for other islands. This feature leads to a surprising result that the
island having a size greater than trimer size can move more easily as
compared to the trimer. Another reason for this immobility is the lattice
mismatch which reduces the island diffusion barrier and hence reduces the
energy cost of bringing the Cu atoms closer together.
It is observed that the trimer island atoms gained different symmetries
at the (111) surface during diffusion at different temperatures. The detailed
analysis of different symmetries shows that the number of surface contacting
atoms with the island varies for different orientations of the island. When
three atoms of the island are connected to each other in such a way that two
atoms lie on one line and the third one lies on another line and that they
occupy the ideal fcc/hcp sites, then the number of surface contacting atoms
with the island atoms is six. The number of the atoms contacted with the
surface increases from six to seven when the island atoms lie in a single
straight line during diffusion. In one of the orientations of the island, one
atom resides at the saddle point position, while the other two atoms are at
fcc/hcp sites, the number of surface contacting atoms becomes five. For two
saddle point positions and one hcp/fcc position of the atoms of trimer island,
the number of surface contacting atoms is four. However, fcc/hcp
configuration of island atoms when connecting with each other making a
triangle shape, shows a stable configuration as compared to the other
105
configurations of the island. Therefore, the island spends most of the
simulation time in this configuration.
The comparison of Cu and Ag trimer diffusion on Ag(111) and
Cu(111) surfaces, leads to the conclusions that only in case of Ag3/Cu(111)
does one atom separate (dissociate) from the island as shown in Fig. 6.5(a). In
all other cases, the island opens in different directions at different
temperatures, but the island atoms are not observed to separate from each
other. The rate of diffusion is increased with increasing temperature in each
case. Near the island fissures have also been observed at 300 K in all four
cases of diffusion. Except for Cu3/Cu(111) at 500 K in addition to fissures,
dislocations are also observed at 500 and 700 K. In contrast, a vacancy is only
formed in the case of Cu3/Ag(111) at 700 K. The formation of vacancy in this
case is due to the high temperature which leads to a strong increase in
disorder which we ascribe to pre-melting induced by the vacancy formation.
The trimer has a lesser rate of diffusion having more surface
connecting atoms as compared to tetramer and larger islands. The triangular
stable symmetry of the trimer having a large lattice mismatch between Cu
and Ag enables it to stay for a comparatively long time on the surface
connecting atoms of the substrate (beneath the island), which may be a cause
of anharmonicity at the surface. So, for different symmetries of the trimer
island the ratio of the number of island to surface connecting atoms varies
from 4 to 7, maintaining almost triangular geometry.
106
By increasing the size of island the ratio of the number of surface
connecting atoms per atom of the island decreases with the increase of size of
the island. This decreasing trend of surface connecting atoms is relatively less
for trimer. In other words, the trimer island has a higher ratio of island to
surface connecting atoms as compared to tetramer and larger islands. Among
different symmetries of the trimer island, it maintains relatively stable
triangular symmetry showing more compatibility/less mobility relative to
larger size island.
The comparison of diffusion of Cu and Ag islands on both Cu(111) and
Ag(111) surfaces, clearly elaborates that the Cu island (due to smaller
comparative atomic size), has a higher diffusion rate on both surfaces as
compared to the Ag island. Therefore, anharmonic effects produced by the Cu
island are higher, prominently confirmed by the creation of a vacancy and
further adsorption of an atom from the Cu island into the vacant site at (111)
surface.
6.4: Conclusions
The anharmonicity on Ag(111) and Cu(111) surfaces during Cu and Ag
trimer diffusion have been studied through the MD simulation technique. The
constant energy MD simulation reveals different mechanisms of diffusion for
all cases of diffusion studied here {i.e. Ag3/Ag(111), Ag3/Cu(111),
Cu3/Ag(111) and Cu3/Cu(111)}, and shows that concerted motion is less
pronounced than the rotation. The concerted motion appears at 700 K and the
107
diffusion rate increases with increasing temperature. The change of
temperature causes very interesting anharmonic features of the island’s
diffusion mechanism. In the case of a Ag island on the Ag(111) surface (at 300
K) a fissure is formed, while at 500 and 700 K (in addition to the fissure) a
dislocation is also observed. The separation of one atom (dissociation) from
the island (at 300 and 500 K) and the separation of all the three atoms of the
island (at 700 K) are observed, in the case of diffusion of Ag3/Cu(111). For the
Cu3/Ag(111) surface at 300 K fissure is observed, while at 500 K a fissure and
a dislocation are both observed. Prominently at 700 K, a vacancy is also
generated along with a fissure and dislocation. In the case of the Cu3/Cu(111)
surface, the island opens in different directions at the above temperatures. A
fissure is found at the above temperatures, while a dislocation is observed
only at 700 K. However, the close similarities bond strengths and bond
lengths of Ag-Ag, Ag-Cu, Cu-Ag and Cu-Cu bonds promote off-lattice sites
and establishes a competition between the optimization of these two trimers
(including dissociation at 700 K), and subjects the substrate to alternate types
of bonds in the form of an in-plane Cu-Cu vibration {in case of Cu3/Ag(111)
study}, which assists the kinetics of strain-release motion.
The average estimated distances for fissures ranges from 1.5 to 4 Å and
for dislocations from 1 to 7Å from the island's position at Cu- and Ag(111)
surfaces at the three temperatures. In the case of Cu3/Ag(111), a vacancy
formed at 700 K means that the dislocation mechanism being active for Cu on
Ag(111). This is not found for the rest of diffusion mechanisms. Obviously,
108
the dislocations reduce the yield stress of bulk metals from their theoretical
values, and in turn, reduce the island diffusion barriers. It is concluded that
for the Cu homo case the island does diffuse easily {for Cu3/Cu(111)}, while
diffusion of the hetero case of Ag is relatively more favorable {for
Ag3/Cu(111)}. Conclusively, it is found that the diffusion trend is faster on the
Cu surface as compared to the Ag surface.
109
Fig. 6.1: Structure of Cu(111) surface.
110
Fig. 6.2: Simple structure of Ag(111) surface.
111
(a)
(b)
Fig. 6.3: (a) Mechanism of trimer Ag/Ag(111) adatom diffusion observed
during molecular dynamics simulations. Left to right: in the 1st picture, the
two atoms of the island are at an fcc site; in the 2nd picture two atoms of the
island are at an hcp site; in the 3rd picture island is at saddle point. (b) From
left to right the snapshots represent anhormonic effects at the Ag(111) surface
in the presence of a trimer Ag island at 300, 500 and 700 K, respectively. The
circle represents fissure and the rectangle represents a dislocation. At 300 K,
only fissures are seen (in left hand snapshot). Both fissures and dislocations
are observed in the middle snapshots at 500 and 700 K.
112
Fig. 6.4: Trace of center of mass of Ag trimer on Ag(111) surface at 500 K for
2000 ps.
113
(a)
(b)
Fig. 6.5: (a) From left to right, snapshots representing the breaking effects of
Ag(111) surface in the presence of the trimer Cu island at 300, 500 and 700 K,
respectively. (b) From left to right, snapshots representing anhormonic effects
at Cu(111) surface in the presence of the trimer Ag island at 300, 500 and 700
K, respectively. The circle represents fissure, the rectangle represents a
dislocation. At 300 K only fissure seen (left hand snapshot). Both fissures and
dislocations are observed in the middle snapshots at 500 and 700 K.
114
Fig. 6.6: Trace of center of mass of Ag trimer on Cu(111) surface at 500 K for
2000 ps.
115
(a)
(b)
Fig. 6.7: (a) Mechanism of Cu3/Ag(111) adatom diffusion observed during
molecular dynamics simulation. Left to right: in the 1st picture, the adatom is
at an hcp site, in 2nd picture it is at a Saddle point; in the 3rd picture it is at an
fcc site. (b) From left to right, snapshots representing anhormonic effects at the
Ag(111) surface in the presence of the trimer Cu island at 300, 500 and 700 K,
respectively. The circle represents a fissure, the rectangle represents a
dislocation, and the square represents a vacancy. The pink color atom is a
surface atom which pops up from the Ag(111) surface/substrate during
vacancy generation at 700 K. At 300 K, only fissure is seen (left hand
snapshot). Both fissures and dislocations are observed in the middle snapshot
at 500 K. The right hand snapshot has a dislocation, fissure and vacancy at 700
K.
116
Fig. 6.8: Trace of center of mass of Cu trimer on Ag(111) surface at 500 K for
2000 ps.
117
(a)
(b)
Fig. 6.9: (a) Mechanism of opening of island for Cu3/Cu(111) surface at
different directions for trimer case. (b) From left to right snapshots
representing anhormonic effects at Cu(111) surface in the presence of the
trimer Cu island at 300, 500 and 700 K, respectively. The circle represents a
fissure, the rectangle represents a dislocation. At 300 K, only fissures are seen
(left hand snapshot). Both fissure and dislocation are observed in the middle
snapshot at 500 and 700 K.
118
Fig. 6.10: Trace of center of mass of Cu trimer on Cu(111) surface at 500 K
for 2000 ps.
119
Chapter 7
THE VACANCY GENERATION AND ADSORPTION OF COPPER ATOM AT Ag(111) SURFACE
7.1: Introduction
The structural changes and dynamics of surfaces caused by variation in
temperature, pressure and sticking of adparticles at the surfaces, are
parameters of keen interest. The change of these parameters produces
vacancies, fissures and dislocations at the surfaces [234-237]. These structural
defects may also migrate from one position to another, and attract or repel
each other with change of these parameters. The adsorption of adparticles on
metallic surfaces is a key to understanding processes like heterogeneous
catalysis and chemisorptions [238-241]. Vacancies also play an important role
in surface morphology [242,243]. Presently, the attention is focused on
understanding the dynamics of an adsorbed copper trimer island on silver
surface and the resulting morphological changes at the surface.
The process of vacancy migration in crystals is fundamental to
understanding solid phase transformations, nucleation and defect migration
[244,245]. The vacancy generation at the surfaces or defects in crystals both
exist at a certain concentration in all materials. This concentration depends
upon the formation energy of the vacancy and temperature [245]. The
temperature dependence of concentration C of vacancies is given by:
TK
E
B
v
eC (7.1)
120
Where Ev is the formation energy of the vacancy,T is the temperature and KB is
Boltzmann’s constant. In the same way, the self-diffusion coefficient SD also
depends upon the formation energy and migration energy of the vacancy,
described by the relation:
TK
EE
SB
mv
eD (7.2)
Where Em is the migration energy. The existence of vacancies in metals and
intermetallic compounds plays an important role for the kinetic and
thermodynamic properties of materials. In this connection, the formation
energy of a vacancy is a key concept in understanding the processes occurring
in metals and their alloys (compounds) during mechanical deformation or
heat treatment.
The value of substitutional adsorption energy is always negative and
less than vacancy formation energy. Theoretical investigations have shown
that the main reason for this substitutional adsorption is the unusually low
energy required to create surface vacancies at the (111) surface [246]. The
substitutional adsorption of K or Na has been observed on the Al(111) surface
for the surface unit cell [243,247].
In case of surface vacancy, the formation the energy of a vacancy is the
sum of energy required to create a vacancy and the energy required for the
diffusion of an adatom of Ag to overcome the barrier at the surface. The
difference between the vacancy formation energy indicates that the
interaction of vacancies at long distances is repulsive and rather small for
121
different metals [248]. Since the formation of a vacancy requires the
destruction of bonds between an atom and its surrounding atoms, therefore,
the energy needed to form a vacancy in different metals is different [249]. This
chapter is focused on the vacancy generation (in the presence of Cu-trimer)
and the adsorption of the Cu-atom into the substitutional site at the Ag(111)
surface during time evolution of a constant energy MD simulation.
7.2: Computational Details
We used the molecular dynamics package/dyn86 given by Foiles et al.
[4] based on embedded atom method potentials [2,4]. Simulations are carried
out by solving Newton's equations of motion using Nordsieck's algorithm
with a time step of 10-15 s [200] A system of 2400 Ag-atoms is simulated in the
form of a slab with 6 layers of (111) geometry. Statistics are recorded after
every 0.05 ps to monitor the Ag surface in the presence of a Cu-trimer. A
reasonable MD simulation involving several thousand steps has been
performed to study the process of vacancy generation. The fluctuations in
calculated results are compensated by taking an average of statistical
quantities. The preliminary simulation is carried out under the condition of a
constant number of 256 atoms, pressure and temperature (NPT ensemble) to
attain the required temperature for the system. The Bulk Ag is simulated by
using a periodic cubic super cell. The experimental value of lattice parameter
of 4.09 (Å) is used to generate a fcc lattice for Ag at 0K, and potentials are built
at this lattice parameter. Interatomic potentials with a cut-off distance of 5.55
Å are used, thus taking into account the interactions among third and fourth
122
nearest-neighbors. The atoms are first relaxed to attain the minimum energy
configuration using the conjugate gradient method [206].
7.3: Results and Discussion
The lattice parameter obtained at three different temperatures(300, 500
and 700 K) is used to develop a slab like structure with (111) surface
geometry. The calculated values of lattice parameter for Ag at 300, 500 and
700 K are 4.114, 4.132 and 4.151 Å, respectively. The lattice parameter as a
function of temperature has been calculated and these values are found in a
good agreement with experimental values [250]. they also show only a small
departure from the values reported by Kallinteris et al. using tight binding
potentials [251].
The 3-atom copper (Cu3) island termed as “trimer” is adsorbed on a
Ag(111) substrate to see the effect of adparticles on the dynamics of surface
atoms and the subsequent relevant parameters. The diffusion coefficient is
calculated at three different temperatures i.e. 300, 500 and 700 K to determine
the energy barrier. All other information about the vacancy generation, the
vacancy migration and the adsorption of atom, are observed at 700 K.
The simulation begins by placing a 3–atom island of Cu, in a randomly
chosen configuration, on the Ag substrate. In Figs 7.1 and 7.2 the blue-colored
atoms are substrate atoms, whereas the red-colored atoms are the island
atoms. The island atoms are initially placed on fcc sites, which are hollow sites
having no atoms underneath them in the layer below. The pink-colored atom
123
in Figs 7.1 and 7.2 represents the popped-up atom from the surface, which
now becomes an adatom over the surface. The green color is used to represent
the Cu substitution atom. The yellow-colored atom is responsible for the
migration of the vacancy.
During the Cu island diffusion on the Ag(111) surface, the island
changes its shape, size and position in different directions and orientations.
During these rotations, orientations and shape changes, the island changes its
positions into different sites which causes some astonishing features i.e.
fissures, dislocations, the generation of vacancy, the migration of vacancy and
change in the positions of fissures and dislocations.
During the course of diffusion of the island on the Ag(111) surface, the
island's atoms are found to be in correlated positions to a high degree. As a
result, the perturbation remains local and the hollow sites are observed near
the island’s position, which are shown in the snapshots of Figs. 7.1(a)–7.1(d)
of our simulations. These hollow sites may result in the formation of fissures
and dislocations, and other dynamics along with the pop-up of the surface
atom. Notice that the terms “fissures” and “dislocations” mean merely the
rupture of the Cu–Ag and Ag–Cu bonds along a line in the lattice rather the
crystallographic defect at the surface. These ruptures at the surface result into
the vibrations of the substrate atoms; whose displacement patterns expand
and contract the Cu–Ag and Ag–Cu bonds along a line in the lattice which are
the cause of formation of fissures and dislocations at the substrate surface
[235].
124
During the time evolution of the Cu3 island on the Ag(111) surface, two
fissures and two dislocations are observed on the surface. Initially the fissures
are observed at the distances of ∼1.5 and ∼ 4.1 Å and the dislocations are
observed at the distances of∼ 14.35 and ∼ 20.5 Å, from the island's position.
These fissures are ∼ 6.15 Å from each other, while the dislocations are located
at the distance of∼ 10.25 Å apart, as shown in Fig. 7.1(a). But after few ps,
these fissures and dislocations reorient their positions from the previous one
to new positions which lie at a distance of∼ 0.5 and ∼ 6.9 Å (for fissures) and
∼ 3.75 and ∼ 7.7 Å (for dislocations), respectively from the island's position
simultaneously, as shown in Fig. 7.2(b). It is noticeable to make it clear that
fissures and dislocations migrate during the constant energy run and show a
tendency to attract each other [252,253].
The “fissures” and “dislocations” refer to the dynamical fluctuations of
the substrate atoms rather the crystallographic defects at the surface as a
result of rupture of the Ag–Ag bonds at a point and/or along a line in the
lattice. The presence of foreign adparticles are responsible for the vibrations of
the substrate whose displacement patterns expand and contract the Ag–Ag
bonds along a line in the lattice.
It is one of the recognized facts that for lattice mismatched systems at
high temperatures disorder appears in the layers of the substrates which can
be the cause of the creation of vacancies at the surfaces. So, in our case of the
diffusion of a copper island on the Ag(111) at 700 K, the disorder appeared in
the first layer causes a vacancy at the Ag surface. As a result, the silver surface
125
atom is popped-up to form an adatom, nearly at about 4 Å from the position
of the Cu3 island. The vacancy generated by the pop-up of an Ag surface atom
also migrates∼ 2.05 Å toward the island, from its previous position, by self-
diffusion of the surface layer atom.
Thermal vibrations of the surface atoms increase with increase of the
lattice temperature. At 700 K, lattice parameter of Ag is increased by 6.1 pico
meter (pm) as compared to its value at 0 K. With a large lattice parameter,
thermal vibrations of atoms at high temperature leads to weak interactions
among the atoms. Thus, an increase in the thermal dynamics, due to the large
gaps among the atoms, is helpful to generate a vacancy at the surface. One of
the atoms of the substrate from the top layer is popped-up and after 798 ps it
became the part of the Cu island formation "mixed-tetramer" (Cu3–Ag).
The energy of vacancy generation at the Ag surface in the presence of
the Cu-trimer is found to be 1.078 electron-volt (eV). On the other hand, the
vacancy energy calculated by ab-initio methods and in simple model
calculations is 0.67 and 0.73 eV, respectively [254], while the experimental
value is 1.11 eV [255]. The comparison of our simulated result with the
experiment shows an average deviation of only 2.7% in reasonable agreement
with experiment.
The calculated diffusion coefficient D for a Cu trimer island during the
diffusion process on Ag(111) surface is 3.82×1011 Å2/s at 700 K, and the value
of energy barrier Eb is 0.12721±9 eV. The trace of the center of mass of the Cu
trimer on the Ag(111) surface is given in Fig. 7.3. This plot shows that the
126
island diffuses ∼ 17 Å along the x-axis and ∼ 11 Å along the y-axis during 798
ps.
During the diffusion process of the Cu island on the Ag(111) surface,
one of the Ag surface atoms gains energy of about Ev = 1.078 eV and popped-
up on the Ag surface. This increase in the energy of the Ag substrate atom is
contributed to by the thermal dynamics of the neighboring atoms due to the
ambient temperature (700 K) as well as by Cu–Ag bonds developed between
the island and nearby substrate atoms. Moreover, the popped-up atom
diffuses separately (and is termed a monomer). The XY-plot of the said Ag
monomer on Ag(111) surface at 700 K is given in Fig. 7.4. This plot shows that
the monomer diffuses ∼ 65 Å along the x-axis and ∼ 76 Å along the y-axis
during a time of 798 ps. The calculated diffusion coefficient D for Ag
monomer on Ag(111) surface is 2.30×1012 Å2/s which is higher in magnitude
as compared to the value (7.16 × 1011 Å2/s) calculated by Self Learning Kinetic
Monte Carlo (SLKMC) simulations [256]. The energy barrier Eb for the Ag
monomer is 0.082 ± 9 eV.
The popped-up monomer from the Ag(111) substrate joins with the
copper trimer island after 798 ps and forms the mixed-tetramer (comprising
of three Cu- and one Ag-atom). The calculated diffusion coefficient D for this
Cu3–Ag island during the diffusion process on Ag(111) surface is 3.515×1011
Å2/s at 700 K and found energy barrier Eb is found to be 0.1882 ± 9 eV.
The trace of center of the mass for the mixed-tetramer is given in Fig.
7.5, inferring that the island diffuses ∼ 14 Å along the x-axis and ∼ 20 Å along
127
the y-axis during 765 ps. Occasionally, all three Cu atoms start rotating
around the Ag adatom. After the period of 765 ps, one of the copper atoms of
the island from the mixed-tetramer is adsorbed in the surface layer of silver as
a substitutional atom and loses ∼ 0.813 eV of its energy to adjust into the layer
which is the substitution energy Es. This is an exchange of a Cu-atom with an
Ag-atom at Ag(111) surface.
The remaining island is again a mixed-trimer comprising of two
copper atom and one silver atom. The XY–plot of this Cu2–Ag island
represents the stay of the island over Cu substituted atom in Ag(111) surface
is given in Fig. 7.6, for the duration of 437 ps. This plot shows that, for most of
the time, the island's center of mass resides in the area of almost 1.2 Å2.
The size of the copper atom is less than that of silver by 13.8%. The
substitution of a Cu atom in the vacant site of Ag(111) surface results in a kink
which is named as a “lapsed well”. This kink at this smooth Ag(111) surface is
formed due to the difference in the sizes of Ag and Cu atoms. Now, the kink
of height 4.0×10-1 Å, the adsorbed copper atom having six surrounding atoms
of Ag substrate. So, the energy barrier for the mixed-trimer is increased from
the expected value of the barrier (0.12721 ± 9 eV) and sufficient energy is
required for the island to overcome the barrier. Thus, the mixed-trimer stays
comparatively more time on the Cu adsorbed atom, which is clearly
elaborated in Fig. 7.6.
Sometimes, the silver atom tries to move away, while the remaining
two copper atoms are compelled to rotate near the exchanged copper atom
128
which now became a part of the substrate. After escaping from this lapsed
well developed by adsorbed Cu-atom, the mixed-trimer diffuses ∼ 36 Å along
the x-axis and ∼ 38 Å along the y-axis during 2.0×103 ps (which is separately
shown in Fig. 7.7). The calculated diffusion coefficient D and energy barrier Eb
for mixed-trimer are 4.24×1011 Å2/s and 0.134 ±9 eV, respectively.
Since the vacancy generation energy at surfaces is less than bulk due to
lower nearest neighbors, so a smaller amount of energy is required to
generate the defects or pre-melt the surfaces [257]. As a result, increase in
temperature (up to 700 K) breaks the Ag-Ag bonds and generates a vacancy
easily at the Ag(111) surface. By increasing the temperature the rate of
diffusion increases due to the increase in thermal vibrations which leads to a
change in the dynamics of the surface atoms and foreign adparticles. Hence
the temperature plays an important role in the generation and migration of a
vacancy at the surfaces.
7.4: Conclusions
The Molecular dynamics simulations provide a strong basis for
understanding the dynamics of interacting atoms. In our case, the inelastic
collision among the atoms of the substrate made the bonding weak in the
neighborhood of the vacancy generating atom at the surface. Meanwhile, the
copper island provides a strong adhesive force to the vacancy generating
atom. Thus, the energy of the vacancy creating atom Ev has increased to 1.087
eV from its normal value, which in turn overcomes the surface energy of the
silver substrate. These factors are strongly responsible for the generation of
129
the vacancy at Ag(111) surface in the presence of Cu3 island. The simulated
energy of vacancy generation deviates from the experimental value only by
2.7%, in good agreement with experiment with in the error limit of our
calculations. It is found that the energy of adsorption of Cu-atom (-0.813 eV)
is less than the energy of vacancy generation.
130
(a) (b) (c) (d)
Fig. 7.1 (Colored on line) The circle represents fissure, the rectangle represents
dislocation, and the square represents vacancy. The pink-colored atom is a
surface atom, popped-up from the Ag(111) surface during vacancy generation
at 700 K. From left to right snapshots are representing instantaneous
anharmonic effects at Ag(111) surface in the presence of the trimer Cu island
at 300, 500 and 700 K, respectively. (a) At 300 K only a fissure seen (left hand
snapshot). (b) Both fissures and dislocations are observed in the middle
snapshot at 500 K. (c) Snapshot representing the vacancy at 700 K. (d) At 700
K, Green-colored atom is the exchanged atom of Cu with the vacancy at the
Ag(111) surface, which was previously part of the Cu-trimer.
131
(a)
132
(b)
Fig. 7.2 (Colored on line) Snapshots show the instantaneous migration of the
vacancy. The pink-colored atom represents the popped-up atom on Ag the
surface. The yellow-colored atom, which is a neighbouring atom to the
vacancy, changes its position during the time of 0.05 ps in these snapshots.
Snapshot (a) shows surface before the shift of vacancy, while the snapshot (b)
shows the shifted vacancy. Meanwhile, the fissures and dislocations
(represented by circles and rectangles, respectively) are also shifted at the
surface.
133
Fig. 7.3 Trace of center of mass of Cu-trimer on Ag (111) surface during 798
ps
134
Fig. 7.4 XY-plot of Ag monomer on Ag(111) surface during 798 ps.
135
Fig. 7.5: Mixed-tetramer (comprising of three Cu- and one Ag-atom) trace of
center of mass of on Ag(111) surface during 765 ps.
136
Fig. 7.6 XY-plot of the mixed-trimer island (comprising of two Cu- and one
Ag-atom) represents the position of the island over Cu substituted atom in
Ag(111) surface about 437 ps. Most part of the time, the center of mass of the
island remains in the area of 1.2 Å2.
137
Fig. 7.7 Trace of center of mass of mixed-trimer (Cu2-Ag island) at Ag(111)
surface during 750 ps.
138
Chapter 8
SUMMARY AND CONCLUSION
This dissertation contains a comprehensive comparative study of the
anharamonic effects for surfaces of Ag and Cu in the presence of adparticles
(islands). The role of interatomic potential is elaborated for generating
anharmonicity on the metallic surfaces. The multilayer relaxations for (311)
and (210) Cu surfaces are presented. The maximum interlayer relaxation for
the Cu(311) surface is -9.17 % for ‘d12’ (contraction) and +3.94 % for 'd23'
(expansion). The interlayer relaxation of Cu(311) shows a uniform damping
in magnitude away from the surface into the bulk with an alternating
oscillatory order of (-, +, -, +, . . . ). The interlayer relaxation near Cu(210)
shows random order such as (-,-,+,-, +, . . .) with a non-uniform damping in
magnitude. The interlayer relaxation near Cu(311) shows that relaxation
decreases with increasing plane number. The interlayer and atomic relaxation
of Cu(210) are affected by the local force fields. These relaxations have been
compared with recent low energy electron diffraction data [177,178 ]. The
most striking feature in the calculated relaxations of Cu(210) is pronounced
expansion of the interlayer distance between the 3rd and 4th layer which leads
to a non-uniform damping magnitude for the multilayer relaxation of
Cu(210). The average deviation in interlayer relaxations using EAM potentials
and low-energy electron diffraction calculations is 1.94% for Cu(210), and for
the registry relaxation it is 0.74%.
139
In a diffusion study of the Cu pentamer on the Ag(111) surface, the
calculated effective energy barrier Ea was formed to be 0.20525 eV and
diffusion prefactor D0 was formed to be 5.549×1012 Å2/s for the Cu pentamer.
The results are in excellent agreement with experimental findings [204]. The
high barrier for a Cu pentamer on Ag(111) is due to lattice mismatch, since the
Cu pentamer on Ag(111) diffuses through hops ~ 8% longer than those they
exhibit for the homo-case which makes them detach significantly from other
Ag nearest neighbors at the transition state. It is inferred that the rate of
diffusion increases with increase in temperature. For the pentamer, our
calculations indicate that the short fcc-hcp configuration with its relative low-
energy triggering processes may act together with those involving the long
fcc-hcp sites, to establish an efficient intercell zigzag diffusion. It has been
found that significant change in the size depends on the variations of
diffusion characteristics of the islands after including concerted motion
mechanism. It has also been found that these small-sized islands (5-atoms)
diffuse primarily through concerted motion with a small contribution from
single atom processes, even though for certain cases the frequency of single
atom processes is large because of lower activation energies.
The anharmonicity at Ag(111) and Cu(111) surfaces during Cu and Ag
trimer diffusion has been explored. The constant energy MD simulation
reveals different mechanisms of diffusion for all the cases studied here {i.e.
Ag3/Ag(111), Ag3/Cu(111), Cu3/Ag(111) and Cu3/Cu(111)} and shows that
concerted motion is less pronounced than rotation. The concerted motion
140
appears at 700 K and the diffusion rate increases the increase in temperature.
The change of temperature causes very interesting anharmonic features of the
island’s diffusion mechanism. For Ag trimer island on Ag(111) surface (at 300
K) a fissure is formed, while at 500 and 700 K (in addition to the fissure) a
dislocation has also been observed. The separation of one atom (dissociation)
from the island (at 300 and 500 K) and the separation of all of the three atoms
of the island (at 700 K) are observed in the case of Ag3/Cu(111). In the case of
diffusion of Cu3/Ag(111) at 300 K a fissure is observed, while at 500 K a
fissure and dislocation, are both observed. Prominently at 700 K, a vacancy is
also generated along with a fissure and dislocation. In the case of
Cu3/Cu(111), the island opens in different directions at the three
temperatures studied. The fissure is found at all three temperatures, while a
dislocation is observed only at 700 K. However, in the case of a trimer, the
close similarities in bond strength and bond length between Ag-Ag, Ag-Cu,
Cu-Ag and Cu-Cu bonds promote off-lattice sites and establish a competition
between the optimization of these two trimers (including dissociation at 700
K) and subject the substrate to an alternate “types of bonds in the form of an
in-plane Cu-Cu vibration {in case of Cu3/Ag(111) study}, which assists the
kinetics of strain-release” motion. The average estimated distances for fissures
ranges from 1.5 to 4 Å and for dislocation it is 1 to 7 Å from the island's
position at Cu- and Ag(111) surfaces at the said temperatures. For
Cu3/Ag(111), a vacancy formed at 700 K meant that the dislocation
mechanism being active for Cu on Ag(111), which has not been observed for
141
other island diffusion mechanisms. Obviously, the dislocations reduced the
yield stress of bulk metals from their theoretical values, and in turn, reduced
the island diffusion barriers. It has been concluded that for Cu homo case the
island diffuses easily {i.e. Cu3/Cu(111)}, while diffusion of the hetero case of
Ag is relatively more favorable {i.e. Ag3/Cu(111)}. Conclusively, it is found
that the diffusion trend is faster on Cu surface as compared to Ag surface.
The Molecular Dynamics simulation provides a strong basis for
understanding the dynamics of interacting atoms. In our case, the inelastic
collision among the atoms of the substrate made the bonding weak in the
neighborhood of the vacancy generating atoms of the substrate. Meanwhile,
the copper island provides a strong adhesive force to the vacancy generating
atom. Thus, the energy of the vacancy generating atom Ev increased to 1.087
eV from its normal value, which in turn overcomes the surface energy of the
silver substrate. These factors are strongly responsible for the generation of
the vacancy at Ag(111) surface in the presence of Cu3 island. The simulated
energy of vacancy generation deviates from the experimental value only by
2.7%, in good agreement with experiment with in the error limit of our
calculations [254,255 ]. It is found that the energy of adsorption of a Cu-atom
(–0.813 eV) is less than the vacancy generation energy. In summary, our MD
simulations have given us new insights into the diffusion mechanisms for
adsorbates on surfaces. We have been able to separate the regions in which
anharmonic effects become dominant. These calculations have set the limits of
validity of the harmonic approximation.
142
RECOMMENDATIONS
Vicinal surfaces have more interlayer and registry relaxation as compared
to the compact surfaces. Therefore smooth/compact surfaces are suitable
for surface diffusion.
The presence of adparticles at the surfaces generate anharmonicity at the
surfaces. Therefore the presence of adparticles during cutting and welding
may cause changes in the dynamics of the surfaces. Thus these processes
should be carried out carefully.
The diffusion prefactor is independent of temperature in the range of 300-
700 K.
Anharmonicity is always dominant in case of hetero diffusion of the
island.
Defects such as dislocations and fissures always have a trend of attraction
to each other.
ACKNOWLEDGEMENT
I feel privileged to acknowledge the generous financial support of
higher education Commission (HEC), Government of Pakistan, for
Completion of this research endeavor. The financial support was provided
under Indigenous PhD Scholarship-500 Program of HEC.
143
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