Surface-morphology-of-ti-and-ti-6al-4v-bombarded-with-1 0-mev-au-ions

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UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
SURFACE MORPHOLOGY OF TI AND TI-6AL-4V BOMBARDED
WITH 1.0-MEV AU IONS
TESIS
PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS (FÍSICA)
PRESENTA:
MIGUEL ÁNGEL GARCÍA CRUZ
TUTOR:
DR. JORGE EDUARDO RICKARDS CAMPBELL
INSTITUTO DE FÍSICA, UNAM
MIEMBROS DEL COMITÉ TUTOR
DR. LUIS RODRÍGUEZ FERNÁNDEZ, IF-UNAM
DR. ALEJANDRO CRESPO SOSA, IF-UNAM
CIUDAD UNIVERSITARIA, ENERO 2017
 
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Contents
Abstract v
Resumen vii
Nt’ut’ant’ofo ix
Acknowledgments xi
Dedication xiii
1 Introduction 1
1.1 Technological Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Physics Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Interdisciplinary Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Surface and Interface Growth 9
2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Ginzburg-Landau Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Experimental Approaches: A Review 16
3.1 Initial Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
i
CONTENTS
3.2 Normal Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Oblique Incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 High-energy Irradiation of Ti . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Theories of Ion Induced Surface Growth 23
4.1 Sigmund Theory of Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Bradley-Harper Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Makeev-Cuerno-Barabási Model . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Kuramoto-Sivashinsky Model . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5 Muñoz-Cuerno-Castro Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5.1 The 1-D & 2-D Effective Model . . . . . . . . . . . . . . . . . . . . . 39
4.6 Bradley-Shipman Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.7 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.7.1 Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.7.2 Non-linear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Experimental Techniques 52
5.1 Ti and Its Alloy Ti-6Al-4V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Ion Implanter Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.2 Ion Implantation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.3 Ion Beam Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Au Ion Implantation of Ti and Ti-6Al-4V . . . . . . . . . . . . . . . . . . . . 60
5.4 Surface Induced Stress on Ti and Ti-6Al-4V . . . . . . . . . . . . . . . . . . 60
5.5 Microscopy Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.6 X-ray Photoelectron Spectroscopy Technique . . . . . . . . . . . . . . . . . . 64
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CONTENTS
6 Results 66
6.1 IBS of Ti and Ti-6Al-4V at 8° & at 45° Angles . . . . . . . . . . . . . . . . . 67
6.2 IBS Evolution for Ti and Ti-6Al-4V at 45° Angle . . . . . . . . . . . . . . . 69
6.2.1 Large-scale Morphologies . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2.2 Small-scale Morphologies . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 IBS Incidence Angle Dependency for Ti and Ti-6Al-4V . . . . . . . . . . . . 74
6.3.1 Large-scale Morphologies . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3.2 Small-scale Morphologies . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.4 Micro-indentation of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.5 Ripple Elemental Composition . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.6 XPS Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.6.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.7 IBS Ion-atom Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Discussion 95
7.1 Experiment and Simulation: Au Ion Implantation of Ti and Ti-6Al-4V . . . 96
7.2 Atomic Damage and Energy Loss Processes . . . . . . . . . . . . . . . . . . 97
7.3 Bradley-Harper Type Theories Considerations . . . . . . . . . . . . . . . . . 100
7.3.1 Profile Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3.2 Atomic Processes: Surface Erosion . . . . . . . . . . . . . . . . . . . 106
7.3.3 Atomic Processes: Surface Diffusion/Relaxation . . . . . . . . . . . . 108
7.3.4 Ion-atom Combination . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.4 Bradley-Shipman Type Theories Considerations . . . . . . . . . . . . . . . . 114
7.4.1 Intermetallic Compound Formation . . . . . . . . . . . . . . . . . . . 115
7.5 Asymptotic Non-linear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.6 Overview of Hydrodynamic Models . . . . . . . . . . . . . . . . . . . . . . . 117
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CONTENTS
7.7 Applications of Surface Structures in the Medical Industry . . . . . . . . . . 119
8 Conclusions 121
9 Outlook 124
Appendix A: Sputtering yield 126
Appendix B: Linear stability analysis 129
Appendix C: Linear and nonlinear corrections 132
Epilogue 134
Bibliography 135
iv
Abstract
This dissertation centers on changes in the surface morphology of titanium and its alloy Ti-
6Al-4V bombarded with high energy gold ions. Surface modification of metallic materials has
generated a great interest due to the possibility of producing many intricate shapes, initially
believed to be only available in semiconductors under very specific conditions. In particular,
the formation of surface ripples in titanium (Ti) and the alloy (Ti-6Al-4V) came as a huge
surprise and has become a starting ground for future technological applications. Formation of
micrometer-size ripples within a few hours of implantation suggests their possible description
from a continuum model approach. In the present case, the growth of these rippling patterns
is studied experimentally at the high energy limit and one of the main purposes is to test
the validity of ion-induced surface patterning theories.
1.0-MeV Au+ ions are implanted in Ti and Ti-6Al-4V at a 45° angle of incidence. Im-
planted ions reach the near surface region inducing a surface modification that shows up as
surface ripples. Depending on the experimental parameters, certain morphologies may be
obtained; for instance by changing the angle of incidence a transition from near-flat into
ripples may be achieved. A critical angle of incidence exists before any surface structure can
develop. This dependence is explored and studied systematically with varying ion fluence.
Observed surface morphologies are studied with the help of surface analysis techniques such
as SEM, AFM, RBS and XPS.
Moreover, experimental data have been utilized as input to compare continuum theories’
v
Abstract
predictingpower for the surface evolution under our ion implantation conditions. Under
the Bradley-Harper classical theory, surface ripples are roughly predicted but cannot fully
describe their asymptotic behavior. This is resolved by considering non-linear models, where
surface saturation and coarsening are seen to occur. Other advanced models are reviewed,
such as those considered in coupled two-field models of Muñoz-Cuerno-Castro and Bradley-
Shipman.
Interest in this type of work comes from its possible applications in the medical field;
ion implantation of metallic orthopedic materials in principle may enhance the adhesion of
associated biomolecules of the bone. Surface modification by noble ion implantation may not
affect the bio-compatibility of Ti-based materials and may eventually be adapted to common
use in medical applications.
vi
Resumen
La presente tesis se centra en los cambios en la morfoloǵıa superficial de Ti y Ti-6Al-4V
bombardeados con iones de oro a altas enerǵıas. La modificación superficial en materiales
metálicos ha generado un gran interés debido a la posibilidad de producir varias formas com-
plejas, a sabiendas que en semiconductores esto sucede bajo condiciones experimentales bien
espećıficas. En particular, la formación de ondulaciones superficiales en sustratos de titanio
(Ti) y de su aleación (Ti-6Al-4V) resultó interesante, sugiriendo aplicaciones technológicas
en un futuro próximo. La formación de ondulaciones de tamaño micrómetrico dentro de
unas cuantas horas de implantación sugiere su posible descripción a partir de los modelos
del medio continuo. En este caso, el crecimiento de patrones ondulatorios es estudiado ex-
perimentalmente a un ĺımite de altas enerǵıas y uno de los principales objetivos es validar
teoŕıas de la formación de patrones inducida por iones.
Iones de Au+ a 1.0-MeV son implantados sobre Ti y Ti-6Al-4V a un ángulo de inci-
dencia de 45°. Los iones implantados son depositados en una región cercana a la superficie
induciendo su modificación a una forma ondulatoria. De acuerdo a lo anterior y dependi-
endo de los parámetros experimentales, se pueden obtener ciertas morfoloǵıas; por ejemplo
cambiando el ángulo de incidencia se obtiene una transición de plano a ondas. De igual
manera; existe un ángulo cŕıtico de incidencia antes de que se forme alguna estructura su-
perficial. Esta dependencia es explorada y estudiada sistemáticamente variando la afluencia.
Las morfoloǵıas superficiales obtenidas fueron estudiadas con la ayuda de instrumentos de
vii
Resumen
análisis de superficies tales como SEM, AFM, RBS y XPS.
Además, los datos experimentales obtenidos son utilizados como parámetros de entrada
para explorar las predicciones teóricas del modelo continuo. A partir de la teoŕıa clásica de
Bradley-Harper, se predicen ondas superficiales, pero no se puede describir completamente
su comportamiento asintótico. Esto se resuelve teniendo en cuenta modelos no lineales que
describen la saturación de la superficie y el ensanchamiento de la estructura. Aśı mismo, se
revisan otros modelos a dos campos acomplados, tales como los de Muñoz-Cuerno-Castro y
Bradley-Shipman.
El interés en este tipo de trabajo se deriva de sus posibles aplicaciones en el campo de
la medicina; en la implantación de materiales metálicos en principio se espera que mejore
la adhesión de biomoléculas asociadas al tejido óseo. La modificación superficial por la
implantación de iones puede no afectar la biocompatibilidad de materiales basados en Ti, y
que podŕıan algun d́ıa ser adoptados a uso común en aplicaciones médicas.
viii
Nt’ut’ant’ofo
Nuna b’efi di udi ñä tengu ra mpadi ra titanio (hñähñu dra hmä: boja xa me ne hinxa hñu)
ne ra hñäts’i Ti-6Al-4V nge’a puni nts’edi ha ra hmi ko xi ra t’uka zeki ra k’ast’aboja. Ra
nthoki ra hmi dige’a ya nt’ote dega boja xa dähä ndunthi ra mudik’at’i ra nge’a tsa da t’ot’e
mar’a ya m’ai, fädi ge ha ya r’ani ra ts’edihuei nuna nt’ote thogi n’andi hä n’andi hinä.
Nsokä sehe, ra thogi ha ya t’uki pents’i ha ya hmi ra titanio (Ti) ne ra nthänts’i (Ti-6Al-4V)
ra boni maske hingi thoni ena nge’ä m’efa dra thähä ya nthoki. Ya nthoki ya m’o dega m’o
ra t’uki zeka t’enima ra pents’i ha ra n’a ra hneb’u ha r’a ya nt’enipa dige’a ra nthuts’i adi
ra tsapi nt’ot’e ri fudi nu’a ri mudi dige’a ra nzot’e nthoki. Nub’u, ha ra te ya pents’i k’oi
di nxadihu habu xa tsapi xa thoki, ngeb’u juadi habu ts’e ra hyodi ne n’a ra m’etsazu pa da
mä mäjuäni ra nthoki ya mudi pents’i tsits’i ya t’uka zeka ra k’ast’aboja.
N’a ra - MeV ra k’ast’aboja kut’i ha ra Ti ne ra Ti-6Al-4V n’a ra nkahmi ha ra n’ate ma
kut’a t’eni. Ya k’ast’aboja thuts’i getbu ha n’a ra xena hyodihai hmi tsits’i ra mponi ha n’a
m’ai xa mpets’i. Ra nge’ä ma mfädihu, da za ga petsi ndunthi ya m’ai ya k’oi; ra nt’utate
da nponi ra nkahmi da za ga petsihu ra mpungi dige’a ra mänjuäntho pa ra npents’i. Ja
n’a ra nkahmi met’o da nja n’a nthoki ngetbu ha ra hmi ya k’oi. Nuna ra b’efi di nxadihu
ngetbu di thuts’i nu’a mä mäjuäni ra ndunthi ya t’uka zeka ya k’ast’aboja. Ya ntsa mpadi
xa ts’udi ne xa nzadi ko ya mfatsi ya mpefi pa da fädi xahño tengu ra nthoki ya hmi hyodi
ngu SEM, AFM, RBS ne ra XPS.
Nehe, ya nt’udi xa thähä xa thä ngu ra t’eni pa da yut’i ne tsa dra hmä nuna te da
ix
Nt’ut’ant’ofo
nja ngu feni nuna nt’ot’e. Ra fudi b’efi mfeni ra Bradley-Harper bi ena ge ri hneki ya tuka
pents’i, pe hingi tsa dra mä gatho ra nzot’etho nuna thogi ha nuna m’efa da nja. Nuna tsa
da thoki di petsi ra guenda ya nt’utate petsi ya hñe’i mä ra nzot’e dige’a ra hmi hyodi ne
ra nxiki dige’a ra m’aint’ot’e. Di handihu man’a ya ntut’ate ha ra nyoho ya hyodi xa nzeti
ngu bi udi ra Muñoz-Cuerno-Castro ne ra Bradley-Shipman.
Ra mudik’at’i ha nuna ra b’efi ri ñ’ehe dige’a ya tsapi nt’ot’e ha ra ofo dige’a ra nt’othe
ya hñeni xa nkum’i o xa huaki ya ndoy’o; ha ra nthuts’i nuya nt’ot’a boja ha ra mudi dra
tom’i ge dra thoki xahño ra nzot’a xahño habu ra ñ’u ha ra ngok’ei dige’a ra zeki habu dra
t’ot’e da ñäni. Ra mpadi dige’a ya k’ast’aboja tsa hinda ot’e ra ñ’u dige’a ya nt’ot’e ra boja
xa me ne hinxa hñu ne tsa ge n’a ra pa dra thä habu dra ot’e ra b’edi pa da mfaste da petsi
ra ehya ya jä’i.
Glossary of terms - t’uka nxadi1
gold ions - xi ra t’uka zeki ra k’ast’aboja
bombardment - puni
surface - hmi
titanium - boja xa me ne hinxa hñu
morphology - k’oi
ripples - pents’i
incidence angle - nkahmi
1Jamädi rá zi xahnäte Prof. Raymundo Isidro Alavez.
x
Acknowledgments
I would like to express my sincere appreciation to my advisor Dr. Jorge Rickards Campbell.
Thank you for encouraging me on this research project and for allowing me to grow as an
individual researcher. I appreciate your teachings and guidance.
Many thanks to my advisory committee members; Dr. Luis Rodŕıguez Fernández and
Dr. Alejandro Crespo Sosa for their valuable suggestions and their accelerator time for our
experiments. Also, thank you Dr. Luis Rodŕıguez for letting me be your teaching assistant.
I would like to thank all our technicians who helped us along the way on sample prepa-
ration (Mr. Melitón Galindo), irradiation (Mr. Karim López, Engr. Francisco Jaimes and
Engr. Mauricio Escobar) and characterization (M. Sc. Rebeca Trejo Luna, Engr. Marcela
Guerrero, M. Sc. Jaqueline Cañetas Ortega, Dr. Luis Ricardo De la Vega, M. Sc. Juan
Gabriel Morales and Dr. Luis Lartundo Rojas).
My sincere appreciation to Dr. Rodolfo Cuerno Rejado for the most welcoming three-
month stay at Universidad Carlos III de Madrid. Thank you for the few hours spent talking
about surface growth models. Moreover, your visit at the Instituto de F́ısica was enlightening
and encouraging. I owe you visits to other archaeological sites.
Many thanks to the staff at the library “Juan B. de Oyarzábal” of the Instituto de F́ısica
for their friendship, the countless copies and books I borrowedthroughout the years.
Thank you Professor Raymundo Isidro Alavez for helping into remembering my roots
and teaching me a few things on the Hñähñu language.
xi
Acknowledgments
I would like to thank all my friends for spending time with me. Sorry if I cannot name
you all, but my list would not fit on a few lines.
Special thanks to my family. Words cannot express how grateful I am to my grandmother,
aunts, sisters, brother and dad for all of the sacrifices they have made on my behalf, I love
you all. Mom, your blessings are all that I have, I miss you and I will never forget you.
Acknowledgements are due to Universidad Nacional Autónoma de México (UNAM), In-
stituto de F́ısica (IF), Posgrado en Ciencias F́ısicas (PCF) and Consejo Nacional de Ciencia
y Tecnoloǵıa (CONACyT). The completion of this reseach was possible through the financial
support of the laboratory in the Departamento de F́ısica Experimental through CONACyT
reseach contracts 102937 and 222485 along with DGAPA-UNAM under PAPIIT projects
IN113-111, IN108-013 and IN110-116. Lastly, a residence research scholarship was awarded
from Programa de Movilidad Internacional of the Coordinación de Estudios de Posgrado
(CEP) during the visit at Universidad Carlos III de Madrid.
xii
For Lućıa Pedraza Dı́az
xiii
Chapter 1
Introduction
The main focus of this work is to increase our understanding of the physical processes oc-
curring at the near surface region of materials being subjected to an energetic ion-beam.
Ion-beams produced by particle accelerators are widespread due to their availability and fre-
quent use in industry for the modification and analysis of materials. Technological advances
in the semiconductor industry (e.g. integrated circuits, photoelectronic devices and surface
analysis techniques) are basically due to these methods and are continually improving. In
this work, an ion beam is utilized for the surface modification of Ti-based biomaterials; this
is done in order to improve their surface properties. Additionally, often due to ion irradia-
tion, curious morphologies can be produced, becoming an effective top-down technique for
pattern formation.
Motivation for this work is based on the premise of consolidating a general knowledge of
the effects of ion-induced bombardment of materials. This chapter is divided in three topics;
first we give a brief overview of known technological advances due to ion beam interaction
with matter (§1.1), followed by the physics motivation in view of a prevalent interest in the
understanding of ion-atom interactions in (§1.2), and finally we relate other natural ocurring
phenomena of pattern formation due to external environmental/biological factors (§1.3).
1
Chapter 1. Introduction
1.1 Technological Advances
The use of a particle accelerator is important for the fabrication of composite materials fre-
quently used in the technology of semiconductors [1]. Irradiation of materials with ion beams
with well-controlled parameters can generate atomic defects which allows changes in the near
surface region, modifying its physical and chemical properties [2]. Applications based on this
modification method have been employed in our everyday use of technology. Cellphones,
computers and related technologies are commonly composed of integrated circuits where the
processing of semiconductors is an important step toward their production.
During ion irradiation of materials, it often occurs that the material erodes; this in fact
causes the emergence of interesting surface morphologies [3]. Ion beam sputtering (IBS) is a
technique that induces the erosion of surfaces [4] and is known to change surface layers of a
material drastically. As a surface is bombarded by an ion beam, many physical effects are
known to occur depending on the parameters of the experiment. It is known now that IBS
experiments produce many intricate surface structures, like ripples, holes and dots. These
periodic or quasi-periodic structures could one day be employed in many important appli-
cations, the medical industry being one possibility of interest in this work [5,6]. Moreover, in
surface analysis techniques using ion beams, like secondary ion mass spectrometry (SIMS),
erosion is its primary mechanism of operation. Advances in irradiation techniques have
helped this field to develop tremendously, and better control systems are now easily accessi-
ble. The accumulated knowledge of the physical mechanisms near the irradiated surface has
helped this development and will probably continue to grow in the near future.
In the industry of orthopedics, the surface physical and chemical behavior of biomaterials
is a very important issue [7,8]. Common human orthopedic implants are made of Ti and from
other elements; the alloys Ti-6Al-4V, Ti-6Al-7Nb and Ti-13Nb-13Zr are three good examples.
The surface of Ti and its alloys poses a favorable behavior with human bone and tissue,
2
Chapter 1. Introduction
due to their bio-compatibility [9,10]. Recent studies suggest that better surface treatments
are necessary in order to cope with the wear of the metal interface over time [11]. Medical
examinations have noted the formation of thrombus along the interface of the metal implant
and the bone, possibly caused by the mobility of surface residues [12]. It is therefore imperative
to be able to control the surface properties of the material. Surface treatments by plasma
immersion, chemical treatments and noble ion implantation [US patent No. 4,137,370] [13]
have been proposed as possible solutions. It is believed that the use of ion implantation [14,15]
could modify the near surface layer allowing a full integration of the metal implant with the
human bone and tissue [5,6,16].
Ion implantation of noble ions, may possibly even enhance the adhesion of associated bio-
molecules of the bone to the implanted metal interface without affecting its bio-compatibility [17].
Furthermore, the production of a well defined surface structure could be used for the attach-
ment and growth of bones cells onto the metal implant. In the work of Riedel et al. [6] 700-eV
and 1100-eV Ar ions were irradiated onto the medical titanium alloy surface (Ti-6Al-4V-ELI,
ELI - Extra-Low-Interstitials) with a resulting favorable roughness. Attachment tests per-
formed on treated surfaces show better adhesion in comparison to those of untreated surfaces;
thus for example the growth of rat mesenchymal stem cells is favored.
Interestingly, in our work, atomic damage near the surface of Ti and on its alloy produces
surface structures resembling ripples. These structures have been studied extensively in other
materials such as silicon, due to their possible impact on the semiconductor industry [4].
Much of the experimental and theoretical work is trying to understand possible mechanisms
of ripple formation with different substrates and laboratory conditions (see Chapter 3) for
low to medium energies bombardment of semiconducting materials).
3
Chapter 1. Introduction
1.2 Physics Motivation
Ion beam irradiation of materials produces atomic damage, consequently changing the initial
structure of the material. The incident ion beam interacts with atoms of the bombarded
material giving rise to energy loss processes and to the slowing down of ions. This rapid
dissipation of the initial kinetic energy of the incoming ions within the near surface region of
materials has been described in terms of individual ion-atom impacts [1,18]. A major challenge
emerged when a macroscopic description was desired. Consequently, continuum models have
been proposed based upon a large number of atomic collisions within a certain volume [3].
Thus in a sense, large space-time scales access to macroscopic observable phenomena may in-
deed be possible. A coarse-grained approximation of the atomic damage has been established
as a natural and efficient way to describe the ion implantation process [19].
Thecontinuum theory constructed by Bradley-Harper (BH) [3] and later revised by Makeev-
Cuerno-Barabási (MCB) [20], revived interest in the generation of surface ripples by ion im-
plantation. Experimental work on the formation of patterns indicated the possible inde-
pendence of the material for the formation of surface structures, promising numerous ap-
plications. However many of these experiments reveal other physical effects not reproduced
by Bradley-Harper type theories [21]. These deficiencies were later corrected with the intro-
duction of general effective models [22,23,24,25]. Coupled two-field models describe complicated
materials highlighting the generation of ripples, dots and holes that had not been previously
accounted for [26,27]. Therefore not only can surface structures be obtained in semiconductors,
but also in metals and multi-elemental targets (alloys) [28], which could in principle lead to
possible generalizations.
In particular, the recent work of Muñoz-Garćıa et al. [23] has introduced a formally derived
theory of surface patterning based upon a coupled two-field model. It has been pointed out
that this introduces natural occurring mechanisms which other theories have incorporated
4
Chapter 1. Introduction
by an ad-hoc method. The study of this model within an effective one-field approximation
brings about additional non-linear terms that accounted for other phenomena observed in
experiments [29,30].
For this work, the formation of surface ripples is reviewed from Bradley-Harper type the-
ories and extensions thereof. Known to be a first approximation, where the surface evolution
is known to erode based upon the energy deposition function and a surface geometry depen-
dence. Along this, careful considerations of the experimental conditions is highlighted; being
the ion energy, ion type, angle of incidence, fluence and target material important in our
studies. Other coupled two-field models advanced by Shenoy-Chan-Chason [24] and Bradley-
Shipman [25,31] are known to encompass a higher parameter variation, as these models take
into account the description of multi-elemental targets.
The principal interest is to understand the physical processes that occur at the near
surface region of the implanted material. The production of various morphologies is explored
and believed to be explained within the mentioned theoretical models.
1.3 Interdisciplinary Sciences
The dynamics of pattern formation in material sciences is a well known area for study, often
in simple systems where generalizations to other fields of science are expected. In the field
of materials science, some growth processes are believed to be governed by simple universal
laws [32]. These are often explained by simplified models where external induced dynam-
ics, out-of-equilibrium phase transitions and reaction-diffusion equations can play the role
of an instability generator [33,34]. Yet actual physical process are extremely complicated be-
cause of their nonlinear character as a consequence of their interactions. These phenomena
observed from nano-structures to macro-structures and even to entire galaxies in the uni-
verse has greatly interested scientists over the last sixty years. Because of the cross-over
5
Chapter 1. Introduction
into other branches of science, pattern formation can be investigated by common analytical
techniques [19,35,36], simplifying their possible descriptions and further motivating their study.
Outside the realm of nano and micro-scale materials science, patterns can also appear
in other cases of everyday experience. For this we turn to macroscopic systems, with scales
ranging frommm to km lengths. Patterns appearing at the macro-scale, include those of sand
dunes and mountains; see Figure 1.1 for an image of sand dunes on Mars (a) and mountains
on Earth (b). These, are related to changes of the weather (pressure and currents) and
the effects of transport properties of the material [37,38]. Interestingly transport phenomena
mechanisms have also been utilized to describe snow crystals (c) and the coffee-ring effect
(d). [39] In many cases, these phenomena have been explained by common continuum models,
and experimentally reproduced utilizing controlled conditions like temperature, pressure and
concentration.
In the animal kingdom, the appearance of patterns on the fur of animals also occurs;
stripes in zebras and tigers, spots on jaguars and cheetahs, hexagons on giraffes and dots on
chameleons (see Figure 1.2). The interesting aspect of these patterns has become a paradig-
matic issue arising from continuum models often associated with instabilities [34]. Initial
studies were performed by British scientist Alan Turing, who coined the word “morphogen-
esis” associated with reaction-diffusion equations [40]. Numerical simulations have been able
to reproduce similar morphologies as those observed in animal fur [33]. Pattern formation in
plants also occurs, as geometrical structures resembling mathematical functions. For some of
these are often related to fractals and circular ordering alike broccoli and sunflower kernels
for example (see Figure 1.2).
These interesting aspects of pattern formation has attracted interest in the study of scale
invariance for out-of-equilibrium systems. Much work remains to be done and the unique
opportunity of study for pattern formation processes has been highlighted recently; general
processes appear to be of universal character [32,36,41,42].
6
Chapter 1. Introduction
(a) Mars’ sand dunes observed from NASA’s
orbiter. A barchan structure formed from
erosion and the motion of material.
(b) Skiing Sochi, mountain terrain observed
from space. This fractal structure repeats
itself within the km scale.
(c) A six-fold radial symmetry snow crystal. (d) Coffee-ring effect.
Figure 1.1: Common natural pattern formation on macroscopic scales. (a) Sand dunes form-
ing in Mars by erosion and motion of surface material from NASA’s Mars Reconnaissance
Orbiter on July 30, 2015, © NASA/JPL-Caltech/University of Arizona. (b) Skiing Sochi,
a view of the town of Krasnaya Polyana and the ski facilities for the XXII Olympic Games,
NASA’s Earth Observing-1 (EO-1) satellite on February 8, 2014, © NASA Earth Observa-
tory image by Jesse Allen and Robert Simmon. (c) Snow crystals form when water vapor
converts directly into ice without the liquid phase © Kenneth G. Libbrecht, California Insti-
tute of Technology. (d) A coffee-ring effect is observed when coffee grounds particles diffuse
on a surface © Google Images.
7
Chapter 1. Introduction
(a) (b)
(c) (d)
Figure 1.2: Common natural pattern formation on macroscopic scales. Living organisms
(animals and plants) also present patterns upon their growth. Animal on their skins and
plants on their leaves and seeds; (a) zebra with stripes, (b) jaguar with spots, (c) Romanesco
broccoli’s fractal structure, and (d) sunflower kernels with a semi-dotted circular pattern.
© Google Images.
8
Chapter 2
Surface and Interface Growth
Surfaces and interfaces are important part of everyday life, the former being associated with
the uppermost layers of physical objects while the latter concept links them together at
their borders. The physical description of surfaces and interfaces alone is a topic of current
research, as Barabási and Stanley have questioned: “How can we describe the morphology
of something that is smooth to the eye, but rough under a microscope?” [32]. Then in this
particular case, by examining the topmost layers of physical materials, we get a general
insight on the physical processes that occur during their growth.
This chapter provides a few key elements utilized in the study of surfaces and interfaces.
We introduce in section §2.1, initial concepts. A proper definition of surfaces and interfaces
along with typical physical measured quantities is mentioned, i.e., the average surface height
and the global roughness is reviewed.For each of these quantities, the growth dynamics
may be defined statistically by functions, i.e., power functions that change with time and
system size. In section §2.2, a brief overview of common surface modeling techniques by
considering experimental results is given. Lastly section §2.3, mentions a general overview
of the approach of surface growth by the free energy functional, motivating the study of
continuum models. Fully developed in the chapter on theories of ion-induced surface growth.
9
Chapter 2. Surface and Interface Growth
2.1 Definitions
In the case of experimental and theoretical studies, one begins by defining surfaces and their
associated interfaces through single-valued functions. From a mathematical point of view, a
suitable approximation is to consider a surface as a continuous height function [32]: h(x, y, t).
Therefore, the height value of the surface is defined from a two-dimensional plane system
that changes in time (see Figure 2.1).
x
y
z
h(x,y,t)
h(x,t)
x
In
te
rf
a
c
e
 P
ro
fi
le
∂h ⁄∂t
Figure 2.1: (Left) In growth experiments, the surface height value increases in the z-direction
with respect to an initial flat configuration. A two-dimensional surface may be mapped to
a one-dimensional function by a one-dimensional horizontal or vertical scan. Example of an
individual profile scan performed on the x-direction (see image on the right).
As a simple system (see Figure 2.2), take for example an initially flat surface, where an
increment in the surface height function is modeled by the addition of individual particles
being dropped at random positions (x, y) and at time t. This deposition process may be
expressed as a partial differential equation of the height function h = h(x, y, t), which evolves
dynamically in time:
∂h(x, y, t)
∂t
= η(x, y, t) (2.1)
where η(x, y, t) on the right hand side describes a random deposition process, modeled after
a Gaussian white noise with zero mean and uncorrelated in space and time. This stochastic
differential equation describes the surface evolution with respect to a deposition process that
10
Chapter 2. Surface and Interface Growth
occurs for example in growth experiments of thin-films, crystals and biological cells [32]. This
defines a height function which describes the surface or interface in time as further particles
are deposited on or attached to.
0 50 100 150 200 250
1.5
2.0
2.5
3.0
X(a.u.)
1
In
te
rfa
ce
 P
ro
fil
es
 (a
.u
.)
1.5
2.0
2.5
3.0 2
1.5
2.0
2.5
3.0 3
Figure 2.2: Schematic representation of surface atomic deposition. (Left) Simulation of ran-
dom spherical particle deposition on a two-dimensional surface. (Right) A few vertical profile
scans (from top to bottom) are performed on the surface, illustrating a general overview of
the surface at different position cuts.
In addition, certain physical phenomena are roughly approximated by implementing
interacting rules on the surface [43]. That is; general physical processes that are known
to be present during surface growth may be included by probability and/or conservation
rules [44,45,46]. In ion bombardment experiments for example, surface effects include those of
erosion [47], relaxation [48,49] and transport [50], modeled by one or more terms in a continuum
equation.
In the mathematical description of surfaces, marcroscopic physical observables are often
studied from their statistical properties of that particular growth system. The global rough-
ness w(t) value, also labeled as the interface width, is defined as an average height function
above an initial flat surface (x − y plane for a 2D system). Strictly speaking, the interface
11
Chapter 2. Surface and Interface Growth
width is defined as the RMS fluctuations in the height with respect to time and written
as [20,32]:
w(t) =
〈 1
L2
∫
[h(~x, t)− h̄(t)]2d~x
〉
(2.2)
where w(t) defines the average growth of the surface at a time t, and h̄(t) is the average
height function which is written as:
h̄(t) =
1
L2
∫
h(~x, t)d~x (2.3)
where this integral is taken over an area size L2 of the system of study, L × L for a 2D
system. For simple systems, the interface width may follow simple scaling laws that depend
on the physical process that occur during growth. In some cases; growth processes develop
rough surfaces and are modeled after continuum equations like random deposition (RD) [32],
Mullins-Herring (MH) [48,49], Edwards-Wilkinson (EW) [51], Kardar-Parisi-Zhang (KPZ) [52]
and other look-alike models. These models will be reviewed in Chapter 4, where particular
growth equations of ion induced surface phenomena are similar as those mentioned.
2.2 Modeling
Our understanding of growth phenomena begin from simple approximations, from discrete
atomic [1] approximations to the continuum limit [3]. In the case of ion-atom interactions,
the use of discrete models is seen to be adequate in the treatment of ion sputtered surfaces
(where atomic collisions between ions and atoms of a target material results in the gener-
ation of atomic displacements resulting in surface erosion). For our particular purpose at
hand, ion-atom interactions may be modelled by individual and multiple binary collisions.
Consequently, surface growth by particle deposition (inverse of surface erosion) is a very
12
Chapter 2. Surface and Interface Growth
simple phenomenon where a coarse-grained physical approximation may be employed. This
bridges the gap between microscopic rules and macroscopic surface evolution phenomena.
Growth dynamics of a two-dimensional surface can be modelled by a simple conservation
equation arising from the so called hydrodynamic approach. One and two-field models have
been implemented by considering the effects that arise from the ion-atom interaction of
materials [32]. In general, the description of surface growth through stochastic differential
equations is often achieved whenever external perturbations on a target material exists.
For ion irradiation of surfaces, changes of the surface height and roughness are exper-
imentally obtained from ex-situ atomic force microscopy (AFM). These experimental data
then may be used as input parameters into continuum models hopefully predicting their
early and late behaviors. This allows a comparison between experimental and theoretical
studies.
2.3 Ginzburg-Landau Approach
An important derivation method for surface growth models is taken by considering a Ginzburg-
Landau free energy functional [53]. This is a coarse-grained approximation of a discrete system
in which a continuum model is developed helping to bridge the gap between atomic processes
to macroscopic observed effects.
The basic idea behind this description is due to the external induced effects which tries
to minimize the surface height value e.g., in a diffusive atomic process. This approach for
example motivates the study of continuum model of ion sputtered surface due to a dissipation
of the initial kinetic energy with ion-induced surface effects.
The time-dependent Ginzburg-Landau (TDGL) equation is written as [53]:
∂h
∂t
= −δL0
δh
+ η (2.4)
13
Chapter 2. Surface and Interface Growth
where ∂h/∂t describes the evolution of the surface height through minimization of its height
value through a Landau free energy functional; L0. This free energy functional may be
expressed in terms of a curvature dependent term and a temperature-dependent relaxation
mechanism term (second term):
L0 =
∫
dd~x
[
1
2
ν(∇h)2
]
+
∫
dd~x
[
1
2
B(∇2h)2
]
=
∫
dd~x
[
1
2
ν(∇h)2 + 1
2
B(∇2h)2
]
(2.5)
Substituing the above relation into the TDGL equation leads to the growth equation:
∂h
∂t
= ν∇2h− B∇4h + η (2.6)
This equation is a Bradley-Harper type equation for surface growth (see section §4.2, with
ν > 0). Consequently, this stable linear equation smooth out a surface by two terms, a
diffusion term with a surface tension coefficient ν and a relaxation term with a temperature
dependentcoefficient B. In contrast, the unstable (ν < 0) equation predicts a surface
structure that grows without bound.
However, during surface growth as considered in the Bradley-Harper type model, de-
veloping fronts grow in the normal direction of the surface, the emergence of a nonlinear
geometrical term may be applied by considering the Kardar-Parisi-Zhang contruction [52].
This growth process is known to occur in the normal direction and given by the geometrical
approximation (see Figure 2.3):
δh
δt
≈ ∂h
∂t
= v
√
1 + (∇h)2 ≈ v
[
1 +
1
2
(∇h)2
]
= v +
λ
2
(∇h)2 (2.7)
14
Chapter 2. Surface and Interface Growth
v td
dh
h(x,t)
x
g
-∂h ⁄∂t
h
x
Figure 2.3: Representation of surface growth with an initial surface curvature. (Left) Surface
growth in the normal direction adds a nonlinear term to the equation of motion. (Right)
Consecutive profile cuts of the surface which develops shock waves due to gradients of the
surface. Adapted from Kardar et al. [52] original article.
In short, incorporating both linear and nonlinear terms and after a change into the comoving
frame, the final form of the growth equation is written as:
∂h
∂t
= ν∇2h− B∇4h+ λ
2
(∇h)2 + η (2.8)
where this equation is also called the noisy Kuramoto-Sivashinsky (nKS) equation and de-
scribes the height profile of a surface that undergoes fluctuations. The noise term η(x, y, t)
is a Gaussian white noise with zero mean and uncorrelated in space and time:
〈η(~x, t)〉 = 0 & 〈η(~x, t)η(~x′, t′)〉 = 2Dδ(~x− ~x′)δ(t− t′) (2.9)
When ν < 0, the system is linearly unstable, describing the surface evolution of systems that
initially undergo an exponential growth before saturation. This being succesfully applied
to ion-sputtered of surfaces represented by a faster erosion of troughs than crests. Further
analysis is carried out in the theory chapter (see section §4.4) while other correction to both
linear and nonlinear terms are given in Appendix C for this particular example.
15
Chapter 3
Experimental Approaches: A Review
The systematic study of surface patterning by ion beam sputtering has developed consid-
erably since its conception. This is partly due to their possible applications in integrated
circuits and catalytic applications. Aside from technological impacts, the vast number of
experiments on the subject has played an important role in the advancement of theories,
from the early Bradley-Harper linear theory to associated non-linear theories. Irradiation of
silicon-based materials falls under the common targets used in the study of pattern forma-
tion. Further progress has been achieved by adding natural occurring mechanisms leading
to the Muñoz-Cuerno-Castro (MCC) and Bradley-Shipman (BS) coupled two-field models.
The understanding of ion-induced pattern formation has been successfully achieved in
large due to a large set of experiments. Further work is continually being performed, high-
lighting general interest in the generation of all kinds of shapes. In this chapter, we review
a set of initial experiments that paved the way in the exploration of ion-induced effects by
ion beams (§3.1), followed by a summary of experiments performed at normal and oblique
incidence (§3.2 and §3.3, respectively) and in particular for high-energy ion bombardment
of Ti-based targets (§3.4).
16
Chapter 3. Experimental Approaches: A Review
3.1 Initial Framework
The first study of ripple formation was due to Navez et al. [54] back in the early 1960s. In their
work, 4-keV ionized air was accelerated onto glass surfaces at 30° angle forming corrugated-
ripples. What was initially intended as a surface cleaning technique led to the discovery of
surface ripples. Further experimental work was able to produce other shapes in all sorts of
materials, leading to revolutionary insights for ion-atom interactions.
The easy accessibility of low-energy ion beams allowed the systematic study of a large
number of cases. Parameters such as the angle of incidence, energy and fluence played a
major role in early studies. The effects often appeared to be independent of ion and target
material combination. This is believed to happen as long as the ion energy is in the nuclear
stopping power range forming a thin layer of amorphous material [30]. The surprise emergence
of surface ripples in non-amorphous materials required the revision of some general concepts,
but it is believed to be described by other theories like the recently developed coupled two-
field model of Bradley-Shipman [25].
Two main categories exist in the formation of surface structures: (I) Normal incidence
experiments with co-deposition of impurities and (II) oblique incidence angles with and
without co-deposition of impurities. In these cases different shapes can be obtained. A brief
summary of important results found in the literature is mentioned below, silicon being the
best example with many results [26].
This is a rich field of study within the science of materials, as many questions have
not been fully resolved. It was later found that the formation of surface dots, holes and
ripples on silicon at normal incidence could only be explained considering surface impurities.
Experiments of co-deposition of impurities have also been well accounted for by theoretical
descriptions [24,25,26,31].
On the other hand, the formation of patterns on metallic surfaces has only been studied
17
Chapter 3. Experimental Approaches: A Review
recently [55,56,57,58,59]. Although these materials were expected to behave similarly, they be-
have differently [21]. One impediment was the experimental observation of re-crystallization
of the damaged region after ion irradiation. This in some cases revealed the lack or forma-
tion of a surface structure. This was highlighted only recently by the possible formation of
chemical compounds of the ion and target combination [25,31].
3.2 Normal Incidence
A formal study of normal incidence experiments was initiated by Facsko et al. [60] where irra-
diation of GaSb(100) by a 4.2 keV Ar+ ion beam revealed ordered dots. Regular hexagonal
dot structures were formed, as a preassumed Bradley-Harper mechanism. A simple promise
of this work was its possible generalization to other materials like InSb and Ge whose surfaces
are seen to acquire similar behaviors. The authors argued that preferential sputtering of Sb
existed leading to the accumulation of Ga atoms on the surface, forming the pattern. This
inspired further exploration at normal incidence with other commonly known materials [61].
In the work of Gago et al. [62] a 1.2 keV Ar+ ion beam is incident on Si(100) at normal
incidence. The formation of ordered hexagonal dots closely resembling those seen in the
bombardment of GaSb came as a surprise. In this case, the authors believed that the for-
mation was due to surface erosion, a coupling between diffusion and relaxation mechanisms,
in a similar fashion to the Bradley-Harper instability.
In recent developments the formation of patterns at normal incidence is found to depend
on the substrate unintentional contamination [63,64,65]; experiments with the deposition of
foreign atoms were able to replicate early experiments in the formation of dots on Si [66].
These experiments were not understood until the effect was attributed to impurities present
on the surface of the sample under irradiation. This result led to an important breakthrough
of Bradley-Harper type theories since it predicted surface ripples at all angles of incidence
18
Chapter 3. Experimental Approaches: A Review
including near normal incidence. This phenomenenon is seen to have a degree of universality
in the sense of material and ion type combination.
Furthermore, experiments at normal incidence together with substrate rotation led to
the formation of nano-dots and holes. This has been studied from in correspondance to
binary compounds such as GaSb and InP [61]. A formally derived theory including substraterotation has been elaborated in terms of the two field model of Muñoz-Garćıa and collab-
orators [67]. Also binary target materials have been studied theoretically by Bradley and
collaborators [68,69,70]. Both of these theories are reviewed in the next chapter, corresponding
to generalized coupled two-field modeling of IBS experiments.
3.3 Oblique Incidence
For oblique incidence studies the production of surface ripples is easily recognized. In the
well-known work of Carter and collaborators [71], surface structures were produced by Ar
ion bombardment of silicon surfaces. Subsequent work include the variation of angle and
energy [72,73,74,75]. For example, Ar+ irradiation of silicon surfaces for energies ranging between
10-40 keV at 45° angles formed well ordered ripples [76]. Ripples with increasing wavelength
with respect to the ion fluence were attributed to the accumulated atomic damage on the
near surface region. The failure to obtain surface ripples at normal incidence became an
important aspect of this work (possibly by the presence of an initial rippling structure [77]);
thus ripples were believed to be absent up to a certain critical angle of incidence, θc.
The large amount of work put into the study of surface ripples appears in many excellent
reviews, like those of Carter [4], Makeev et al. [20], Chan and Chason [21] and recently Muñoz-
Garćıa et al. [26]. These are usually focused in low and medium energies using noble ions in
semiconducting materials. It is important to note that these reviews focus on the formation
of ripples in silicon and associated binary semiconductors, due to their potential technological
19
Chapter 3. Experimental Approaches: A Review
applications.
Meanwhile in the case of metallic materials, for instance the initial work of Valbusa and
collaborators [55,56] and recently by Ghose and collaborators [57,58,59], show the formation of
ripples. Further important results include ripple formation in single crystalline and for thin
metallic films targets [58] (see also the review of Chan and Chason, Ref.[21]).
A brief overview of ion irradiation in semiconducting materials at 45° angle incidence
is summarized in Table 3.1. These few experiments being performed at low and medium
energies giving rise to nano and micrometer wavelenghts.
Ion Type Material Angle (degrees) Ion Energy (keV) Ripple Wavelength (µm) Ref.
Xe+ Si 45° 10-40 0.4 [76]
Ar+ SiO2 45° 0.5-2.0 0.2-0.55 [78]
Au2+ SiO2 45° 1800 2.5 [79]
Table 3.1: Ripple formation on various non-metallic materials upon IBS experiments at 45°
angles.
As mentioned previously, the presence of impurities, either by accident or intentionally,
affects the formation of surface structures. This also happens when bombarding with oblique
incidence angles. Madi et al. [80,81] performed extremely clean experiments, reaching the
conclusion that a critical angle of incidence existed for which ripple formation occurs. A
morphological diagram was constructed; from normal incidence to 48° there is a flat stable
region, from 48° to 85° parallel mode ripples form, and finally for θ > 85° perpendicular
ripples exist. This same conclusion appears in other studies [82,83], a cutoff angle in the
formation of surface structures. This display is reminiscent of a continuous phase transition.
For pattern forming systems, this is of type II [34], where the ripple wavelength diverges at a
critical angle of incidence.
20
Chapter 3. Experimental Approaches: A Review
Meanwhile in the case of compound materials, a segregation of the atomic phases re-
verses this behavior allowing pattern formation at normal and low incidence angles. This
morphological arrangement has been taken care of in the theories of Bradley-Shipman and
collaborators [31]. This determines the surface composition and the dependence of the surface
sputtering of the target material. This is seen to occur in semiconductor compounds and
other materials that may form compounds upon interacting with an ion beam.
3.4 High-energy Irradiation of Ti
A majority of studies found in the literature are associated with low energy irradiation. This
leaves high-energy phenomena unexplored. As mentioned previously, low energy irradiation
reduces penetration depths, so processes occur on the top most layers of a material. In the
case of high energy irradiation (dE/dxelec ≥ 1.0-keV/nm), high penetration depths implies
small sputtering yields [85,86]. For high energy experiments, ions are implanted into the ma-
terial, therefore the use of the term ion implantation. This in fact allows other important
mechanisms like: ion-induced viscous flow, plastic deformation and mass redistribution.
In terms of the model developed by Trinkaus and Ryazanov [85], a flow of material is
produced during implantation by heating of the material, along with a relaxation of the
atomic stress producing a fluid-like state that freezes soon after the ion beam is removed. In
simple words, this viscoelastic flow of material occurs when electronic excitations of the atoms
are coupled to phonons producing a rise of extremely high temperatures in the material.
Additionally, the production of shear stress relaxes within cylindrical thermal spikes regions
induced by thermal dilatation and a freeze-in afterwards [87]. This behavior is noticeable for
irradiation at high energies leading to high displacement of the irradiated material [88,89].
In the case of Au ion implantation, ion energies in the hundred MeV’s (around or above
1.0-MeV/u) have been used to study heavy-ion interaction with metallic materials [90]. In the
21
Chapter 3. Experimental Approaches: A Review
study of Mieskes et al., Ti substrates were implanted with 109-MeV, 230-MeV and 270-MeV
Au ions with different charge states. The main results point out an increase in sputtering
yields of 4.3 atoms/ion to 7.6 atoms/ion resulting from 11+, 13+ and 15+ charge states. These
have been attributed to the electronic excitations produced during the slowing down of ions.
On the other hand, lower energy implantations like 1.0-MeV Au ions (∼0.005 MeV/u) belong
to a region where nuclear stopping dominates, accounting for higher sputtering yields from
nuclear collisions [91]. Interestingly 1.0-MeV Au ions remove about 6.1 atoms/ion from Ti [92],
similar to those measured in very high-energy implantations.
In contrary, other associated experimental work in titanium is particularly of interest
at low energies for biomedical applications (see section §1.1). In the work of Riedel and
collaborators [6], 7-keV and 1.1-keV Ar irradiation of Ti-6Al-4V-ELI at normal incidence,
etched surfaces are rough, but nano-scale ripples appear with an average wavelength of
λ = 20 nm. Additional low and medium energies include the work of Fravventura [93], where
irradiated Ti with 10-keV Xe ions at an angle of incidence of 60◦ with a fluence of 7 × 1017
ions cm−2. The formation of ripples is evident with nano-scale wavelengths and roughness
values. The appearance of ripples at normal, oblique in the nano-scales at high fluence
experiments are indeed comparable to low and medium energy studies.
It is hoped that the present our work on titanium can shed light on the physical processes
occurring during high-energy heavy ion implantation. Adquired surface roughness of tita-
nium and their alloys may become good examples of surface modifications at the nano and
micrometer scale. The use of continuum models utilized in similar pattern forming systems
may help us to describe the macroscopic ion induced bombardment of titanium and its alloy
Ti-6Al-4V at high energies even in the presence of additional surface effects.
22
Chapter 4
Theories of Ion Induced Surface
Growth
The formation of surface structures is known to be influenced by the collective behavior
of ions colliding with the atoms of the target material. This is understood in terms of a
continuous height function from a theory developed by Bradleyand Harper [3]. In this model,
Bradley and Harper utilized the Sigmund result [18] in order to propose an approximate linear
partial differential equation for the surface evolution. The height function determined by
the Bradley-Harper (BH) model depends on the geometry of the experiment and on the
properties of the target material. A comparison with experimental results revealed certain
inconsistencies with this theory; further improvements were needed. The starting point for
the study of surface structures has been this pioneer work.
It is by now known that structures formed by ion beams include ripples, dots, holes and
other quasi-periodic structures. The use of continuum equations to describe the experimental
findings has been fruitful and has been improving since the proposal by Bradley and Harper.
Meanwhile, other Bradely-Harper type models are considered as improvements with similar
asymptotic behaviors, including the well-known anisotropic Kuramoto-Sivashinsky (aKS)
23
Chapter 4. Theories of Ion Induced Surface Growth
equation [20,94,95]. Its non-linear term corrected deficiencies of the Bradley-Harper linear the-
ory, accounting for experimental observations.
In this chapter, we give a brief overview of the main theories of ion-induced pattern
formation. We start with a review of the Sigmund theory of sputtering (§4.1), followed
by the single-field theory of Bradley-Harper (§4.2), including higher-order and non-linear
model extension proposed by Makeev-Cuerno-Barabási (§4.3), then give a comparison to the
associative model of the anisotropic Kuramoto-Sivashinsky (§4.4), follow up with a review
of the recent developed coupled two-field model of Muñoz-Cuerno-Castro (§4.5) and that
of Bradley-Shipman (§4.6), finally a few numerical simulations of single-field models are
performed (§4.7). In all of the mentioned models, they have been able to reproduce ripples
and other morphologies by varying certain parameters. Their applicability is formally due
to contributing factors in the surface evolution of surfaces during ion irradiation.
4.1 Sigmund Theory of Sputtering
A formal theory of sputtering originated in the late sixties when Thompson [96] published an
article explaining that surface erosion results from the ion bombardment of materials. This
effect had been observed and studied many years before by Navez et al. [54]. Within a year
after Thompson’s work, Peter Sigmund published an article [18], deriving a theory of atomic
sputtering from the collisions of ions within a target material. The Fokker-Planck derived
formula of atomic transport explained many experimental results obtained previously, in
terms of the relationship between the number of incoming ions and those that are expelled
from the material. This relationship is known as the sputtering yield (Y ), also called the
erosion coefficient [1]:
Y =
Ne
Ni
(4.1)
24
Chapter 4. Theories of Ion Induced Surface Growth
where Ni and Ne are the average number of incoming ions and emitted atoms, respectively.
This relation depends on the implantation energy distribution and on the parameters of
the target material. The value of the sputtering yield Y (E) is then given in terms of the
deposited energy distribution in the near surface region of the implanted material and given
by the formula:
Y (E) =
3
4π2
F (E)
ΛU0
, (4.2)
where F (E) is the deposited energy distribution that depends on the incident energy, as-
sumed generally as a Gaussian distribution; Λ is a constant that depends on the atomic
density and on the effective interaction potential and U0 is the surface binding energy of the
material under bombardment. Consider Figure 4.1(a): an ion beam enters on the x−z plane,
traveling a distance a with stragglings σ and µ in the parallel and perpendicular direction,
respectively. A schematic representation of the deposition function is given by ellipsoidal
contour lines of Figure 4.1(b).
In the case of non-planar geometries, the ion deposition function depends on the local
surface curvature. The sputtering process of a surface now depends on the geometry of the
surface, with positive and negative curvatures [3] (see Figure 4.2). Thus, higher exposed areas
may not easily erode in comparison to valleys. This is described by the larger distances (solid
lines) that must travel atoms to the surface point A’ (concave geometry) in comparison to
the point A (convex geometry).
In general, the form of the deposition energy distribution depends on the material and
parameters of the experimental study. Low energy (10eV -10keV) ions may be described by
a Gaussian distribution:
F (~r) =
ǫ
(2π)3/2σµ2
exp
[
− z
′2
2σ2
− x
′2 + y′2
2µ2
]
(4.3)
25
Chapter 4. Theories of Ion Induced Surface Growth
q
z
y
x
ion
beam
a)
q
n
z
x
b)
sm
asurface
n
Figure 4.1: Illustration of a coordinate system for surface ion implantation. a) In the case
of an ion beam entering in the x− z plane, the x− y plane defines the surface of the target
material. b) The incident ion beam penetrates the surface a distance a with straggling lengths
σ and µ in the parallel and perpendicular direction, respectively. An erosion (growth) of the
surface is represented by height decrease (increase) on the z-axis. The angle of incidence is
taken to be with respect to the surface normal of the surface.
where ǫ represents the ion energy, σ and µ are the ion distribution width in the parallel and
perpendicular directions, respectively. As the sputtering yield depends on the energy, two
regimes exist; one due to nuclear interactions and one due to electronic interactions. Low
energy implantations are well described by nuclear interactions and those at high energies
by electronic interactions [1,2]. Atomic displacements occur due to nuclear collisions while
ionization is due to ion-electron collisions. The contribution from both processes may lead
to both erosion and diffusional processes near the surface [3].
The calculated energy dependence of surface erosion of Ti by Au ion bombardment is
given in Figure 4.3. For comparison the Matsunami et al. [97] and Yamamura-Tawara [98,99]
semi-empirical calculations at normal incidence are shown (see left image of Figure 4.3). The
sputtering yield increases as a function of the energy reaching a maximum before decreasing
for high energies. This is consistent with high energy ions penetrating higher depths (see
right image of Figure 4.3).
The sputtering yield dependence on the angle of incidence is also explored. TRIM simu-
lations (binary collision Monte Carlo Method, SRIM-2008.04 [91]) and computed values using
26
Chapter 4. Theories of Ion Induced Surface Growth
ion beamion beam
a) convex b) concave
A
A’
Figure 4.2: The schematic of the origin of surface erosion for non-planar surfaces. Sur-
faces erode according to the energy deposition functions where convex (concave) erode faster
(slower). This surface instability is generated by erosion due to ions travelling smaller dis-
tances at A in comparison at A’. This diagram has been adopted from Makeev et al. [20]
original article.
a closed form equation (Yamamura-Tawara formula) are shown in Figure 4.4. As the angle
of incidence is increased the sputtering yield increases up to a maximum value then drops
off at near grazing angles [100]. This angle of incidence dependence has been corrected by
utilizing an inverse cosine function (see Appendix A).
During implantation, ion-atom collisions create atomic displacements [101], thus generating
vacancies; if the energy is sufficient atoms will be displaced with no option of returning to
their initial configuration. This process moves atoms in the general direction of the ion beam.
The ion-target mass ratio of four in this work induces large erosion yields. This is seen from
TRIM simulatios, where Au ion implantation into Ti gives a shift of the deposited ion
distribution toward the surface. Furthermore Au ion implantation also erodes implantedAu
as well, as shown in the implanted profile analysis [92]. The combination of ionic displacement
and erosion creates new surface morphologies. A description of the effect of atoms colliding
with the surface of a target material is possible with a continuum model if the length scale
27
Chapter 4. Theories of Ion Induced Surface Growth
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
1
2
3
4
5
6
7
8
Sp
ut
te
rin
g 
Yi
el
d 
(a
to
m
s/
io
n)
Energy (MeV)
 Matsunami et al. (1984)
 Yamamura et al. (1996)
Figure 4.3: Graphic representation of the energy dependence of the sputtering yield for Au
ions into Ti (left plot). Above 0.5-MeV yields decrease as the ion energy is increased. Energy
contour plot for 1.0-MeV Au ions into Ti at 45◦ (right plot).
is comparable to that of the penetration depth [53]. With a continuum approach, numerical
simulations of surface morphology are similar to those observed experimentally, strongly
advocating its use. Note that the hydrodynamical approximation in numerical simulations
disregards crystal structure.
4.2 Bradley-Harper Model
In the theory of Bradley and Harper [3], the formation of surface ripples is generated by
a morphological instability produced by surface erosion and a relaxation mechanism. The
erosion rate of a surface is approximated by a continuous equation derived by considering
erosion in the direction normal to the surface and a factor of curvature [103]:
∂h(x, y, t)
∂t
≈ −v(θ, Rx, Ry)
√
1 + (∇h)2 (4.4)
In this equation, h = h(x, y, t) describes the surface height function as evolving from an
28
Chapter 4. Theories of Ion Induced Surface Growth
0 10 20 30 40 50 60 70 80 90
0
10
20
30
40
50
60
70
80
90
100 SRIM Simulation
 Yamamura et. al. (1996)
Sp
ut
te
rin
g 
Yi
el
d 
(a
to
m
s/
io
n)
Angle( ) [Degrees]
0 10 20 30 40 50 60 70 80 90
0
10
20
30
40
50
60
70
80
90
100 SRIM Simulation
Sp
ut
te
rin
g 
Yi
el
d 
(a
to
m
s/
io
n)
Angle( )[Degrees]
Ti
Al
V
Figure 4.4: Angle of incidence dependence for the sputtering yield for 1.0-MeV Au ions into
Ti (left) and Ti-6Al-4V (right) substrates. The Yamamura-Tawara semi-empirical formula is
the solid line and the dots are the SRIM/TRIM simulation, no theoretical line is yet possible
for alloys. The maxima of the sputtering yield of Ti atoms is located both at θ ≈ 85◦.
initial assumed flat configuration, where θ is the angle of incidence with respect to the
surface normal, and Rx and Ry are the radii of curvature of the local surface. In this case,
the curvature dependent surface erosion assumes that troughs erode faster in comparison
to crests [3]. As a consequence the Bradley-Harper (BH) model is strongly dependent on
the surface geometry during the erosion process. In this continuum approach it is assumed
that (1) the surface curvature is much greater than the ion penetration depth and (2) the
surface curvatures may obtain their maximum values either in the x or y directions excluding
cross terms [103]. A Taylor expansion of the geometrical square root factor in equation (4.4)
has been performed, thereof, the height evolution of ion-sputtered surfaces is given within a
linear approximation by [3]:
∂h
∂t
= −v0(θ) + γ(θ)
∂h
∂x
+ νx(θ)
∂2h
∂x2
+ νy(θ)
∂2h
∂y2
− B(T )∇4h (4.5)
where the coefficients of the Bradley-Harper equation have been well approximated by Ma-
keev and collaborators [20] and given as:
29
Chapter 4. Theories of Ion Induced Surface Growth
v0 = Fc (4.6)
γ = F
s
f 2
[
a2σa
2
µc
2(a2σ − 1)− a4σs2
]
(4.7)
νx = Fa
a2σ
2f 3
[
2a4σs
4 − a4σa2µs2c2 + aσaµs2c2 + a4µc4
]
(4.8)
νy = −Fa
c2a2σ
2f
(4.9)
where F is written as [104]:
F =
jǫΛa
σµN
√
2πf
exp
[−a2σa2µc2
2f
]
(4.10)
and the reduced energy deposition depths have been defined as:
aσ =
a
σ
, aσ =
a
σ
, s = sin θ, c = cos θ, f = a2σs
2 + a2µc
2 (4.11)
If one considers the incoming ion beam direction as that situated on the x − z plane,
this linear partial differential equation determines the surface height which evolves in time
according to the following terms: v0(θ) describes the angle dependent erosion of a flat sur-
face, γ(θ)∂h/∂x allows the surface to move along the projected direction of the ion beam.
The terms νx(θ)∂
2h/∂x2 and νy(θ)∂
2h/∂y2 describe the curvature dependent surface diffu-
sion, where (νx and νy) are the surface tension coefficients generated by the erosive process
along the x and y directions, respectively. Finally the fourth order term ∇4h represents
a temperature-relaxation mechanism with B(T ) being the coefficient of an Arhenius-type
temperature relation of Mullins-Herring [48,49]:
30
Chapter 4. Theories of Ion Induced Surface Growth
B(T ) =
D0γν
n2kBT
exp
[−∆E
kBT
]
(4.12)
where D0 is the surface diffusion probability constant, γ the surface free energy per unit
area, ν the areal density of diffusing atoms, n the number of atoms per unit volume, ∆E the
activation energy and T the absolute temperature. This bi-harmonic, relaxation temperature
activated term, relaxes the surface by allowing particles to move to energetically favorable
sites [32].
A linear stability analysis of the Bradley-Harper equation considering a height function
h(x, y, t) = −v0t + A exp [i(qxx+ qyy) + ω(qx, qy)t] leads to a dispersion relation ω(qx, qy) =
−iγqx − νxq2x − νyq2y − B(q2x + q2y)2. The real part; Re[ω(qx, qy)] describes the growth of
ripples along a specific direction while the imaginary part; Im[ω(qx, qy)] relates its mode
velocity on the surface (see also Appendix B). The maximum value of the growth rate is
found to be at a particular value for the wave vector given by qmaxx,y =
√
νx,y/2B associated
to a characteristic length scale, lc = 2π/q
max
x,y . As has been pointed out in many other
works [20,32,103], this describes the wavelength of surface ripples. The direction is dictated
by the greatest negative-value surface tension coefficient, a depiction of the Bradley-Harper
instability [3,20].
For the Bradley-Harper model, an unbound exponential growth of surface ripples is pre-
dicted; this mathematical result is not consistent with experimental observations. In the
work of Park et al. [95] an inherent non-linear model is studied, taking into account interface
saturation. In this case, the authors studied the behavior of the anisotropic Kuramoto-
Sivashinsky (aKS) equation being a Bradley-Harper type model due to the appearance of
the second and fourth order terms, in which the inclusion of the KPZ non-linearility, and a
non-correlated noise term supports the saturation of the interface width and the concept of
a random arrival of particles at the surface [102]. Likewise, Cuerno and Barabási [103] and then
31
Chapter 4. Theories of Ion Induced Surface Growth
Makeev and Barabási [105] have suggested the evolution of surface ripples is due to initial
rough [106,107] and undulated surfaces [108,109]. Surface roughness is often present in experi-
mental set-ups. In practice, it is common to assume that an initial flat surface exists, but
experimentally and in numerical simulations an initial rough surface is important [110].
Non-linear models have been recently considered to play important roles in the devel-
opment of features seen in experimental observations. These features were advanced in the
work of Makeev et al. [20], and predict interesting results which are reviewed in the following
section and thereafter its connection to the anisotropic Kuramoto-Sivashinsky (KS) equation.
4.3 Makeev-Cuerno-Barabási Model
Advances on the work for the description of the growth of surface ripples were obtained from
a general continuum equation considering higher linear and non-linear terms. The work of
Makeev, Cuerno and Barabási [20] considers a higher order Taylor expansion of the erosion
velocity geometrical factor along the local surface normal of the bombarded material. The
natural addition of a non-correlated Gaussian white noise accounts for the random arrival
of ions atthe surface of the material. Considering up to fourth-order terms, the equation is
written as [20]:
∂h
∂t
= −v0 + γ
∂h
∂x
+ νx
∂2h
∂x2
+ νy
∂2h
∂y2
+ λx
(
∂h
∂x
)2
+ λy
(
∂h
∂y
)2
+ Ω1
∂3h
∂x3
+ Ω2
∂3h
∂x∂y2
−Dxy
∂4h
∂x2∂y2
−Dxx
∂4h
∂x4
−Dyy
∂4h
∂y4
− B∇4h + η (4.13)
The same terms that appeared on the BH model are seen to be contributing to the
equation of motion, with the addition of higher order terms up to fourth-order, non-linearities
with coefficients λx and λy, and relaxation self-diffusion ion-induced terms with Dxy, Dxx
32
Chapter 4. Theories of Ion Induced Surface Growth
and Dyy coefficients often labeled as “ion-induced effective surface difusion”. Moreover, a
stochastic term η = η(x, y, t) is added representing the random arrival of ions on the surface
of the solid. These coefficients have been fully defined in terms of experimental parameters
and given in the Makeev et al. [20] article. The behavior of these additional terms are well
documented and seen to exhibit a behavior similar to that of the BH model. Major changes
occur when non-linear terms are included, briefly mentioned in the next section.
In this treatment, the Makeev-Cuerno-Barabasi (MCB) model considers a large set of
parameters which inhibits a careful analysis of the surface evolution. In the present case
(1.0-MeV Au+ ions into Ti), at approximately an angle of θ ≈ 45° incidence, the values
for the linear coefficients are; v0(θ = 45) = 0.016 Å/s, γ(θ = 45) = 0.387 Å/s, νx(θ =
45) = −154.193 Å2/s, νy(θ = 45) = −43.654 Å2/s, Dxx(θ = 45) = 1.05 × 10−25 cm4/s and
Dyy(θ = 45) = 1.54×10−26 cm4/s. Further analysis is presented in the discussion section (see
section §7.3), where a larger parameter space is explored, leading to a better understanding
of the underlying physics, always taking into account that this is a linear approximation.
Additional arguments given by Makeev et al. [20], Barabási-Stanley [32] and Cuerno et
al. [103] acknowledge the competition between the surface tension and its relaxation leading
to the appearance of a characteristic length scale. This length scale is usually associated
with the wavelength and given by a simple relation between the surface tension and the
self-diffusion coefficient [20]. If thermal and ionic relaxation terms are taken into account
K = B +Dxx,yy, a linear stability analysis yields a wavelength for surface ripples given by:
λ = 2π
√
2K
min(|νx, νy|)
(4.14)
In this case both temperature and ionic relaxation mechanisms contribute to the estab-
lishment of the ripple wavelength. Again taking into account the maximum of the negative
surface tension coefficient, 1.0-MeV Au ions penetrate a Ti surface with an average depth
33
Chapter 4. Theories of Ion Induced Surface Growth
a = 0.11µm, with longitudinal (σ = 0.03µm) and lateral (µ = 0.07µm) stragglings. The
calculated wavelength has a value of λ = 0.13µm, below the value of the wavelenght that is
often observed in our experiments.
It is of no surprise to observe a disagreement with measurements, since Bradley-Harper
type theories are usually applied to low-energy ion bombardment of materials. Only linear
terms of the MCB model have been considered here, which is far from being the true nature
of ion bombardement at high energies. Further mechanisms are reviewed in the discussion
section, leading to better agreement with experimental results. Non-linear terms are dis-
cussed, which produce important asymptotic effects of the surface evolution different from
those of the linear theory. These effects are seen in numerical simulations as the long-time
behavior of the surface morphology.
4.4 Kuramoto-Sivashinsky Model
During ion-beam sputtering experiments, non-linear effects have been regarded as important
mechanisms for the long-term behavior of surfaces and interfaces [94,95,102]. These effects are
seen to be due to fast developing slopes where amplitude saturation, kinetic roughening and
rotation of ripples appear [95]. These nonlinear characteristics can be seen in experiments,
and have been reproduced in numerical simulations. In comparison to the Bradley-Harper
model, an appropriate addition of non-linear terms leads to the noisy Kuramoto-Sivashinsky
(nKS) equation:
∂h
∂t
= ν∇2h−K∇4h+ λ
2
(∇h)2 + η (4.15)
where as usual the surface height, h(x, y, t) = h evolves dynamically in time according to a
diffusive term, followed by the fourth-order relaxation term, a nonlinear “KPZ nonlinearity”
34
Chapter 4. Theories of Ion Induced Surface Growth
term and the stochastic term mimicking the random arrival of ions on the surface.
For this particular model, the non-linear term models a lateral correlation along the
surface and shows up as amplitude saturation (saturation of the interface width) and kinetic
roughening (the time evolution of rough surfaces) of surface morphologies. As before, the
combination of the second and fourth order derivatives generates surface ripples but with
additional non-linear effects [32].
In fact, if one removes the fourth-order term, the Kardar-Parisi-Zhang (KPZ) [52] equation
emerges. The KPZ equation is associated with the description of the interface of non-linear
phenomena such as the burning of a sheet of paper [112], the growth of bacteria colonies [113],
the spreading of a drop of coffee [114] and many other phenomena that can be inserted into
an interface problem. The KPZ non-linear model being associated with out-of-equilibrium
systems has been succesfully utilized in many physical systems of general interest. The
mapping of surface erosion and growth phenomena with partial differential equations is an
interesting application of continuum models [32,44,45,46].
For ion-sputtered surfaces, the important equation is the anisotropic Kuramoto-Sivashinsky
(aKS) model, due to a preferential direction of the ion beam. The aKS equation is written
as:
∂h
∂t
= νx
∂2h
∂x2
+ νy
∂2h
∂y2
+
λx
2
(
∂h
∂x
)2
+
λy
2
(
∂h
∂y
)2
−K∇4h + η (4.16)
where it is assumed that the ion-beam direction is in the x − z plane, leading to the usual
terms from the Bradley-Harper and Makeev-Cuerno-Barabási models. The analysis of this
equation concurs with many non-linear behaviors, like the coarsening of structures, kinetic
roughening and the growth of rotated ripples.
In the work of Park et al. [95] and Drotar et al. [102], a method of separation between linear
and non-linear effects includes a characteristic transition time τ . This transition time has
35
Chapter 4. Theories of Ion Induced Surface Growth
been estimated from the strengths of the linear and non-linear terms [95]. Values of the times
can in principle be small, which may become an impediment for observation. Some transition
times may be too slow or too fast to be noticed, complicating the determination of linear
and non-linear effects. There is the possibility of studying its dependence on experimental
parameters, since non-linear coefficients depend on the incident particle energy, penetration
depth, angle of incidence and flux [102]. The nKS and KPZ asymptotic behaviors have been
recently studied by Nicoli and collaborators [115,116,117]. These effects include non-linear fea-
tures that are seen to appear due to a finite size of the system along with the variation of
the theoretical parameters.
The integration of the noisy KS equation was carried out in order to show a clear sep-
aration of the linear and non-linear behavior [115,116]. The analysis was done studying the
surface width and the erosion velocity of the interface. Explicitly it was found that two
morphologies exist depending on the product of the non-linear coefficients; λx and λy. For
λxλy > 0 (t > τ), non-linear terms destroy the early ripple structure replacing it by a rough
interface, while for λxλy < 0 the replacement takes a characteristic morphology of rotated
ripples also known as cancellation modes (CM) [95,94].
4.5 Muñoz-Cuerno-Castro ModelIn this model the surface morphology and its evolution is formulated by coupling two fields,
one that describes the density of ad-atoms within the topmost layer of the surface and a
second one which corresponds to the static bulk of the substrate. This concept was borne
out of a better fit to experimental results, where natural occurring processes like surface
diffusion, redeposition and transport are important [120,121,122,123,124]. That is in analogy to
sand dunes in deserts where their growth is controlled by surface transport. Formation and
growth of sand dunes were initially described in the studies of Bagnold [125] and only recently
36
Chapter 4. Theories of Ion Induced Surface Growth
advanced in the work of Valance et al. [126] and then fully implemented by Muñoz-Garćıa and
collaborators [121,123] for ion beam sputtering (IBS) experiments.
The coupled two-field model is expressed as:
∂R
∂t
= (1− φ)Γex − Γad −∇ · J (4.17)
∂h
∂t
= −Γex + Γad (4.18)
where the first equation describes the evolution of a thin surface layer R(r, t) = R, while the
second equation describes the height of the static bulk phase h(r, t) = h. The third term
on the right of the first equation considers the surface transport through surface diffusion of
mobile ad-atoms in the form of a continuity equation, ∂R/∂t = −∇ · J, where J = −D∇R.
In this model, the φ coefficient sets certain cases like for example; complete (φ = 0), partial
(0 < φ < 1) and no deposition (φ = 1) which occur due to the variation of sputtering yield
value.
In the special case of metals, Ehrlich-Schowoebel barriers would counteract the diffusive
process in the form of surface steps (e.g. for polycrystalline metals, grain boundaries appear
as surface steps). For this case, additional terms on the surface current are written as
follows [121]:
J(R, h) = KR∇(∇2h) + SESR
∇h
1 + (αES∇h)2
+ vR−D∇R (4.19)
where the surface current is described by the surface relaxation mechanism (first term),
a Ehrlich-Schwoebel barrier (second term) counteracting the relaxation mechanism where
SES is the barrier strength and αES is a characteristic length scale; a drag term (third
term) corresponds to ionic impacts and finally the last term as the usual surface diffusive
mechanism stabilizing the surface.
The first two terms on the right of the coupled-two field equations (4.17) and (4.18)
37
Chapter 4. Theories of Ion Induced Surface Growth
describe the rate at which atoms from the static and the thin layer interchange, Γex and Γad.
The rates of excavation (Γex) and addition (Γad) terms are written as follows
[123]:
Γex = α0
[
1 + α1x
∂h
∂x
+∇ · (α2∇h) +
∂
∂x
∇ · (α3∇h)
+
∑
i,j=x,y
α4ij
∂2
∂i2
(
∂2h
∂j2
)
+
∂h
∂x
∇ · (α5∇h) +∇h · (α6∇h)
]
(4.20)
where Γex is written in terms of the erosion rates with respect to developing slopes, for which
case the small slope approximation has been applied [123]. The coefficients αi = diag(αix, αiy)
are represented by 2×2 diagonal matrices for i = 2, 3, 5, 6 and αix and αiy are Makeev-Cuerno-
Barabási erosion velocity-ratio coefficients (see Muñoz-Garćıa et. al. [123] review article for a
detailed list of these coefficients). The rate of excavation in principle relates how material is
dislodged from the immobile target by irradiation, being dependent upon experimental pa-
rameters; ion beam flux, angle of incidence and properties of the target material. Meanwhile
the rate of addition depends on the nucleation rates of ad-atoms with a proportionality on
the surface curvature [23], describing the rate at which mobile material incorporates back into
the immobile bulk:
Γad = γ0
[
R− Req
(
1− γ2x
∂2h
∂x2
− γ2y
∂2h
∂y2
)]
(4.21)
This rate of addition Γad is described by the behavior of the surface in analogy to Gibbs-
Thompson evaporation condensation. In a sense, this coupled two-field equation determines
the interchange of atoms from those of the static phase that move into an amorphous phase
and vise-versa. Explicit dependence on the physical parameters are fully reviewed on the ar-
ticle of Muñoz-Garćıa et al. [123] with some specific cases relating complete, partial and/or no
redeposition. Furthermore, the reduced 1D and 2D effective growth equations are compared
38
Chapter 4. Theories of Ion Induced Surface Growth
to previous results [20,94,95,103] and prove to be in excellent agreement.
In a reductive calculation (multiple-scale calculation along with an adiabatic approxima-
tion), the coupled two-field model may be written as a single effective equation [121,123]. This
single field equation takes the usual form of a non-linear equation for the description of sur-
face patterning. The analysis of the behavior of individual terms on this effective equation
includes those related to linear, non-linear terms along with conserved and non-conserved
dynamics.
4.5.1 The 1-D & 2-D Effective Model
The effective single-field equation is written as [123]:
∂h
∂t
= γx
∂h
∂x
+
∑
i=x,y
[
− νi
∂2h
∂i2
+ λ
(1)
i
(
∂h
∂i
)2
+ Ωi
∂2
∂i2
(
∂h
∂x
)
+ ξi
(
∂h
∂x
)(
∂2h
∂i
)]
+
∑
i,j=x,y
[
−Kij
∂2
∂i2
∂2h
∂j2
+ λ
(2)
ij
∂2
∂i2
(
∂h
∂j
)2]
(4.22)
where the surface height function h(x, y, t) evolves as usual from an initial flat configuration.
This single-field growth equation depends on the following terms, from left to right; (1)
a surface transport term on the x-direction, (2) diffusive terms either x or y directions,
(3) KPZ non-linearity terms, (4) and (5) third-order terms representing surface dissipative
transport, (6) fourth-order relaxation terms, and finally (7) non-linear conserved dynamics.
This equation is similar to that obtained in Bradley-Harper type theories with higher order
terms relating to conservative properties of the system. Additionally this last term mimics
a natural physical process that appears in erosion experiments and takes into account re-
deposition of sputtered material.
The precise behavior of this general single-field model is carried out in the work of Muñoz-
39
Chapter 4. Theories of Ion Induced Surface Growth
Garćıa et al. [123]. A careful analysis of the dispersion relation shows that odd number deriva-
tives lead to the translation of surface ripples along the x direction, while those of even order
contribute to their growth.
Let us consider the 1-dimensional single-field growth problem. This is done by removing
any terms involving y leading to a 1-D effective field equation and written as:
∂h
∂t
= −ν ∂
2h
∂x2
−K∂
4h
∂x4
+ λ(1)
(
∂h
∂x
)2
− λ(2) ∂
2
∂x2
(
∂h
∂x
)2
(4.23)
rewriting this equation by applying a transformation, x′ = (K/ν)1/2x, t′ = K/ν2t and
h′ = ν/λ(1)h:
∂h
∂t
= −∂
2h
∂x2
− ∂
4h
∂x4
+
(
∂h
∂x
)2
− r ∂
2
∂x2
(
∂h
∂x
)2
(4.24)
where the parameter r is written as r = (λ(2)ν)/(λ(1)K) which is a factor that depends on
other coefficients. This parameter measures the strength of the conservative non-linearity
term. Muñoz-Garćıa and collaborators [123] have studied this equation and explored its be-
havior for different values of the r parameter. Three general stages exist that control the
interface width for this equation, (1) surface ripples appear on the linear regime, after which
(2) the non-linear conserved term ∂2x(∂xh)
2 acts at small scales that allows the coarsening of
structures, thus reaching a characteristic length scale, and finally entering (3) a non-linear
term, the KPZ non-linearity that interrupts the coarsening.
Next, we consider the 2-dimensional effective field equation, in its isotropic version is
written as:
∂h
∂t
= −ν∇2h−K∇4h+ λ(1)(∇h)2 − λ(2)∇2(∇h)2 (4.25)
where h(x, y, t) again is the height interface equation; the two terms on the right describe
the formation of surface ripples. Meanwhile, the competition between non-conserved and
conserved non-linearities is an important aspect of the surface morphology related to re-
40
Chapter 4. Theories of Ion Induced Surface Growthdeposition. Likewise, curious morphologies are obtained by differentiating between isotropic
and anisotropic versions of this equation. The numerical simulations have been able to repro-
duce dots, holes and ripples seen in experiments for most materials that undergo irradiation
under normal incidence (see section §3.2). This implies a non-preferential direction of atomic
diffusion and relaxation mechanisms.
4.6 Bradley-Shipman Model
Ion beam sputtering experiments of composite materials are understood as complicated
systems because preferential sputtering can play an important role in the pattern forming
process [1]. Irradiation of binary, ternary and multi-elemental materials leads to the partial
erosion of atoms and segregation of others on the near-surface region. Knowledge of this
particular atomistic behavior in composite materials is a key ingredient into a theoretical
formulation for pattern formation. Recently the formation of patterns on binary materials at
normal and at oblique incidence angles was formally studied from a compositional coupled
two-field model [24,25,31].
In the work of Bradley and collaborators [25,28,31], the coupling of a compositional field on
top of a static field describes the evolution of the interface width of surfaces being subjected
to ion irradiation. The density of corresponding atoms on this thin layer plays the role in
the development of surface patterns which is coupled to that of the static bulk phase. The
coupled two-field equation is written as [31]:
∂h
∂t
= −Ω(FA + FB +∇ · JA +∇ · JB) (4.26)
∆
∂Cs
∂t
= Ω
[
(cb − 1)(FA +∇ · JA) + cb(FB +∇ · JB)
]
(4.27)
This coupled two-field equation considers the ion irradiation of an initial binary compound
41
Chapter 4. Theories of Ion Induced Surface Growth
consisting of A and B atoms. The atomic volume Ω has similar values for these two atomic
species, and it is assumed that B atoms is preferentially eroded when subjected to an ion
beam of type A. Since B is preferentially eroded, the stoichiometric concentration is altered
with the presence of higher amount of A atoms. In a continuous ion irradiation, the surface
develops a steady thin layer of thickness ∆, of the order of the penetration depth (d ∼ Rp)
of the bombarding ion.
In the above description, the first equation describes the top surface layers evolution due
to a particle net flux of A and B atomic species with their respective surface currents. The
second equation describes the composition change of the thin layer of altered stoichiometric
composition coupled to erosion and to surface currents mechanisms. The relation for the
surface currents of individual atomic species is given by:
Ji = −Dins∇Ci + βTDicinsΩγs∇∇2h− µ∇h (4.28)
where this surface current is described by a diffusion term of the atomic composition, a
fourth-order term because of surface height relaxation, and a term of the surface current.
This last term takes into account momentum transfer from the incident ion beam to the
atomic target [70]; which mimics a diffusion term on the equation motion for the surface
height [76].
This model has been constructed in order to take care of the shortcomings of the irra-
diation of alloys and composite materials. The parameter space of this model outlines the
possible study of individual systems of particular interest, e.g., the titanium alloy (Ti-6Al-
4V) which has been used in this work.
Given the above coupled two-field equations; Bradley-Shipman and collaborators [25,31]
applied a small-amplitude long-wavelength perturbation on the surface height, h = h0−v0t+u
and in the compositional field, cs = cs,0 + φ. This perturbation leads to a non-linear couple
42
Chapter 4. Theories of Ion Induced Surface Growth
two-field equation reduced that solely depends on h and φ. Further approximations include
those of a geometrical surface erosion up to second order, non-linearities of the height and
of the compositional fields:
∂u
∂t
= φ−∇2u+∇4u+ λ(∇u)2 (4.29)
∂φ
∂t
= −aφ + νφ2 + ηφ3 + b∇2u+ c∇2φ (4.30)
where the first equation determines the evolution of the surface while the second describes
the atomic concentration of the thin layer of altered composition.
Surface morphologies obtained through these equations have been examined numerically
in the work of Motta et al. [31] through variation of the theoretical parameters. The parameter
space is large enough to encompass many of the experimental observed morphologies rem-
iniscent of surface ripples, dots and holes. The possibility for dots-on-ripples morphologies
and a continuous transition from ripples to a dotted structure is observed in their extensive
review. Furthermore, a general interest of metallic materials and their alloys has sparked
this huge interest. This is a theory that responds to preferential sputtering data and its
general possible contribution in the theory of ion-beam sputtering experiments.
4.7 Numerical Analysis
Numerical solutions of the Bradley-Harper equation and their associated single-field models
may be carried out. In our case, the numerical simulations were obtained from the software
package “Ripples and Dots” borrowed from Professor Rodolfo Cuerno [110]. Well known con-
tinuum equations like the Edwards-Wilkinson (EW), Mullins-Herring (MH), Kardar-Parisi-
Zhang (KPZ) and the Kuramoto-Sivashinsky (KS) can in principle be solved numerically by
finite differences, and from pseudo-spectral Fourier methods [110,111].
43
Chapter 4. Theories of Ion Induced Surface Growth
Many of these dynamic interface models have been successfully simulated [43,94,95,102]. A
renewed interest has emerged by other authors [46,115,116,117,118,119]. The following simulations
have been computed utilizing the finite difference method with random coefficients. We want
to show the behavior of continuum equations in surfaces and interfaces [110].
Just as a reminder; for a two-dimensional surface, the height function as constructed
by individual particles deposited at a particular position ~x = (x, y) and time t, and is
expressed as a continuum function h = h(x, y, t) = h(~x, t) (see section §2.1). General
physical processes that are known to be present during ion bombardment include erosion,
relaxation and transport. Each of these effects may be modeled by one or more terms in a
continuum equation, as those mentioned in previous sections of this chapter.
In the statistical analysis of the surface height, the interface width, w(t) is defined as the
RMS fluctuations in the height with respect to time, see equation (2.2) and equation (2.3).
In single non-linear field models, the interface width functions behave in two ways; first an
exponential growth followed by a saturated state. These are related by critical exponents,
β (growth exponent) and α (roughness exponent). The relations are commonly given by
w(L, t) ∼ tβ and wsat(L) ∼ Lα, where L is the system size. These functions are related by
the Family-Vicsek relation [32]:
w(L, t) ∼ Lαf(t/Lz) (4.31)
where z = α/β is the dynamic exponent. Critical exponents [36,41] for many important models
in the study of interface growth phenomena are given in Table 4.1. These exponents describe
the interface width evolution with respect to time and system size L.
Information of the critical exponents tells of the average growth but does not describe
the emerging surface structure. For this, a structure factor may be calculated by considering
equation (4.34). This relation says that for all modes greater or close to kmax, an expo-
44
Chapter 4. Theories of Ion Induced Surface Growth
∂h/∂t = Model name α β z = α/β Ref. a
η Random deposition - 1/2 - [32]
∇2h + η Edwards-Wilkinson 0 0 2 [32]
∇2h + |∇h|2 + η Kardar-Parisi-Zhang 0.38±0.01 0.24±0.01 1.62±0.07 [115]
−∇4h+ η Mullins-Herring 1 1/4 4 [32]
−∇2h+ |∇h|2 −
∇4h + η
Kuramoto-Sivashinsky
(early time)
0.75-0.80 0.22-0.25 3.0-4.0 [102]
−∇2h+ |∇h|2 −
∇4h + η
Kuramoto-Sivashinsky
(late time)
0.39±0.01 0.20±0.011.95±0.10 [115]
Table 4.1: Scaling exponents for the various interface models in (2+1) dimensions. These
models have been often used in the description of interface and surface growth phenomena.
aModified from Table II of Drotar et al. [102].
nential growth of the surface structure occurs. In particular, the wavelength is obtained by
computing the structure factor S(~k, t) of the surface function height in k-space, also known
as the power spectral density (PSD) [19,32]. In PSD curves, plots reveal information on the
emergence of a structure with a characteristic size, lc = 2π/k
max. In particular, the structure
factor function S(~k, t) is defined as:
S(~k, t) =
〈
ĥ(~k, t)ĥ(−~k, t)
〉
(4.32)
where h(~k, t) is the Fourier transform of the height function. The spatial Fourier transform
of the surface height is written in k-space as:
ĥ(~k, t) =
1
L
∫
[h(~x, t)e−i
~k·~x]d~x (4.33)
45
Chapter 4. Theories of Ion Induced Surface Growth
Family-Vicsek scaling relation for the structure factor generally follows [32]:
S(~k, t) = k−d−2αs(tk1/z) (4.34)
with a scaling asymptotic function for s(u) given by:
s(u) =







const. if u ≫ 1
u(2α+d)/z if u ≪ 1
In the case of surface ripple generation, PSD curves for the linear models lead to the rela-
tion [20]:
S(~k, t) = −J
2
1− exp[2ω(~k)t]
ω(~k)
= J
exp[2ω(~k)t]− 1
2ω(~k)
(4.35)
where the structure factor depends on the growth rate of the linear theory, ω(~k).
In the following two examples; two-dimensional surfaces have been simulated along with
their PSD and global roughness curves. In general, the data shows the evolution of the
surface in two different important cases (linear and non-linear models).
4.7.1 Linear Theory
In the linear theory of Bradley-Harper, the surface evolution is determined by the second and
fourth order terms, the ripple wave-vector and direction being determined by the greatest in
negative value for the surface tension coefficient (νx, νy) along with the relaxation mechanism
through the coefficient, B. Consider for example the numerical simulation of equation (4.13)
with coefficients: v0 = 0, γ = 0, νx = -1.3, νy = -0.5, Ω1 = 0, Ω2 = 0, λx = 0, λy = 0,
Dxy = 0, Dxx = 0, Dyy = 0 and B(T ) = 1.0, a surface tension coefficient νx > νy, leads
to ripples aligned on the x-direction. A spatial-temporal discretization of ∆t = 0.01 and
46
Chapter 4. Theories of Ion Induced Surface Growth
∆x = ∆y = 1.0 on a L×L = 200× 200 grid system size is performed. The noise amplitude
Dη = 1 is maintained throughout the simulations.
 
 
50 100 150 200
50
100
150
200
(a) t = 1, early surface configuration
 
 
50 100 150 200
50
100
150
200
(b) t = 10, ripples start to form
 
 
50 100 150 200
50
100
150
200
(c) t = 20, ripples ordering increases
 
 
50 100 150 200
50
100
150
200
(d) t = 40, well order ripples
Figure 4.5: Surface morphology evolution of the Makeev-Cuerno-Barabási (MCB) equation
(4.13) for νx = -1.3, νy = -0.5 and B = 1.0. Ripples align on the x-axis are obtained with
wavelength (lc = 7.8).
47
Chapter 4. Theories of Ion Induced Surface Growth
0.1 1
100
101
102
103
104
105
106
107
Po
w
er
 S
pe
ct
ra
 D
en
si
ty
 (P
SD
)
kx
 t = 1
 t = 5
 t = 10
 t = 20
 t = 30
 t = 40
0.1 1
100
101
102
103
104
105
106
107
Po
w
er
 S
pe
ct
ra
 D
en
si
ty
 (P
SD
)
ky
 t = 1
 t = 5
 t = 10
 t = 20
 t = 30
 t = 40
Figure 4.6: Power spectra density (PSD) curves for x and y directions from the linear model.
A single characteristic peak (kx in comparison to ky) appears signaling the emergence of
ripples. In particular ripples emergence on the x-direction in comparison to the y-direction.
In the case of noise; a Gaussian white noise with a zero mean average, and uncorrelated
in space and time is used:
〈η(~x, t)〉 = 0 (4.36)
〈η(~x, t)η(~x′, t′)〉 = 2Dδ(~x− ~x′)δ(t− t′) (4.37)
The surface evolution is shown in Figure 4.5 and identified with a characteristic ripple struc-
ture (lc = 7.8). In this numerical simulation, the ion beam direction would be given from
left to right in analogy to an experimental setup. This particular condition agrees for exam-
ple with experimental results. Time series of the power spectral density (PSD) is given in
Figure 4.6, showing a peak increase associated with a surface structure.
In summary, for a linear theory case PSD curves predict an exponential growth of the
surface structure in the x-direction, and an absence in the y-direction. The global roughness
grows exponentially, see Figure 4.6. In particular, the ripple direction and wavelength are
48
Chapter 4. Theories of Ion Induced Surface Growth
determined by the coefficients given in the linear model.
 
 
50 100 150 200
50
100
150
200
(a) t = 1, early surface configuration
 
 
50 100 150 200
50
100
150
200
(b) t = 10, ripples with disordering
 
 
50 100 150 200
50
100
150
200
(c) t = 20, ripples are further disordering
 
 
50 100 150 200
50
100
150
200
(d) t = 40, kinetic roughening regime
Figure 4.7: Surface morphology evolution of the anisotropic Kuramoto-Sivashinsky (aKS)
equation (4.16) with νx = -1.3, νy = -0.5, λx = 0.1, λy = 0.05 and K = 1.0. Surface
morphology with perpendicular ripples evolve after t = 20 into disorder structures. At
t = 33 surface morphology evolves into a cell-pattern (undergoing kinetic roughening up to
a undefined time).
4.7.2 Non-linear Theory
In the case of secondary effects, a particular model is exemplified by the anisotropic Kuramoto-
Sivashinsky (aKS) equation as a possible description of surfaces that undergo amplitude
49
Chapter 4. Theories of Ion Induced Surface Growth
saturation. Other effects like kinetic roughening is signaled by a coarsening of a characteris-
tic structure, broadening and disordering of the structure. Consider the particular example
given in Figure 4.7 where non-linear terms have been added, these terms are associated to
the disappearance of surface ripples which evolve into a cell-like structure that maintains
for a long time. The numerical parameters in the simulation of equation (4.13) are v0 = 0,
γ = 0, νx = -1.3, νy = -0.5, Ω1 = 0, Ω2 = 0, λx = 0.1, λy = 0.05, Dxy = 0, Dxx = 0, Dyy =
0 and B(T ) = 1.0. Meanwhile the same noise conditions and space and time discretization
parameters have utilized as those given in the linear model.
0.1 1
100
101
102
103
Po
w
er
 S
pe
ct
ra
 D
en
si
ty
 (P
SD
)
kx
 t = 1
 t = 5
 t = 10
 t = 20
 t = 30
 t = 40
0.1 1
100
101
102
103
Po
w
er
 S
pe
ct
ra
 D
en
si
ty
 (P
SD
)
ky
 t = 1
 t = 5
 t = 10
 t = 20
 t = 30
 t = 40
Figure 4.8: PSD curves for x and y directions from the non-linear model. At earlier times, a
characteristic peak appears which later vanishes due to a coarsening of the surface structures.
In summary, for the non-linear model PSD curves show an initial increase in amplitude
in a rippling structure which reduces as the non-linear term takes a greater role. This occurs
after the cross-over into non-linear regime. Furthermore, this surface evolution is related to
experimental set-ups at long times, where the rapid development of slopes are taken into
account. On the other hand, the global surface roughness while initially grows it saturates,
see Figure 4.9 on the right.
50
Chapter 4. Theories of Ion Induced Surface Growth
1 2 3 4 5 6 7 8 910 20 30 40
100
101
102
103
104
105
106
In
te
rfa
ce
 W
id
th
 [w
(t)
]
Time (s)
0 5 10 15 20 25 30 35 40
0
5
10
15
20
25
30
35
In
te
rfa
ce
 W
id
th
 [w
(t)
]
Time (s)
Figure 4.9: Interface global roughness curves with respect to time from linear (left) and
non-linear (right) models.
51
Chapter 5
Experimental Techniques
The materials used in this research are titanium (Ti) and its alloy Ti-6Al-4V. Titanium
is the twenty-second element of the periodic table, classified as a transition metal with
a silvery look, a corrosion resistant element with a density of 4.51 g/cm3, an electronicconfiguration of 1s2 2s2 2p6 3s2 3p6 3d2 4s2 and an atomic mass of 47.867 amu. [127] This
low density metal commonly occupied in the aerospace industry represents an important
material but due to its high production costs its usage has been limited. The alloy, Ti-
6Al-4V (ρ = 4.42 g/cm3) utilized in the medical field is composed of titanium, aluminum
and vanadium and distributed accordingly by weight of 90% of Ti, 6% of Al and 4% of V.
These low density materials, in comparison to stainless steel (S.S. 316L, 7.9 g/cm3) and the
cobalt alloy (CoCrMo, 8.3 g/cm3), have become an important part of orthopedic implant
applications. Their biocompatibility properties make them useful in the medical industry as
total hip and knee joint replacements [7].
This chapter outlines our experimental techniques; section §5.1 reviews sample prepa-
ration, §5.2 describes the ion implanter and ion beam analysis. Later, in section §5.3 we
mention our ion implantation conditions, §5.4 describes the initial surface induced stress
and lastly §5.5 and §5.6 report the microscopy and XPS analysis techniques, respectively.
52
Chapter 5. Experimental Techniques
5.1 Ti and Its Alloy Ti-6Al-4V
For this work, 20-cm length Ti (Grade 4) and Ti-6Al-4V (Grade 5) rods were purchased
from Goodfellow Corporation Inc. [128]. The purity quoted by the manufacturer is 99.6+%
for Ti and the maximum impurity for the alloy is 650ppm of oxygen (see Table 5.1 and
Table 5.2 for other commercially pure titanium (cpTi) grades along with their chemical and
mechanical properties). Purchased rods of 9.5mm (Ti) and 10mm (Ti-6Al-4V) diameter
were cut to a thickness of approximately 3-5mm. Small disks the size of coins are desired,
as usual implantation areas do not exceed 0.5 cm2 and are completely inscribed within the
circular sample area.
Element Grade 1 Grade 2 Grade 3 Grade 4 Ti-6Al-4V a
Nitrogen 0.03 0.03 0.05 0.05 0.05
Carbon 0.10 0.10 0.10 0.10 0.08
Hydrogen 0.015 0.015 0.015 0.015 0.0125
Iron 0.20 0.30 0.30 0.50 0.25
Oxygen 0.18 0.25 0.35 0.40 0.13
Titanium Balance
Table 5.1: Chemical compositions of commercially pure titanium (cpTi) grades and its alloy,
Ti-6Al-4V (adapted from Park and Bronzino [7]).
aAluminum 6.0%, vanadium 4.0%, and other elements 0.4% total. Maximum allowed impurities of grade
level titanium.
Property Grade 1 Grade 2 Grade 3 Grade 4 Ti-6Al-4V
Tensile strength (MPa) 240 345 450 550 860
Yield strength (0.2% offset) (MPa) 170 275 380 485 795
Elongation (%) 24 20 18 15 10
Reduction of area (%) 30 30 30 25 25
Table 5.2: Mechanical properties of commercially pure titanium (cpTi) and its alloy, Ti-6Al-
4V (adapted from Park and Bronzino [7]).
Titanium is quite a difficult material to work with; in its pure form it is soft and easily
scratched. A major challenge exists when polishing, handling and storing. The sample
53
Chapter 5. Experimental Techniques
preparation procedure is as follows: (1) obtain a flat surface with a cutting rotor machine
(2) polishing is done with sand papers (P360, P400, P500, P600, P1000, P1200, P2000 and
P4000; FEPA Grading) with a running water system in order to prevent the heating of
the sample while washing off the removed metal (3) further polishing is performed with a
water-diluted diamond compound using a lapping machine (South Bay Technology, Inc.,
Model 910) with polishing cloths. The diamond compound (Leco Corp.) particle sizes
are 3µm, 1µm and 1/2µm. Finally, mirror finished samples are rinsed with ethyl alcohol
using an ultrasonic cleaner (Branson Cleaning Equipment Co., Model B-12) for 30 minutes.
Scratches smaller that those visible by the naked eye are eliminated by checking regularly on
an optical microscope (even though mirror finished samples are smooth, surfaces are rough at
the atomic scale; see Figure 5.1 for atomic force microscopy (AFM) analysis of the polished
surfaces). Initial surface roughness of W = (8.24± 4.6)nm for Ti and W = (22.9± 12.5)nm
for the alloy.
Figure 5.1: AFM micro-graphs of mirror finished Ti (left) and Ti-6Al-4V (right) polished
surfaces at 25µm×25µm scale (inset 5µm×5µm). At half-micrometer diamond-compound
finished scratches are evident of the polishing procedure.
54
Chapter 5. Experimental Techniques
During and after polishing, the formation of surface oxides on Ti and its alloy is always
present. It is known that Ti is a highly reactive element with the oxygen present in the atmo-
sphere [129,130]. A newly polished Ti surface forms a protective oxide layer of a few nanometers
in thickness, depending on the temperature and the composition of the atmosphere [131]. This
protective layer of around 10 nm of thickness is responsible for the biocompatibility of Ti and
its alloy in orthopedic implants [7,11,131,132]. Furthermore, a sample which has spent a longer
time in ambient conditions tends to grow higher oxide states, consisting of TiO2, Ti2O3 and
TiO from the Ti-bulk substrate as pointed out by Padma et al. [130].
After polishing, samples are implanted. Ion beam sputtering experiments have been per-
formed producing morphologies within a micrometer underneath the initially mirror-polished
surface. Oxidized titanium states on the top surface layers of the sample may recoil-implant
into the sample [133]. On the other hand, oblique-incidence angle ion implantation may also
recoil-implant and remove oxidized states from the surface during the initial implantation.
The surface roughness changes according to implantation conditions and is commonly smaller
(greater) for angles lower (higher) that 45° angle.
It has been found in these experiments that, surface erosion tends to distort the implanted
profile of Au ions. This has been reported in earlier work which refers to the deposition of
Au and Ti atoms on the surface of glass slides [92]. It is believed that the oxide layers are
eroded as well due to the implantation of Au ions on the Ti sample. This cleans the sample
of oxides but they will grow on removal from the implantation chamber, and during storing.
Since some of the implanted Au ions are located near the surface this would in principle
hinder the build up of oxides.
It is estimated that for a vacuum of the order of 10−7 torr, a flux of 3.57 × 1013 cm−2
s−1 oxygen atoms will collide on the surface. On the other hand, it is known that a flux of
5.2×1012 cm−2 s−1 of Au ions interacts with the surface producing of the order of 1013 cm−2
s−1 of Ti atoms leaving the surface. This in principle evens out the number of incoming
55
Chapter 5. Experimental Techniques
Material Thermal
diffusivity
αT (cm
2/s)
Thermal
conductance
κT (W/cm
K)
Volumetric
heat capacity
ρTCT (W
s/cm3 K)
Electrical
resistivity
(Ω-cm) a
Ti-6Al-4V 0.026 0.068 2.6 1.78×10−4
Titanium 0.068 0.16 2.3 6.0×10−5
Iron 0.23 0.80 3.5 7.4×10−5
304 S.S. 0.41 0.16 4.0 7.2×10−5
Silicon 0.53 0.84 1.6 10 - 6000
Aluminum 0.95 2.3 2.4 3.5×10−6
Copper 1.1 3.9 3.4 1.72×10−6
Table 5.3: Thermal and electrical properties of titanium, its alloy and other related materials
(Modified from Nastasi et al. [1]).
aAll data at 20°C conditions. ASM, Aerospace Specification Metals Inc.
oxygen atoms with those that leave the surface, so oxygen atoms may not be able to remain
on the surface. This would happen only during the implantation, since after implantation,
surfaces tend to interact with the atmosphere.
The increase in the temperature of the sample is determined mainly by the coupling
between the sample and holder and vacuum chamber. A simple measurement of the increase
in temperature yields a value of about 150° C during implantation [134], additionally Ti and
its alloy not being good thermal conductors (see Table 5.3). For these two materials, the
thermal diffusivity and thermal conductance values are rather small in comparison to other
metallic materials.
Surface analysis techniques performed on Ti and Ti-6Al-4V bombarded surfaces are Scan-
ning Electron Microscopy (SEM), Atomic Force Microscopy (AFM). Furtheranalysis have
been performed with the help of Rutherford Backscattering Spectroscopy (RBS) and X-ray
Photoelecton Spectroscopy (XPS) providing a better quantification of the ion implantation
profile.
56
Chapter 5. Experimental Techniques
5.2 Ion Implanter Facility
5.2.1 Generalities
Ion implantation and other ion-beam analysis are performed at the Marcos Mazari Acceler-
ator Laboratory located at the Instituto de F́ısica of the Universidad Nacional Autónoma
de México (see Figure 5.2 for a drawing of the Pelletron� accelerator). A 3-MV Tandem
Pelletron (9SDH-2) accelerator built by National Electrostatics Corporation (NEC) is used
both to modify and characterize samples.
SNICS
Alphatross RF
implantation
chamber
RBS
chamber
3-MV Pelletron
TM
accelerator
Figure 5.2: (Raw drawing) The 3-MV Pelletron� accelerator (9SDH-2) located at the Marcos
Mazari Laboratory of the Instituto de F́ısica at Universidad Nacional Autónoma de México.
Ion beam sources are located at the far bottom left corner while implantation and RBS
chambers on the top right corner of the diagram. Adapted from a drawing of L. Rodŕıguez-
Fernández.
57
Chapter 5. Experimental Techniques
Figure 5.3: An overview image of the 3-MV Pelletron� accelerator at Instituto de F́ısica,
Universidad Nacional Autónoma de México. The ion sources are located on the top left
corner, followed by the main tank on the middle and on the far right the experimental
research lines. Ions are accelerated through a three-stage potential difference depending
on the desired kinetic energy which are implanted into our materials. The production,
acceleration and implantation of samples are all carefully performed in a couple of hours.
5.2.2 Ion Implantation
The production of a negative ion beam is obtained from the SNICS ion source. In the SNICS
(Source of Negative Ions by Cesium Sputtering) source a cathode (a small cylindrical cavity
with deposited element) is eroded by heated cesium atoms accelerated through a potential
difference, successfully extracting negative ions. The extracted negative ion beam is then
introduced into the main tank of the accelerator after passing through an electromagnet
selector (see Figure 5.3). The first acceleration stage of the ion beam is performed as it
58
Chapter 5. Experimental Techniques
approaches the high voltage terminal. The high voltage (HV) terminal is obtained from a
pellet charging system [135]. The kinetic energy of the ion beam now has E = qV , where q is
the particle charge [136,137]. In the terminal the accelerated negative charged ion beam passes
through a neutral gas. A molecular nitrogen gas (N2) strips electrons from the ion beam,
effectively changing the ion beam from negative to positive.
The positive ion beam now receives a second boost. The desired kinetic energy of the ion
beam is now written as, E = (1 + q)V , where q is the charge state of the positive ion beam.
Ions exiting the accelerator, have gained the desired kinetic energy. Based on charge and
mass, an electromagnet chooses the right ion. The ion beam is now focused by quadrupole
lenses, as ion beam dispersion often reduces its intensity.
In order to obtain uniform beam distribution, ion beam must be scanned with a high
frequency horizontal and vertical fields. This guarantees a uniform ion beam distribution.
The ion beam is collimated with a square opening, defining the implantation area. In the
implantation chamber, the sample is mounted on an aluminum plate. As ion implantation
occurs, the aluminum plate acts as charge collector. This is coupled to a charge integrator
module allowing the effective measurement of ions being implanted in the sample.
5.2.3 Ion Beam Analysis
The production of negative He ions occurs in the ALPHATROSS source. This is a radio
frequency (RF) source in which a mixture of neutral gases in a quartz bottle is dissociated.
The extracted helium negative ions are then introduced in the accelerator. This He ion beam
is in a similar fashion accelerated through the main tank of the accelerator. Its main use is
for Rutherford backscattering spectroscopy (RBS) analysis of materials [138].
In this spectroscopy technique, materials are bombarded with a helium ion beam, which
is backscattered from the target sample at 167°, detected by a Si surface barrier (SSB)
59
Chapter 5. Experimental Techniques
detector. In a subsequent fitting analysis, the spectra determine the elemental composition
of the material.
5.3 Au Ion Implantation of Ti and Ti-6Al-4V
1.0-MeV Au+ ion implantation of titanium and its alloy, Ti-6Al-4V have been performed.
The ion implantation conditions are; 200-nA of current (flux of 5.12× 1012 ions cm−2 s−1),
0.5 cm2 area and 10−7 Torr vacuum at room temperature. The target materials are position
on an aluminum incline plane for oblique incidence angle experiments (see Figure 5.4). The
generated heat is dissipated along this aluminum plate. Glass slides are position in the
opposite side of the implated material. This collects atoms eroded from the target material.
Ion implantation experiments fluence range from ∼ 1016 ions cm−2 up to ∼ 1017 ions cm−2.
Oblique ion implantation incidence have been performed at 23°, 45°, 49° and 67°. Naked-eye
observation of the glass slides, show the deposition of a thin film from the eroded material
(see right image in Figure 5.5).
5.4 Surface Induced Stress on Ti and Ti-6Al-4V
Utilizing a Vickers� micro-hardness tester (Micro-Hardness Tester (Matsuzawa MHT-2)),
surfaces are indented before implantation. A Vickers� micro-hardness tester is based off the
resistance to indentation of materials. A fixed load is applied on the surface of a material
and by measuring the size of the indenter, a value of the hardness is obtained. Marks left
on the surface form triangular planes of equal size (see Figure 5.6 for an example of the
indenting mechanism with a 45° angle rotation). The value of Vickers� micro-hardness is
quantified by the relation:
HV =
1.854× 103F
d2
(5.1)
60
Chapter 5. Experimental Techniques
Figure 5.4: Experimental setup for IBS experiments. Incline plane that makes a 45° angle
with respect to the surface normal (right image). Aluminum plate for mounting glass slides
has been positioned (left image).
where HV is the value for Vickers� micro-hardness, F is the weight in grams-force and d
is the average diagonal length in micrometers; furthermore a relation between indentation
depth (h) and the diagonal (d) is given by:
h̄ =
1
7
d (5.2)
where the average depth h̄ is given by a 1/7 of the length of the diagonal. The microhardness
of our materials have been previously studied by Trejo-Luna and collaborators [88,89]. The
reference Vickers� hardness values for the titanium alloy (Ti-6Al-4V) of HV = 330 and the
pure titanium (Ti) with HV = 220.
If for example a sample is implanted with an angle of incidence 45°, the triangular mark-
ings make angles of 23°, 49° and 67° with respect to the surface normal of the indented plane
(see Figure 5.6). The formation of a structure on the surface for each of the triangular faces
is therefore independent on size, but only on the angle of incidence. In subsequent analysis,
61
Chapter 5. Experimental Techniques
Figure 5.5: Ti and Ti-6Al-4V sample size in comparison to a 1¢ US coin. Actual experimental
run with mounted sample on the inclined plane. The ion-beam direction is from bottom to
top, atoms eroded from the surface are deposited at the glass slides (visible by a slight grey
smear).
similar single-angle experiments are then compared to the planes of the surface indentation.
5.5 Microscopy Techniques
Surface topography analysis of ion-implanted samples was performed with help of micro-
scopes available at the Instituto de F́ısica. An optical microsope (OM; Olympus, Model
BH2-UMA), a scanning electron microscope (SEM; JEOL, Model SM-5600 LV) and an
atomic force microscope (AFM; JEOL, Model SPM 4210) were used in this work.The
morphological and topographical information obtained on these microscopes explored both
small and large length scales of surface structures.
From the very beginning, examination of Au ion-implanted samples were carried out
with help of an optical microscope (OM). This characterizes surface scales in the hundred-
micrometer sized. Later, scanning electron microscopy (SEM) assists in the identification of
small and large structures. Additionally, SEM along with an electron dispersive spectroscopy
62
Chapter 5. Experimental Techniques
y
x
ion beam
23o
67o
49o 49o
45o
22o
136o
Fg
Figure 5.6: Left: A 45° angle rotated indenter. Right: Ion beam direction from top to
bottom on the x− z plane at an angle of incidence of 45°. Shown angles reflect to those that
form with the incident ion beam direction onto normal of the surface of the various planes.
(EDS) in a secondary electron mode probe individual structure’s atomic concentration at
higher resolutions. The obtained information gives a clear contrast to depth of conductive
solid surfaces due to the electron beam interacting with the near-surface atoms of the target
material.
Meanwhile, in the case of surface topografies, these were obtained by atomic force mi-
croscopy (AFM). This technique probes the surface topography with help of silicon tips
cantilevers which scans the surface by attractive and repulsive electric forces. MikroMasch�
aluminum-coated n-type silicon tips were used in this case. Statistical surface analyses were
later performed with tools of the Gwyddion software version 2.41 [139] and of the WSxM
software [140].
63
Chapter 5. Experimental Techniques
5.6 X-ray Photoelectron Spectroscopy Technique
X-ray photoelectron spectroscopy (XPS) analyses were performed on Ti and on its alloy
(Ti-6Al-4V) after Au ion implantation. The XPS analyses were carried out at the Centro
de Nanociencias Micro y Nanotecnoloǵıas of Instituto Politécnico Nacional (IPN) with a
Thermo Scientific� K-Alpha� equipment. A diagram of the analysis method is shown in
Figure 5.7. The top surface layers of the material are analyzed by monochromatic X-rays
(Al Kα) producing photoelectrons characteristic of the atoms present in the sample [141].
Survey and high resolution spectra of Au 4f and Ti 2p were collected by an hemispherical
detector and later, quantified with respect to the elements binding energies (see Figure 5.8).
The second stage of a XPS machine is then performed by an argon ion beam, eroding the
surface in subsequent analysis. This gives a detailed information of the oxidation states and
composition of the Au-Ti chemical species present in the sample.
-
-
-
-
-
-
-
++
+
++
X-ray source
Detector
sample
Lens
UHV
PC
Figure 5.7: Incident X-rays penetrate the near-surface region kicking photoelectrons off the
material. These photoelectrons are guided by electromagnetic lenses and deposited in a
detector. The detector determines the binding energy based of the photoelectric formula
and are analized by a personal computer.
64
Chapter 5. Experimental Techniques
1s
2s
EB
2p1/2
2p3/2
3s
3p1/2
3p3/2
4s
3d3/2
3d5/2
4p
5p1/2
5p3/2
6s
4f5/2
4f7/2
5d
e-ν
K
L1
EB
L2
L3
M1
M2
M3
N1
M4
M5
N2
O1
O2
P1
N3
N4
O3
e-
a) b)
Figure 5.8: Schematic electronic structure of XPS core-level(a) and Auger(b) electrons (not
analyzed in this work). The main lines of the XPS spectra come from any of the core-level
and from the valence band. The main elemental composition is determined provided the
elemental concentration is high enough.
This technique is based off the photoelectric effect, where high-energy photons (X-rays in
the thousand-eV range) are incident into a sample kicking photoelectrons out. The recorded
signal is then analized in accordance to the binding energy of photoelectrons with respect
to the initial energy; Eb = hν −K − φ, where the binding energy Eb is written as an initial
photon energy (hν) minus the kinetic energy (K) of the photoelectron and work function φ of
the target material. The obtained signals from the XPS aparatus is analyzed with respect to
the elemental content of the target material. High-resolution spectra of individual elements
are then occupied in the characterization of compound formation with our ion implantation
conditions (see section §6.6).
65
Chapter 6
Results
The effects of ion beam sputtering (IBS) experiments of Ti and Ti-6Al-4V surfaces is reviewed
in this chapter. Ion implantation of titanium and the alloy erodes the target material and
surface shapes emerge that depend on the experimental conditions. Surface ripples on Ti
and Ti-6Al-4V are formed during high-energy Au ion implantation at 45° and 49° angles.
Other shapes form at 23° and 67° angles of incidence. In particular for ripples, the initial
formation, growth and the asymptotic behavior of surface ripples is observed and measured
accordingly to the ion fluence.
This chapter is divided into seven sections: (§6.1) a basic comparison between near
normal (8°) and 45° incidence angles; (§6.2) the growth of surface ripples at 45° for Ti and
Ti-6Al-4V is given, then in (§6.3) surface height growth of Ti and Ti-6Al-4V, (§6.4) other
angles of incidence at 23°, 49°, and 67° are explored; (§6.5) relates an initial induced-surface
stress with micro-indentation; (§6.6) surface ripple atomic concentration is explored; (§6.7)
X-ray photoelectron spectroscopy (XPS) analyses are reviewed and lastly in (§6.8) other IBS
experimental studies are mentioned.
66
Chapter 6. Results
6.1 IBS of Ti and Ti-6Al-4V at 8° & at 45° Angles
Figure 6.1: SEM micrographs of 1.0-MeV Au+ ion implantation of Ti (left) and Ti-6Al-4V
(right) near-normal incidence angle (θ = 8°) for Φ = 5.0 × 1016 ions cm−2 fluence. The ion
beam direction is from top to bottom on both micrographs.
Near-normal (θ = 8°) ion beam sputtering (NIBS) and oblique ion beam sputtering (OIBS)
at 45° angles have been performed (see Figure 6.1 and Figure 6.2 for SEM micrograph images
of Ti and Ti-6Al-4V ion implanted surfaces). Surface morphology effects are observed for
these two angles of incidence; at near-normal incidence angles surfaces remain flat while at
higher angles of incidence ripples form. Au ion implantation for both materials shows similar
structures; the absence or appearance of a rippling structure is seen to be independent of
the material type but dependent only on the angle of incidence.
67
Chapter 6. Results
Figure 6.2: SEM micrographs of 1.0-MeV Au+ ion implantation of Ti (left) and Ti-6Al-4V
(right) at 45° incidence angle for Φ = 4.7× 1017 ions cm−2 fluence. The ion beam direction
is from top to bottom on both micrographs.
Figure 6.3: Optical microscope (OM) micrographs of 1.0-MeV Au+ ion implantation of Ti
(left) and Ti-6Al-4V (right) at 45° incidence angle for Φ = 4.7 × 1017 ions cm−2 fluence.
The ion beam direction is from left to right on both micrographs[labeled mistakenly in the
article]. [92]
In analogy to sand dunes formed by air flow, sand particles are eroded away impacting fur-
ther downstream, organizing the sand into a rippling structure [142]. In the Au ion-implanted
Ti and Ti-6Al-4V samples, these surface structures view within an optical microscope (see
Figure 6.3) are reminiscent of sandy dunes of deserts. Crescent-like “barchan” structures
found in sand surfaces are also seen for IBS experiments. The analogy ends here because
68
Chapter 6. Results
the length scales of these systems vary by about seven orders of magnitude [143,144,145]. IBS
experiments, being atomistic in nature, require rather different treatments but often can be
compared to macroscopic phenomena in order to gain insight on the physical mechanisms.
6.2 IBS Evolution for Ti and Ti-6Al-4V at 45° Angle
Ti and Ti-6Al-4V samples have been implanted at 45° angle with respect to the ion beam
direction. The ion implantation performed at high fluence (Φ = 4.7×1017 ions cm−2), results
in highly elongatedrippling structures on the surface (see Figure 6.2). Lengthy implantations
produce small rippling structures joined together to form large terrace-like figures. These
figures are often compared to house rooftops because of the stepwise structure remaining up
to an indefinite time.
The surface evolution for the 45° angle is described in the following stages. The Au ion
implantation initially roughens the surface (an incubation period exists before any surface
structure can develop). Probably preferential sputtering of Ti and Au atoms occurs, inducing
the formation of small mounds on the target material. Surface ripples develop afterwards
evolving into single-mounds that may evolve into ripples. A rippling structure develops for
intermediate fluences of the order of ∼ 1016 ions cm−2 and settling into the terrace structure
at high fluences (∼ 1017 ions cm−2) [146].
6.2.1 Large-scale Morphologies
For large scale observations, a few scanning electron microscopy (SEM) images of implanted
Ti and its alloy (Ti-6Al-4V) are given in Figures 6.4 and 6.5. Surface ripples are clearly
visible which evolve into a disordered rippling structure that may terminate or fuse with
their neighbors [27]. In this rough approximation, broken structures are associated with non-
linear terms in the equation of motion and observed in numerical simulations [110].
69
Chapter 6. Results
(a) Φ = 6.5 × 1016 ions cm−2 (b) Φ = 6.7× 1016 ions cm−2
(c) Φ = 1.17 × 1017 ions cm−2 (d) Φ = 4.7× 1017 ions cm−2
Figure 6.4: SEM micrographs of Au ion implanted Ti surfaces at 45° at a X2000 magnifica-
tion. The ion fluence are as indicated in the legends. The ion beam direction is from top to
bottom on all images.
6.2.2 Small-scale Morphologies
In an attempt to look for similar behaviors (at small-scales) for ion implanted surfaces,
atomic force microscopy (AFM) has been performed on fluence dependent ion implanted
titanium and Ti-6Al-4V samples. Similar ion fluences of 6.5×1016 ions cm−2, 6.7×1016 ions
cm−2, 1.17×1017 ions cm−2 and 4.7×1017 ions cm−2 were performed for both materials, see
Figures 6.6 and 6.7 [147]. Scale comparison is provided by insets of 5µm×5µm on all of these
surface morphologies.
70
Chapter 6. Results
(a) Φ = 6.5 × 1016 ions cm−2 (b) Φ = 6.7× 1016 ions cm−2
(c) Φ = 1.17 × 1017 ions cm−2 (d) Φ = 4.7× 1017 ions cm−2
Figure 6.5: SEM micrographs of Au ion implanted Ti-6Al-4V surfaces at 45° at a X2000
magnification. The ion fluence are as indicated in the legends. The ion beam direction is
from top to bottom on all images.
In particular, the surface morphological evolution at small scales may give a detail analysis
at a couple hundred nanometers. These structures are carefully reviewed in the discussion
section (see section §7.3). A coupling of small and large scales helps in determining the
evolution of the surface structures at higher ion fluences.
71
Chapter 6. Results
(a) Φ = 6.5× 1016 ions cm−2 (b) Φ = 6.7× 1016 ions cm−2
(c) Φ = 1.17 × 1017 ions cm−2 (d) Φ = 4.70 × 1017 ions cm−2
Figure 6.6: AFM micrograph measurements of Ti-sputtered surfaces over 25µm×25µm
(5µm×5µm inset) at 45° incidence angles. The ion beam direction is from top to bottom on
all images. The ion fluences are indicated in the legends.
72
Chapter 6. Results
(a) Φ = 6.5× 1016 ions cm−2 (b) Φ = 6.7× 1016 ions cm−2
(c) Φ = 1.17 × 1017 ions cm−2 (d) Φ = 4.70 × 1017 ions cm−2
Figure 6.7: AFM micrograph measurements of Ti-6Al-4V sputtered surfaces over
25µm×25µm (5µm×5µm inset) at 45° incidence angles. The ion beam direction is from
top to bottom on all images. Ion fluences are indicated in the legends.
73
Chapter 6. Results
6.3 IBS Incidence Angle Dependency for Ti and Ti-
6Al-4V
Angle of incidence dependence is a common parameter of study and found in the existing
literature (see Ref. [26] for the latest review of Si-based Ar ion irradiation pattern formation).
A critical non-zero (θc 6= 0) angle of incidence exists before any surface ripple can develop. In
the present work, ion implantation experiments were performed above the incubation fluence
for ripple formation (Φ = 6.0× 1016 ions cm−2).
6.3.1 Large-scale Morphologies
Experiments carried out at 23°, 45°, 49° and 67° angles reveal the formation of surface
structure that depend on ion fluence and angle of incidence. In essence, similar surface
morphologies are obtained for both materials. The surface structure for angles of incidence
lower than 45° appear flat. For higher angles of incidence a ripple structure develops. The
angle dependence of the formation of different shapes is shown in Figure 6.8 and Figure 6.9
(SEM images) comparing the metal and the alloy.
The variation of the surface morphology in some instances also depends in particular on
ion fluence. This sets a particular minimum ion fluence for surface structure formation at
the different angles in this work.
74
Chapter 6. Results
(a) θ = 23° at Φ = 1.68 × 1017 ions cm−2 (b) θ = 45° at Φ = 1.17 × 1017 ions cm−2
(c) θ = 49° at Φ = 1.33 × 1017 ions cm−2 (d) θ = 67° at Φ = 6.70 × 1016 ions cm−2
Figure 6.8: Scanning electron microscopy (SEM) images of Au ion implanted Ti surfaces at
a X5000 magnification. Angle of incidence and ion fluences are as indicated in the legends.
The ion beam direction is from top to bottom on all images.
75
Chapter 6. Results
(a) θ = 23° at Φ = 1.68 × 1017 ions cm−2 (b) θ = 45° at Φ = 1.17 × 1017 ions cm−2
(c) θ = 49° at Φ = 1.33 × 1017 ions cm−2 (d) θ = 67° at Φ = 6.70 × 1016 ions cm−2
Figure 6.9: Scanning electron microscopy (SEM) images of Au ion implanted Ti-6Al-4V
surfaces at a X5000 magnification. Angle of incidence and ion fluences are as indicated in
the legends. The ion beam direction is from top to bottom on all images.
6.3.2 Small-scale Morphologies
Ion implanted surface morphologies for different angles are also probe by Atomic Force
Microscopy (AFM). Profiles are obtained and compared for both materials. Implantation of
Ti and Ti-6Al-4V at 23°, forms large round shapes and pits. In the case of implantation at
45° and 49° ripples are visible on the surface for both materials. These images confirm that
a minimum ion fluence of Φ = 6.0× 1016 ions cm−2 is needed for ripple formation at 45° and
76
Chapter 6. Results
49°. For implantation at 67° angle, both materials show a ripple structure that is smaller
than those occurring at 45° and 49°.
For comparison, AFM micrographs of Ti and Ti-6Al4-V images 25µm × 25µm and 5µm
× 5µm (inset) are also shown (see Figures 6.10 and 6.12). Profiles were scanned at selected
positions for the average surface height behavior with respect to the angle of incidence. These
profile plots are shown in Figure 6.11 and Figure 6.13 for Ti and Ti-6Al-4V, respectively.
(a) θ = 23° at Φ = 1.68×1017 ions cm−2 (b) θ = 45° at Φ = 1.17×1017 ions cm−2
(c) θ = 49° at Φ = 1.33×1017 ions cm−2 (d) θ = 67° at Φ = 6.70×1016 ions cm−2
Figure 6.10: Atomic force microscopy (AFM) micro-graphs measurements of Ti-sputtered
surfaces over 25µm×25µm (inset scale of 5µm×5µm). Ion-beam direction of incidence is
from top to bottom on all images.
77
Chapter 6. Results
0 5 10 15 20 25
0
250
500
750
1000
23o
H
ei
gh
t P
ro
fil
e 
(n
m
)
x( m)
ion beam
(a) θ = 23° at Φ = 1.68 × 1017 ions cm−2
0 5 10 15 20 25
0
250
500
750
1000
45o
H
ei
gh
t P
ro
fil
e 
(n
m
)
x( m)
(b) θ = 45° at Φ = 1.17 × 1017 ions cm−2
0 5 10 15 20 25
0
250
500
750
1000
49o
H
ei
gh
t P
ro
fil
e 
(n
m
)
x( m)
(c) θ = 49° at Φ = 1.33 × 1017 ions cm−2
0 5 10 15 20 25
0
250
500
750
1000
67o
H
ei
gh
t P
ro
fil
e 
(n
m
)
x( m)
(d) θ = 67° at Φ = 6.70 × 1016 ions cm−2
Figure 6.11: Atomic force microscopy (AFM) profiles obtained from ion-implanted Ti for the
various angles of incidence. Line scans are taken at particular positions of the AFM image,
all from top to bottom in-line with respect to the ion beam direction.
78
Chapter 6. Results
(a) θ = 23° at Φ = 1.68×1017 ions cm−2 (b)θ = 45° at Φ = 1.17×1017 ions cm−2
(c) θ = 49° at Φ = 1.33×1017 ions cm−2 (d) θ = 67° at Φ = 6.70×1016 ions cm−2
Figure 6.12: Atomic force microscopy (AFM) micro-graphs measurements of Ti-6Al-4V sput-
tered surfaces over 25µm×25µm (inset scale of 5µm×5µm). Ion-beam direction is from top
to bottom on all images.
79
Chapter 6. Results
0 5 10 15 20 25
0
250
500
750
1000
23oHe
ig
ht
 P
ro
fil
e 
(n
m
)
x( m)
ion beam
(a) θ = 23° at Φ = 1.68 × 1017 ions cm−2
0 5 10 15 20 25
0
250
500
750
1000
45o
H
ei
gh
t P
ro
fil
e 
(n
m
)
x( m)
(b) θ = 45° at Φ = 1.17 × 1017 ions cm−2
0 5 10 15 20 25
0
250
500
750
1000
49o
H
ei
gh
t P
ro
fil
e 
(n
m
)
x( m)
(c) θ = 49° at Φ = 1.33 × 1017 ions cm−2
0 5 10 15 20 25
0
250
500
750
1000
67o
H
ei
gh
t P
ro
fil
e 
(n
m
)
x( m)
(d) θ = 67° at Φ = 6.70 × 1016 ions cm−2
Figure 6.13: Atomic force microscopy (AFM) profiles obtained from ion-implanted Ti-6Al-
4V. Line scans from top to bottom on AFM images in-line with respect to the ion beam
direction.
Surface statistical analysis was performed on the AFM micrographs. Average surface
height values (distance between minima and maxima of surface structures) are obtained and
compared to initial polished surfaces, see Table 6.1.
80
Chapter 6. Results
Material/Angle θ = 8° θ = 23° θ = 45° θ = 49° θ = 67°
Ti 71.4±17.5 105.0±27.7 354.4±151.1 166.0±39.8 156.7±47.4
Ti-6Al-4V 74.1±16.6 103.4±16.6 412.9±154.8 290.2±49.4 161.3±45.6
Ti 31.2±4.98
Ti-6Al-4V 103.1±14.1
Table 6.1: Maximum-to-minimum surface height difference of ion-implanted Ti and Ti-6Al-
4V over L× L = 25µm × 25µm in comparison to control samples (bottom two values). All
values are given in nanometers.
6.4 Micro-indentation of Surfaces
Another way to explore angles of incidence is through surface indentation. This was per-
formed utilizing a Vickers� micro-hardness tester. 200 grams weight were applied on the
surface of both Ti and Ti-6Al-4V target materials. The marking creates other planes cor-
responding to different angles of incidence in relation to an incoming ion beam direction.
Under customary indentation (CI), a rhomboidal mark is created on the surface. In a 45°
rotated version (RI), a square-like geometry indenter emerges (see Figure 6.14). The two
indenter geometries have been used in this work. In the square-like figure the entry angle is
easily determined for each of the four planes.
Ion implantation of Ti and Ti-6Al-4V with CI geometries are given in Figure 6.15. Similar
behaviors are observed at 32° but different at 62° angles. For ion implantation with RI
geometries see Figure 6.16. Implanted planes of 23°, 49° and 67° angles show similar shapes
for both materials. Ripples are formed at 49° and 67°, but not at 23° angles. The pattern
formation on these individual plane follow Figures 6.8 and 6.9 for Ti and Ti-6Al-4V.
An accumulation of material is observed on the borders of the indentations (see Fig-
ures 6.15 and 6.16). Ion induced surface transport is observed to occur in the direction of
the ion beam. These features are reminiscent of a viscous flow at the top-most surface layers
of the target material as observed in low and medium energy experiments in semiconductors
and insulator targets [148,149].
81
Chapter 6. Results
x
yy
x
ion beam
ion beam
32o 32o
62o62
o
23o
67o
49o 49o
Figure 6.14: Vickers� indentation for customary indentation (CI) and for rotated version
(RI) geometries. An ion beam at 45° incidence angle enters from top to bottom on both
figures. Angles of incidence shown on the triangular planes are shown.
Figure 6.15: Customary indentations (CI) with ripples for Ti (left) and Ti-6Al-4V (right)
surfaces. Ion fluence of Φ = 1.17× 1017 ions cm−2 for Ti and Φ = 2.15× 1016 ions cm−2 for
Ti-6Al-4V. The ion-beam direction is from top to bottom on both images.
82
Chapter 6. Results
Figure 6.16: Rotated indentations (RI) with ripples for Ti (left) and Ti-6Al-4V (right)
surfaces. Ion fluence of Φ = 6.7×1016 ions cm−2 for both materials. The ion-beam direction
is from top to bottom on both images.
6.5 Ripple Elemental Composition
Ion implantation changes the elemental concentration of the target material, depending
on ion fluence. Since surface erosion is known to occur on the top layers of the titanium
target, both titanium and gold concentration change continuously. The alloy also changes
concentration with respect to the content of titanium, aluminum, vanadium and gold.
The electron dispersive spectroscopy (EDS) surface analysis mode of a scanning electron
microscope (SEM) has been used. EDS characterizes a depth of 1-2 µm of the target material.
The analysis show the accumulation of Au on the crest of high fluence ripples (see Figures 6.17
and 6.18). This Au accumulation occurs on the side that faces the incident ion beam, but
the atomic concentration of Ti atoms is uniformly distributed on the target material. For the
alloy case, a similar gold concentration occurs at the peaks of the surface ripple structure. On
the other hand, the titanium, aluminum and vanadium elements are uniformly distributed
on the alloy. It is important to note that these images of the elemental composition are a
representation of a 3D volume from a 2-3 µm of depth of the sample.
83
Chapter 6. Results
(a) Gold (Au) element mapping from
implanted surface.
(b) Titanium (Ti) element mapping
from implanted surface.
Figure 6.17: Micrograph of the surface morphology of 1.0-MeV Au+ ion implanted Ti. Ion
fluence of Φ = 4.7× 1017 ions cm−2. Ion beam direction from left to right on all images.
84
Chapter 6. Results
(a) Gold (Au) element mapping from
implanted surface.
(b) Titanium (Ti) element mapping
from implanted surface.
(c) Aluminum (Al) element mapping
from implanted surface.
(d) Vanadium (V) element mapping
from implanted surface.
Figure 6.18: Micrograph of the surface morphology of 1.0-MeV Au+ ion implanted Ti-6Al-
4V. Ion fluence of Φ = 4.7 × 1017 ions cm−2. Ion beam direction from left to right on all
images.
85
Chapter 6. Results
The average atomic weight percentage concentration of the implanted material is given
in Tables 6.2 and 6.3. Au ion implanted Ti with a fluence of Φ = 4.7× 1017 ions cm−2 show
a 71.2% concentration of Ti and a 28.8% concentration of Au. On the other hand, Au ion
implanted Ti-6Al-4V with the same ion fluence shows a 67.9% concentration of Ti, a 25.1%
concentration of Au, a 3.8% concentration of Al and 3.2% concentration of V.
Element Atomic Concentration (wt.%)
Ti 71.2 ± 0.4
Au 28.8 ± 0.4
Table 6.2: Elemental concentration of Au im-
planted Ti.
Element Atomic Concentration (wt.%)
Ti 67.9 ± 0.4
Au 25.1 ± 0.4
Al 3.8 ± 0.1
V 3.2 ± 0.2
Table 6.3: Elemental concentration of Au im-
planted Ti-6Al-4V.
6.6 XPS Analysis
X-ray photoelectron spectroscopy (XPS) analyses have been performed on Au ion implanted
Ti and Ti-6Al-4V with respect to depth. In our particular work, XPS technique characterizes
the atomic composition of the topmost layers of target materials up to a depth of 10 nm.
With further erosion, steps of 10 nm layers have effectively been measured.
6.6.1 Characterization
X-ray photoelectron spectroscopy (XPS) analyses were performed on Ti and Ti-6Al-4V after
Au ion implantation at Φ = 6.4 × 1016 ions cm−2 and Φ = 1.17 × 1017 ions cm−2 fluences,
respectively. In this particular study, both ion implantations were performed at 45° angles.
The top surface layers of the material are analyzed by monochromatic X-rays (Al Kα)
86
Chapter 6. Results
producing photoelectrons characteristic of the atoms present in the sample. Core level and
Auger electrons are collected by a detector and quantified with respect to their energy [141].
General spectra and elemental profile depth are obtained from both materials (see Figure 6.19
through Figure 6.21). For each element of the implanted material, high-resolution spectra
are obtained where a data fitting is applied.
XPS analysis of Au implantedTi sample
A 16-stage erosion process was performed on the Ti sample, reaching a maximum of 60 nm
beneath the initial undulated surface, up to the ion implantation maximum penetration depth
(calibrated by RBS measurements [92]). An initial accumulation of oxides on the surface was
observed and it diminished as the surface is eroded away. The concentration of Ti decreases,
and then stabilizes. On the other hand, the concentration of Au increases then stabilizes
(see Figure 6.19).
Of the sixteen spectra obtained, six high-resolution peaks are chosen for fitting, where
there is a substantial change in concentration: the first-3 levels, the fourth, twelfth and
sixteenth levels (these are labeled in Figure 6.20 with arrows). The XPS database of the Na-
tional Institute of Standards and Technology (NIST) [152] was used. For all peaks, Gaussian-
Lorentzian (Voigt type) mixed functions have been utilized with help of the Shirley type
background substraction method, similar to that used by Biesinger [141].
The high resolution Au 4f peak deconvolution confirms AuTi3, Au2Ti, Au2O3 compounds
(see Figure 6.24 on the left). Furthermore, the Ti 2p peak deconvolution also reveals TiO2,
TiO, Ti2O3, AuTi3, Au2Ti (see Figure 6.24 on the right). Metallic phases of Au and Ti
are also observed on these deconvolutions [141,153,154]. Native oxides of the titanium target
is observed to decrease as the target is eroded away [155,156]. The atomic compounds are
illustrated with respect to depth in Figure 6.25.
87
Chapter 6. Results
1200 1000 800 600 400 200 0
CLK1 Au5s O2s
Ti5p
Ti3s
Au4p
Ti2s
Au4dAu4pAu4s
OKL1
TiLM2
TiLM3
C1s
O1s Ti2p
C
ou
nt
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
Au4f
Figure 6.19: XPS analysis of 1.0-MeV Au implanted Ti at θ = 45° for Φ = 6.4 × 1016 ions
cm−2 fluence. (left) A 16 stage erosion process was performed on the topmost surface of the
implanted material.
0 10 20 30 40 50 60
0
10
20
30
40
50
60
70
16120 42
at
om
ic
 w
t(%
)
Depth (nm)
Au 4f
Ti 2p
O 1s
1
Figure 6.20: XPS spectrum for each erosion stage showing the principal core-level peaks of
Ti, Au and O. The mark arrows show the levels that were analyzed.
88
Chapter 6. Results
1200 1000 800 600 400 200 0
V2p Al2p
C
ou
nt
s/
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
CLK1 Au5s
O2s
Ti5p
Ti3s
Au4p
Ti2s
Au4d
Au4p
Au4s
OKL1
TiLM2
TiLM3
C1s
O1s
Ti2p
Au4f
Figure 6.21: XPS analysis of 1.0-MeV Au implanted Ti-6Al-4V at θ = 45° for Φ = 1.17×1017
ions cm−2 fluence. (left) A 15 stage erosion process was performed on the topmost surface
of the implanted material.
0 20 40 60 80 100 120 140
0
10
20
30
40
50
60
70
V 2pAl 2p
at
om
ic
 w
t(%
)
Etch Time (s)
Au 4f
Ti 2p
O 1s
Figure 6.22: XPS spectrum for each erosion stage showing the principal core-level peaks of
Ti, Au, Al, V and O.
89
Chapter 6. Results
94 92 90 88 86 84 82 80
Au 4f7/2
C
ou
nt
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
57.6nm
43.2nm
14.4nm
7.2nm
3.6nm
surface
Au 4f5/2
490 485 480 475 470 465 460 455
C
ou
nt
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
57.6nm
43.2nm
14.4nm
7.2nm
3.6nm
surface
Ti 2p3/2
Ti 2p1/2
Figure 6.23: The six erosive levels chosen from the XPS high-resolution Au 4f (left) and Ti
2p (right) core-level spectra for analysis.
90 88 86 84
C
ou
nt
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
Au 4f
simulation
AuTi3
Au2Ti
Au0
Au2O3
468 466 464 462 460 458 456 454
plasmons
satellites
Ti2O3
TiO2
TiO Au2Ti
Ti0
AuTi3
Ti 2p
simulation
C
ou
nt
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
Figure 6.24: Example of the XPS high-resolution Au 4f (left) and Ti 2p (right) deconvolution
of the core-level spectra. These two plots represent the 16 level of the erosion process of
figure 6.19.
90
Chapter 6. Results
0 10 20 30 40 50 60
0
10
20
30
40
50
60
70
80
90
100
at
om
ic
 w
t(%
)
Depth (nm)
AuTi3
Au0
Au2TiAu2O3
0 10 20 30 40 50 60
0
10
20
30
40
50
60
70
80
90
100
at
om
ic
 w
t(%
)
Depth (nm)
AuTi3 Ti Au2Ti
TiO
Ti2O3
TiO2
Figure 6.25: Depth profile curves of Au and Ti compounds identified after peak deconvolution
under XPS high-resolution Au 4f (left) and Ti 2p (right) core-level spectral analysis.
XPS analysis of ion implanted Ti-6Al-4V sample
The Au ion implanted alloy (Ti-6Al-4V) was also subjected to XPS surface analysis. A
15-stage analysis process was performed on the alloy (see Figure 6.21 for the general spectra
and profile elemental concentration). The concentration of the elements on the target alloy
behaves in a similar fashion to that of ion implanted titanium sample. Small concentrations
of aluminum and vanadium are observed from the general XPS spectra. High-resolution
spectra from Au 4f, Ti 2p, Al 2p and V 2p are shown in Figure 6.26. In the high-resolution
spectra, aluminum and vanadium have relatively small concentrations, and are overshadowed
by the background noise.
91
Chapter 6. Results
94 92 90 88 86 84 82 80
Au 4f7/2
C
ou
nt
s/
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
surface
Level 1
Level 2
Level 3
Level 12
Level 15
Au 4f5/2
485 480 475 470 465 460 455 450
Ti 2p1/2
C
ou
nt
s/
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
surface
Level 1
Level 2
Level 3
Level 12
Level 15
Ti 2p3/2
82 80 78 76 74 72 70 68 66
C
ou
nt
s/
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
surface
Level 1
Level 2
Level 3
Level 12
Level 15
Al(0)/Al0
Al(III)/Al2O3
524 520 516 512 508
C
ou
nt
s/
s 
(a
rb
. u
ni
ts
)
Binding Energy (eV)
surface
Level 1
Level 2
Level 3
Level 12
Level 15
V(0)/V0
V(IV)/VO2V(V)/V2O5
Figure 6.26: XPS high-resolution Au 4f (top-left), Ti 2p (top-right), Al 2p (bottom-left) and
V 2p (bottom-right) core-level spectra.
6.7 IBS Ion-atom Dependency
Other target material IBS experiments have been carried out, maintaining the same condi-
tions of energy, fluence and angle of incidence. 1.0-MeV Au+ ion implantation of stainless
steel (SS 316L) is carried out at 45° for Φ = 1.17×1017 ions cm−2. The surface apparently re-
mains flat by the ion implantation (see Figure 6.27 on the left side), but after AFM analysis,
the formation of perpendicular ripples at even smaller scales was observed (see Figure 6.28
92
Chapter 6. Results
on the left side). Additionally, 1.0-MeV Ag+ ion implantation of Ti at 45° for Φ = 3.0×1017
ions cm−2 is carried out. In the case of silver ion implantation of Ti, the formation of a
structure similar pine bark-like shapes is seen to occur (see Figures 6.27 and 6.28 on the
right side). Profile analysis was carried out using the Gwyddion software, see Figure 6.29
corresponding to the profile cuts shown in Figures 6.28 for Au and Ag ion implantation of
SS316L and Ti, respectively.
Figure 6.27: SEM micrographs of 1.0-MeV Au+ ion implantation of stainless steel (S.S.
316L) at 45° for Φ = 1.17 × 1017 ions cm−2 (left) and 1.0-MeV Ag+ ion implantation of Ti
at 45° for Φ = 3.0 × 1017 ions cm−2 (right). The ion beam direction is from top to bottom
on both images.
The ion-atom variability surface morphology observed in these experiments highlight the
possible dependence of other physical effects. These effects may take into account distinct
critical angles of incidence which determines the formation or absence of surface structures.
Recent approaches due to the ion induced stress is able to determine the variation of the
critical angle of incidence [157,158], thus a large difference of material type.
Although the formation of surface structure appears to be universal in respect to the
ion-atom target combination. The varying degree of shapes is believed to be influenced by a
number of issues that require further study. In regard to these issues, our results point out
the interest at high energies for metallic samples.
93
Chapter 6. Results
Figure 6.28: AFM micrograph of 1.0-MeV Au+ ion implantation of SS 316L at 45°(see left
plot). AFM micrograph of 1.0-MeV Ag+ ion implantation of Ti at 45°(see right plot).The
ion beam direction is from top to bottom on both images.
0 5 10 15 20 25
20
40
60
80
H
ei
gh
t P
ro
fil
es
 (n
m
)
x( m)
20
40
60
80
20
40
60
80 1
3
2
0 5 10 15 20 25
150
300
450
600
750
900
H
ei
gh
t P
ro
fil
es
 (n
m
)
x( m)
150
300
450
600
750
900
150
300
450
600
750
900
3
2
1
Figure 6.29: Height profile scans of ion implanted stainless steel (SS316L) on the left and
titanium surfaces at θ = 45° corresponding to the top views shown in Figure 6.28 on the
right. The ion beam direction is from left to right on both plots.
94
Chapter 7
Discussion
Formation of surface structures on Ti and Ti-6Al-4V occurs after 1.0-MeV Au+ ion implan-
tation at 45° angles for fluences above Φ = 6.0× 1016 ions cm−2. The formation and growth
of ripples and surface structures depend on the experimental conditions including the angle
of incidence, fluence and target material.
Changes of the surface morphology observed in this work are similar to those found in
the published literature at low and medium energy experiments of semiconducting materi-
als [26]. These low up to medium energy experiments of semiconductors are well described
by continuum models. It is conjecture that similar physical mechanisms may be invoked
in order to describe surface structures at high-energies. A basic assumption is the scaling
nature of the size of the surface structure with respect to the ion energy.
The interpretation of our experimental results is addressed in this chapter and orga-
nized as follows: (§7.1) experiment and simulation comparison; (§7.2) near-surface atomic
damage and energy loss processes; (§7.3) Bradley-Harper type theory considerations; (§7.4)
Bradley-Shipman type theory considerations and (§7.5) asymptotic non-linear effects of ion
implantation. The last two sections are devoted to an overview of the continuum model
approach (§7.6) and applications of surface structures of Ti and Ti-6Al-4V (§7.7).
95
Chapter 7. Discussion
7.1 Experiment and Simulation: Au Ion Implantation
of Ti and Ti-6Al-4V
TRIM simulations [91] of 1.0-MeV Au ion implantation of titanium and its alloy were carried
out for normal and at 45° angles (see Figure 7.1 for Au ion implantation of Ti). At normal
incidence, Au ions are implanted at an average depth of Rp = 0.16 ± 0.04 µm on both
materials. On the other hand, ions are implanted at a depth of Rp = 0.11 ± 0.03 µm for
45° angles of incidence. Even though, implantation ranges are nearly similar, higher erosion
yields occurs for the 45° incidence angles (differing on their ion distribution). Comparable
Au ion implantation statistics occurs whenever the titanium alloy (Ti-6Al-4V) is simulated.
Figure 7.1: Full-cascade simulations of 1.0-MeV Au ion implantation of Ti at normal (left
image) and at 45° (right image) angles. The ion beam direction is from left to right on both
images. The ion implantation and damage region is approximately localized at half of a
micrometer of depth.
For instance, during ion beam analysis experiments Rutherford backscattering (RBS)
measurements revealed that the ion distribution of the implanted material is located in
the first 300nm of depth. A comparison between RBS and TRIM simulations is shown in
Figure 7.2. The ion distribution is similar for both materials, see Ref. [92]. In general,
96
Chapter 7. Discussion
distorted Gaussian distributions of the implantated ions are seen to occur for both materials
associating the fluence of the bombarding ion. Deformed distributions also hint on the limit
of ions being implanted into a target material, which is of the order of ∼ 1017 − 1018 ions
cm−2 for semiconducting materials [146].
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Au
 C
on
ce
nt
ra
tio
n
Depth ( m)
RBS
SRIM 
0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42
0.00
0.04
0.08
0.12
0.16
0.20
0.24
0.28
Au
 C
on
ce
nt
ra
tio
n
Depth ( m)
SRIM
RBS
Figure 7.2: Rutherford backscattering spectroscopy (RBS) spectra of the Au ion concentra-
tion with respect to depth for 1.0-MeV Au ion implantation of Ti at normal (left image) and
45° angles (right image).
7.2 Atomic Damage and Energy Loss Processes
During ion implantation, the near-surface region of the target generates atomic defects in-
cluding vacancies and interstitials. Atomic displacements, kicked out atoms (erosion), and
ion-atom relaxation mechanisms are also known to be caused by the ion implantation process.
The atomic damage is quantified from binary collision approximation (BCA) simulations
(TRIM simulations), as mention above in §7.1. The ion distributions resemble ellipsoidal-
shapes often utilized in theoretical studies at low energies [1,101](see Figures 4.2 and 4.3). The
spread of the initial ion kinetic energy is also known to occur through secondary effects (e.g.
phonon and plasmon generation).
97
Chapter 7. Discussion
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0
50
100
150
200
250
300
Au ions
Ti recoils
Io
ni
za
tio
n 
(e
V/
A/
io
n)
Depth( m)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0
1
2
3
4
5
Ph
on
on
s 
(/A
 /i
on
)
Depth( m)
Ti recoils
Au ions
0
100
200
300
400
0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.0
0.1
0.2
0.3
0.4
0.5
Va
ca
nc
ie
s 
(/A
/io
n)
Depth( m)
Ti
Au ions
0
2
4
6
8
10
Figure 7.3: TRIM calculations of ion-atom ionization (top-left image), phonon (top-right
image) and vacancy production (bottom image) of Au ions and Ti atoms with respect to
depth during 1.0-MeV Au ion implantation of Ti. The red-dotted curve represents those
from titanium atoms while the blue-dotted curve from incident Au ions.
The ion energy percentage losses have been obtained from simulation data; ionization,
vacancies and phonons (see Figure 7.3). In particular for a 1.0-MeV Au ion into a titanium
target, the energy losses are approximately given by ionization with ∼ 51.1%, phonons with
∼ 45% and vacancies with ∼ 3.8% (see Table 7.1). Ionization is generally viewed as a drag
force between the ion and the sea of electrons of the target. Phonons result from ion-atom
collisions generating oscillations of the atomic lattice. Finally, vacancies result from ion-atom
collision which displaces atoms of their original positions unable for them to return [1].
98
Chapter 7. Discussion
For these high energy ions a two-step proccess occurs: (1) high ionization decreases the
initial kinetic energy of the ion to medium range (∼ 500-keV) afterwards (2) slow down is
mediated by vacancy production, i.e. nuclear stopping is the dominant process at this stage,
similar to those ocurring in medium and low energies. This latter steps produces the largest
ion-induced atomic damage of the target material with additional thermal losses through
phonons. A continuum approach takes a higher role at these energies [157].
At 1.0-MeV, the stopping powers of Au ions in Ti are dE/dxnucl. = 10.31 MeV cm
−2
mg−1 and dE/dxelec. = 4.65 MeV cm
−2 mg−1 for nuclear and electronic, respectively. The
nuclear-stopping power contribution is twice the electronic making it the important factor
in the slowing down of ions (see Figure 7.4 for a comparison between nuclear and electronic
stopping power of Au ions into titanium at MeV energies).
In general, energy loss effects are centered on the near-surface region of the target material
being twice of order with respect to the ion range. This in consequence may change the
surface morphology of the material on the first top layers due to atomic displacement effects.
% Energy loss Ions Recoils Total energy loss
per incident ion
Ionization 23.11 28.06 511.7 keV
Vacancies 0.06 3.77 38.3 keV
Phonons 0.15 44.85 450.0 keV
Table 7.1: TRIM simulated energy loss processes from 1.0-MeV Au ion implantation of
Ti. Ionization and phonons have approximatelly higher contributions in the energy loss in
comparison to the atomic generated vacancies by the ion implantation process.
During these ion-atom processes, an increase oftemperature is expected. As for metals,
there appear to be a high tendency to follow activated-temperature processes [21]. Further-
more, for general cases ionization and phonons generates out-of-equilibrium conditions that
lead to the formation of compounds as those occuring at low and medium energy ion irradia-
tion (see section §6.7). These explicitly couple special thermodynamic conditions that favors
99
Chapter 7. Discussion
the nucleation of compounds [1]. Meanwhile, in the collective behavior of vacanies, atoms are
kicked out of the near-surface region of the target material accounting for instance in relative
small fraction of the energy loss by the erosion of the target material.
0 1 2 3 4 5 6
0
2
4
6
8
10
12
St
op
pi
ng
 P
ow
er
(1
03
 M
eV
 c
m
2 /g
)
Energy (MeV)
Electronic
Nuclear
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
R
an
ge
 (
m
)
Energy (MeV)
Rp( m) = 0.15
0.95
Figure 7.4: Nuclear and electronic stopping power of Au ions in titanium at a few MeVs of
energy. A power law fit is performed on the ion projected penetration depth, confirming a
scaling with energy as d ∼ E2m, where m ≈ 0.47. [160]
7.3 Bradley-Harper Type Theories Considerations
In Bradley and Harper type theories (see sections §4.2 through §4.5), the formation of surface
structure is seen to occur gradually. This is given by an initial rough surface which changes
as ion implantation proceeds. The emerge of surface ripples occurs before saturation of the
interface height at high fluences (see sections §6.2 and §6.3 for SEM and AFM micrographs,
respectively). A qualitative analysis follows and performed based on the obtained AFM mi-
crographs, demonstrating the evolution of surface structures occurring at 45° angles for both
materials. This followed up is due to the consideration of a continuum field approximation,
as the crystalographic phase (grain size ≈ 100µm for Ti and ≈ 10µm for the Ti-6Al-4V α
phase surrounded by the β phase) of the target material is absent in the surface evolution [92].
100
Chapter 7. Discussion
7.3.1 Profile Analysis
For the system of study (both titanium and the alloy), formation of surface structures requires
a minimum fluence of Φ = 6.0 × 1016 ions cm−2, this has been labeled as the incubation
fluence, see Ref. [147]. Two types of surface structures develop, nearly symmetric (small
structures) and antisymmetric (large structures). Defined as follows: nearly symmetric
shapes have similar slopes, while asymmetric shapes differ in their uphill and downhill slopes.
In the case of asymmetric shapes, uphill slopes tend to be smooth while downhill slopes fall
sharply with respect to the direction of the ion beam.
In analogy to sand-dune formation on deserts, two type of structures are often observed
in experimental field studies [143,144]; small rippling undulations and large surface structures
often labeled as barchan dunes [125,142,145]. The former emerging at the beginning of sand
erosion while the latter appears when large enough surface undulations develop shadowing
small structures, this in turn promotes higher disordering and barchan-like dunes grow in
height and width.
In order to gain insights of the development of surface structures in titanium and its
alloy, profile analysis is performed on developed ripples. This is seen to resemble desert
ripple surface formation, where small surface structures develop then being overshadowed
by larger structures at high enough ion fluence.
The statistical analysis was performed with help of the Gwyddion software [139] along with
WSxM [140]. Numerical tools including the 1-Dimensional Power Spectral Density Function
(1D PSDF), and 2D Auto-Correlation Function (2D ACF) were used. These were used in
order to corroborate the size and ordering of measured surface structures. Periodicity and
height of surface structures of ∼ µm may be defined albeit their disordering behavior.
101
Chapter 7. Discussion
(a) Φ = 6.5× 1016 ions cm−2 (b) Φ = 6.7× 1016 ions cm−2
(c) Φ = 1.17 × 1017 ions cm−2 (d) Φ = 4.70 × 1017 ions cm−2
Figure 7.5: AFM micrograph measurements of Ti-sputtered surfaces over 25µm×25µm at
45° incidence angles. The ion beam direction is from top to bottom on all images. The ion
fluences are indicated in the legends. The numbered vertical solid lines on each top-view
indicate the location of the corresponding line profiles shown in Figure 7.6.
102
Chapter 7. Discussion
0 10 20
0
200
400
600
3
2
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t P
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 (n
m
)
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(a) Φ = 6.5× 1016 ions cm−2
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x( m)
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1000
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(c) Φ = 1.17 × 1017 ions cm−2
0 10 20
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x( m)
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ei
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t P
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 (n
m
)
0
500
1000
1500
2000
(d) Φ = 4.70 × 1017 ions cm−2
Figure 7.6: Height profile scans of ion implanted titanium surfaces at θ = 45° corresponding
to the top views shown in Figure 6.6. The ion beam direction is from left to right.
In the case of Ti samples, Figures 7.5(a) and 7.6(a) surface symmetric shapes have a
correspoding characteristic width size of ≈ 1.0 micrometer where in comparison to asym-
metric shapes with a varying size in the range of ≈ 2.5µm and height of 200nm (see also the
profile cuts given in Figure 7.6 for each of the mention AFM image). The increase of large
(asymmetric) surface features is observed for the following fluence, Figure 7.5(b). Symmetric
shapes have a width ≈ 0.7µm and heigth of 80nm while asymmetric with ≈ 2µm of width
and height 300nm. After this ion fluence, single structures appear to emerge laterally to form
103
Chapter 7. Discussion
large enlongated chain-like waves. In Figure 7.5(c), large (asymmetric) structures populates
the surface with ≈ 1.5µm width and height of 500nm. Finally in Figure 7.5(d), asymmetric
structures fill the surface with ≈ 3µm of width and height at ≈ 1µm. Saw-tooth profiles
with flat tops have developed at this ion fluence (see Figure 7.6(d) for the surface profile).
(a) Φ = 6.5× 1016 ions cm−2 (b) Φ = 6.7× 1016 ions cm−2
(c) Φ = 1.17 × 1017 ions cm−2 (d) Φ = 4.70 × 1017 ions cm−2
Figure 7.7: AFM micrograph measurements of Ti-6Al-4V sputtered surfaces over
25µm×25µm at 45° incidence angles. The ion beam direction is from top to bottom on
all images. Ion fluences are indicated in the legends. The numbered vertical solid lines on
each top-view indicates the location of the corresponding line profiles shown in Figure 7.8.
104
Chapter 7. Discussion
0 10 20
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100
150
200 3
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m
)
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50
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200
(a) Φ = 6.5× 1016 ions cm−2
0 10 20
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 (n
m
)
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 (n
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1000
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2000
0
500
1000
1500
2000
(d) Φ = 4.70 × 1017 ions cm−2
Figure 7.8: Height profile scans of ion implanted Ti-6Al-4V surfaces at θ = 45° corresponding
to the top views shown in Figure 7.7. The ion beam direction is from left to right.
On the other hand, the titanium alloy (Ti-6Al-4V) behaves slightly different. Initially an
absence of surface structure is observed, see Figure 7.7(a), belonging to a lower or near value
to the incubation fluence. In Figure 7.7(b), small (nearly symmetric) surface structures
start to appear with characteristic width size of ≈ 0.7µm and height ≈ 70nm and large
(asymmetric) with ≈ 1.5µm and heigth of ≈ 300nm. As ion implantation proceeds, large
(asymmetric) surface structures grow with respectto their widths and heights. Figure 7.7(c),
has been populated by large (asymmetric) structures of width ≈ 1.5µm and height ≈ 600nm.
105
Chapter 7. Discussion
Subsequently at higher fluences, Figure 7.7(d), asymmetric shapes have completetly filled
the surface. This is similar to the observed in the case of titanium surfaces but with the
absence of flat tops. Shapes of width ≈ 2.5µm and height 1000nm are measured.
In regard to the surface evolution at 45°, specifically the fomation of a ripple structure
appears to be almost universal in respect to the target material. For our particular case, this
is similar to low and medium energy experiments of semiconducting materials, even those at
high energies, while their size size appear to scale with respect to the ion energy.
7.3.2 Atomic Processes: Surface Erosion
For normal-incidence ion implantation experiments, ions penetrate higher depths (small
erosion yields) in comparison to oblique incidence angles. In particular for 1.0-MeV Au
into Ti at normal-incidence erodes approximately 5 titanium atoms per incident ion. At
increasing angles of incidence, the ion distribution is closer to the surface, inducing higher
erosion yields. At an angle of 45° each 1.0-MeV Au ion erodes approximately 11 titanium
atoms per incident ion (see Table 7.2). In the case of the Ti alloy, initially, it behaves
slightly different due to aluminum and vanadium content. Each elemental component erodes
depending on the atomic concentration of the target material and binding energies [97,98,99].
The erosion yield of Ti and Ti-6Al-4V at the various angles of incidence is given in
Table 7.2 and Table 7.3, respectively. Erosion yields increase with angle of incidence as also
seen in Figure 4.4 of the theory section. This erosion of surface atoms is known to correlate
to changes in the top surface layers (at least in Bradley-Harper linear type theories), where
the emerge of surface topographies are often observed that depend on the ion and target
material combination.
As experimentally observed in our work (see section §6.3) when the angle of incidence is
increased the ripple structure decreases in wavelength along with the surface height (distance
106
Chapter 7. Discussion
between maxima and minima) in comparison to that at 45°. This decrease in wavelength
is not what is expected if erosion were the only pattern forming mechanism. This rather
opposite effect has been explored in other works [161,162,163], suggesting additional mechanisms
driving the pattern, e.g. mass redistribution and/or viscous flow. Thus, even in the presence
of irregularities of the structure, the appearance of a pattern occurs regardless of the angle
of incidence.
Angle/Yield (ion/atom) Ti
0° 5.1
23° 5.8
45° 10.8
49° 12.4
67° 27.2
Table 7.2: Sputtering yield of 1.0-MeV Au ion implanted Ti surfaces from TRIM simulations.
Number of sputtered atoms per Au ion at various angles of incidence.
Angle/Yield (ion/atom) Ti Al V
0° 4.6 0.3 0.2
23° 5.9 0.4 0.2
45° 10.4 0.7 0.4
49° 12.0 0.8 0.5
67° 26.7 1.7 1.1
Table 7.3: Sputtering yield of 1.0-MeV Au ion implanted Ti-6Al-4V surfaces from TRIM
simulations. Number of sputtered atoms per Au ion at various angles of incidence.
An initial ripple structure with length scale∼ 1µm is seen to evolve into larger asymmetric
shapes (see previous section §7.3.1). 1D PSD analyses (see review article Ref. [147] and online
107
Chapter 7. Discussion
supplemental material) of the AFM images were performed, thus confirming its growth and
ordering. The evolution of surface ripples may be described by similar low-medium energy
theories in respect to an initial ion-induced instability with the exception to nonlinear terms
that simulate irregularities of the pattern.
Following the previously mention analogy, sand ripple and dune formation in deserts,
sand particles are transported by air impacts of both large and small particles affecting
the density of the top surface layers. Two physical mechanisms are known to control the
surface evolution; surface erosion and re-deposition. These surface properties are displayed
by crawling and saltation as particles move and bounce along the surface. The density of
crawling in comparison to saltation sand particles leads to a surface instability with the
emergence of a rippling structure. If air blow is high enough, dunes are formed which vary
in size and shape that depend on the air direction [125].
In contrast, Au ion erosion of Ti and its alloy, the mass of the particle plays the role of
the variation of the particle size. Its effect rest on the variability of the diffusive behavior of
the impacting ion. In particular, particle mass contributes to momentum transfer as those
investigated in ion stress-induced models. This leads to a re-distribution effect analogous to
diffusive terms that accompanies the formation of a surface structure [161]. Additionally, the
alloy has a greater diffusivity in the presence of additional atoms, aluminum and vanadium.
This has been observed as a slightly faster growth rate of the ripple structure in case of the
alloy [147].
7.3.3 Atomic Processes: Surface Diffusion/Relaxation
The combination of erosion and relaxation mechanisms give rise to a rippling pattern on the
surface of the target material (see section §4.2). This is a competing effect of roughening due
to the ion beam and smoothing by diffusive and relaxation mechanisms [3]. In pattern forma-
108
Chapter 7. Discussion
tion theories [34], the prevalence of instabilities is a general hallmark that underlines growth
mechanisms. This concept is represented in this work by ion-induced atomic processes.
The formation of surface ripples where their wavelength is roughly characterized by the
relation between surface tension and relaxation coefficients (see equation (4.14) of section
§4.3). Note that this occurs whenever a linear Bradley-Harper (BH) type theory is consid-
ered, however this is only good for high-temperature experiments [21] as considered in our
work [92].
Surface Stabilization
The concept of competing mechanisms of surface erosion (instability) and a thermal activated
Herring-Mullins relaxation (stabilizing) mechanism [32] highlight generic growth dynamics [34]
by a instability-stability relation. The growth mechanism in the case of ion-sputtered surfaces
is labeled as the Bradley-Harper instability and known to appear when a surface curvature
dependent erosion is present while a fourth order relaxation mechanism smooths out the sur-
face [3]. Recent studies have been questioning the importance of a curvature erosive process,
as near normal incidence experiments appear to contradict the results of the Bradley-Harper
model [161,164]. Other experimental results also differ in surface morphology for oblique inci-
dence angles [63,64,82].
Utilizing the Makeev-Cuerno-Barabási (MCB) theory, we computed the effective surface
tension coefficients relating the instability of ion-sputtered surfaces where the working mech-
anism is assumed to be surface erosion. For Au ions implantated in titanium surfaces at
1.0-MeV, the ion-induced surface tension coefficients are given in Figure 7.9. The greatest in
negative-value for the surface tension determines the direction of surface ripples (in our case
the x-direction). A transition from perpendicular ripples to parallel is predicted to occur at
an angle of incidence θ ≈ 62°. This has not been observed in our work, instead implantation
at 67°, ripples of smaller sized are observed (see surface morphologies at 67° in section §6.3).
109
Chapter 7. Discussion
0 15 30 45 60 75 90
0.00
0.05
0.10
0.15
0.20
0.25
0.30
v 0
(A
/s
)
Angle( )[Degrees]
0 15 30 45 60 75 90
-200
-150
-100
-50
0
50
100
150
200
x,
y(A
2 /s
)
Angle( )[Degrees]
x
y
Figure 7.9: Left plot: surface erosion velocity with respect to the angle of incidence. Right
plot: surface tension coefficients; (red line) νx(θ) coefficientand (blue line) νy(θ) coefficient.
At θ ≈ 62° angle of incidence a ripple rotation should occur for the present experimental
conditions.
Meanwhile, the surface erosion velocity behaves as expected from the MCB theory (in-
creasing for oblique incidence angles). For the surface tension coefficients, the higher in-
stability occurs in the x-direction as experimentally observed, signaled by the formation of
surface ripples with a wave-vector in the direction of ion beam.
Along this similar discussion, a natural question was recently raised: “Is sputtering rel-
evant for ion-induced self-organized pattern formation?” In the review of Hofsäss et al. [165];
mass re-distribution and surface relaxation play the role in the pattern formation. A coop-
erative behavior between erosive and mass re-distribution effects occurs for certain angles of
incidence. At small angles of incidence, a surface remains flat (stable), after 45° angle, sur-
face ripples (unstable) appear. Additional arguments poised by Davidovitch et al. [164], Madi
et al. [161] and Norris et al. [163,167,168]; a mass re-distribution term like the Carter-Vishnyakov
(CV) mechanism [76] can in principle destabilize the surface for certain angles of incidence
leading to the formation of surface structures. A surface patterning occurs for angles higher
than 45° and absence for lower angles, i.e. in the case of near-normal incidence angles the
110
Chapter 7. Discussion
mass re-distribution smooths out the surface.
The ion-induced mass re-distribution of Au ion-sputtered titanium surfaces is given in
Figure 7.10. These are approximated according to the theory of Davidovitch-Madi (DM)
mass re-distribution contribution [161,162,163,164,166] to ion-sputtered titanium surfaces.
0 15 30 45 60 75 90
-200
0
200
400
600
800
S x
(A
2 s
-1
)
Angle( )[Degrees]
 Erosive
 Composite
-8
0
8
16
24
32
 Redistributive
0 15 30 45 60 75 90
-200
0
200
400
600
800
S y
(A
2 s
-1
)
Angle( )[Degrees]
 Erosive
 Composite
-8
0
8
16
24
32
 Redistributive
Figure 7.10: Effective ion-induced surface tension coefficients from erosive and mass redis-
tribution contributions. Sx (left) and Sy (right) surface tension coefficients from Au ion
implantation of titanium at 1.0-MeV of energy.
Additionally this mass re-distribution effect contributes to a surface current in the ion
beam direction. A flow of material has been observed experimentally by the accumulation
of material on the borders of Vickers’ indentations (see section §6.4 and Ref. [147]). This
transport phenomenon has been observed in other systems [148,149], where surface material
travels on the direction of the ion beam. The top surface layer in other words behaves like
a fluid [150,151].
As seen grafically in Figure 7.10, there exists a stability-instability transition (x-direction)
at an angle of incidence 45°, angles lower than this remain stable while those higher are
unstable. Meanwhile in the y-direction case (see image on the right of Figure 7.10), the
mass re-distribution always contributes to stable modes. A similar analysis was carried out
by Carter and Vishnyakov [76] leading to a diffusion-like term on the equation of motion
111
Chapter 7. Discussion
generated by atomic recoils of the target appearing as a diffusional term on the surface
evolution equation.
On the other hand, a growth rate of similar characteristics may lead to equivalent effects
but distinct origins, erosion versus mass-redistribution effects. Sometimes, these are related
to small length-scale structures due to shifts of higher values of k. [157,158,159] Other consider-
ations must be consistent with hydrodynamical approaches of the bombarded material. The
effects observed from experimental results point out these additional effects that emerge only
when considering the stress generated. In the model advanced by Castro and Cuerno [160,169]
and recently by Moreno-Barrado and collaborators [158], where the generated stress from ion
irradiation leads to a general relation for the growth rate and surface transport of ripples.
The model predicts higher critical angles of incidence which have been observed in the case
of semiconducting materials.
One interesting analogy is the flow of lava in volcanic eruptions [170]. A high viscous
fluid is formed due to molten rock which starts out in a liquid-state before cooling down
and crystallizes. The lava dynamics is controlled by its composition, temperature, crystal
and bubble content. This results in a time dependent system which evolves with respect to
heat losses, and material transport bounded by the underneath static surface morphology.
As aparticular case, basaltic type lava (pahoehoe lava; being a smooth, shiny or swirled
surface of Hawaiian origin) erupt at 1100 °C with initial viscosity ηinitial ≈ 102 − 103 Pa·s
incrementing at η ≈ 109 Pa·s at long times leading to ropy surface morphologies.
7.3.4 Ion-atom Combination
The combination of ion-atom target leads to differences of experimental results. This differ-
ence for example is observed after ion implantation of stainless steel (SS316L) where surface
structures resembling perpendicular ripples occur (see Figure 6.28) in comparison to the
112
Chapter 7. Discussion
titanium samples. This result contradicts the usual Bradley-Harper predominant erosive at
oblique incidence angles. Additionally, the formation of tree bark-like structures in titanium
after silver implantation differs from the usual rippling structure in the direction of the ion
beam (see image on the right of Figure 6.28). This similar induced effect of ion type change
has also been observed in semiconducting materials [157].
One of the most pronounced effect is the variation of the critical angle of incidence, θc.
For instance in the case of ion bombardment of silicon, heavier ions induce higher values
of θc. Irradiation of silicon with Ar and Xe ions at 500-eV increases the critical angle from
θc = 46° to θc = 58° for argon and xenon ions
[157], respectively. Atomic generated stress
is higher for oblique incidence in comparison to 45° experiments. This behavior seem to
highlight the applicability of hydrodynamic models and coupled by the variation of the ion
type.
At any rate, Bradley-Harper type theories ion-induced pattern formation of bombard-
ment of materials is known to underestimate the experimental measured wavelength. This
failure of the continuum model is due to various factors including the assumptions of phys-
ical interactions and experimental parameters. From the stand point of a complete theory,
an atomistic description is desired but deem impossible (limited by space-time simulation
scales), a continuum model has become a natural choice of assessment.
A variation of the near-surface residual stress at oblique incidence angles occurs and
competes with other ion-atom processes. These would inhibit certain surface effects and
contradictig previous results for the same ion and energy (see section §6.7). Furthermore,
the idea of a surface current generated by a surface mass redistribution instability at certain
conditions was further investigationed by Hofsäss and collaborators [65,165]. This actually
raised a feasible argument whether other external conditions could destabilize the surface as
the ion implantation proceeds. A specific emphasis was put forward suggesting that surface
impurities could destabilize the surface. As a matter of fact, the presence of foreign atoms as
113
Chapter 7. Discussion
those considered in codeposition experiments [63,64] induce a surface instability that modifies
the surface response. Surface ripples have been seen to evolve with respect to the metallic
deposition in accordance to the theory of Bradley [28].
Molecular dynamic (MD) simulations for instance can only probe fast-time processes
of order of picoseconds and small-length scales of nanometers. In the case of experimental
studies, large- scalesare desired, requiring massive dimensions inaccessible through numerical
simulations. Rough approximations that have been employed in our work include; atomic
collisions as collective effects characterized by general deposition parameters of the volume,
which cannot explain small deviation of the atomic concentration of the evolving target
material [124].
7.4 Bradley-Shipman Type Theories Considerations
Difficulties have arised in the study of ion bombardment of composite solid surfaces like alloys
and binary semiconducting materials. The surface response is a complicated system to study
due to the target material. In the case at hand, metallic materials suffer additional consid-
erations; (1) possible formation of atomic compounds of associated Au ion and titanium
atoms, (2) recrystallization of the target material during ion implantation and (3) the role
of impurities and/or preferential sputtering of the target metallic material. Theories devel-
oped by Bradley and collaborators [25,31] have advanced our understanding of multi-elemental
surfaces, specially the case of alloys.
In theories consistent with the formation of atomic compounds (see section §4.6), a
coupling of the varying thin layer of altered stoicheometric, Cs on top that of a static field
underneath is known to occur. The develpment of surface structures is seen to depend largely
in the atomic damage (akin to Bradley-Harper type models) but also from secondary effects
like mass redistribution. As the top surface layers change, the atomic concentration may also
114
Chapter 7. Discussion
depend on developing slopes such that additional effects of the target material may need to
be accounted. This kind of analysis requires the inspection on the formation of compounds
associated by the ion and target atom combination.
7.4.1 Intermetallic Compound Formation
In X-ray photoelectron spectroscopy (XPS) studies performed on ion-implanted Ti (see sec-
tion §6.6.2), the formation of Au2Ti, AuTi3, TiO2, TiO and Au2O3 compounds are found
to depend on the atomic concentration of the bombarding ion and thermodynamic avail-
able conditions [150,151]. These oxides and intermetallic compounds suggest other important
pattern forming mechanisms of Ti.
A coupled two-field model may be utilized in order to describe the behavior of the target
material [24]. One field accounts for the mean surface height growth while a second field
describes the atomic composition of the top layers (see section §4.6). The variation of the
atomic density of a small layer of thickness ∆ is approximated in the theory of Bradley and
collaborators [25,27,28,31,68,69].
The formation of compounds and its subsequent dynamics can effectively describe the
evolution of the near surface region of binary and composite targets. For example in the
plots given below (see Figure 7.11) both Au-associated and Ti-associated compounds have
been utilized to compute the approximate atomic number density of the bombarded target
with respect to depth (from the computation of the intermetallic concentration with depth).
A stoicheometric variation of the near-surface layers of our implanted material indeed
occurs, i.e, an apparent higher atomic density of associated Au-compounds in the top layers,
which decreases as the substrate is slowly eroded away. This is given by the formation of
intermetallic compounds in the case of Au ion implantation of Ti, consequently changing
the dynamics of the surface which is coupled to that of the stoicheometric variation leading
115
Chapter 7. Discussion
to a highly nonlinear coupled two field equation, see equations (4.29) and (4.30) of section
§6.6.2.
0 10 20 30 40 50 60
0.0
1.5
3.0
4.5
6.0
At
om
ic
 D
en
si
ty
 (1
02
2 a
to
m
s/
cm
3 )
Depth (nm)
0 10 20 30 40 50 60
0.0
1.5
3.0
4.5
6.0
At
om
ic
 D
en
si
ty
 (1
02
2 a
to
m
s/
cm
3 )
Depth (nm)
Figure 7.11: Atomic number density (number of atoms/cm3) of general intermetallic com-
pounds for both Au (left) and Ti (right) high-resolution peaks of the XPS analysis (see
section §6.6.1). The horizontal solid line represents the initial titanium number density.
The use of coupled two-field models have recently taken a greater relevance in the study
of alloys [173,174]. In the work of Bharathi and collaborators, a component-dependent segrega-
tion effect exists that manifests itself as a compositional along with morphological changes.
Erosion and diffusive mechanisms are component dependent which evolve with respect to
ion irradiation conditions.
7.5 Asymptotic Non-linear Effects
As ion implantation experiments proceeds at higher fluences, different shapes develop on the
surface. Long time effects that are observed include the formation of terrace-like structures.
These structures are considered to be non-local effects due to changes in the surface geom-
etry [77,82]. There are two types of mechanisms that may explain their origin: shadowing
and a reflection of ions Hauffle mechanism (see for example Figures 7.5(d) and 7.7(d) for
116
Chapter 7. Discussion
high-fluence ion implantation).
In the shadowing effect, an obstruction of the initial rippling structure occurs. This effect
shadows part of the bombarded surface, and a ripple structure evolves into a terrace like
structure. An asymmetric rippling structure develops; the lee slope grows at a greater rate
in comparison to the side facing the ion beam. This leads to a flattening of surface ripples.
In the Carter theory [77], a limiting ratio value W/l for shadowing not to occur is given by:
tan
(
π
2
− θ
)
≤ 2πW
l
(7.1)
where θ is the angle of incidence, W is the initial ripple amplitude and l is the ripple
wavelength. The ratio between the initial amplitude and the wavelength is labeled as the
aspect ratio. For an ion beam at an angle of 45° with respect to the normal, a ratio W/l =
0.16 is obtained in comparison to our experimental data of W/l ∼ 0.2. Since this value
exceeds the limiting value, the planarization of the rippling structure is expected to occur.
On the other hand, the Hauffle mechanism leads to a rapid coarsening of the structure due
to a reflection of ions. While both mechanisms may occur on materials, shadowing is believed
to occur first followed by the reflection of ions. Large saw tooth wave-like structures develop
after a fluence ∼ 1017 ions cm−2. Indeed, this effect has been observed in the experiments
of Xe irradiation of Si at oblique incidence angles [82].
7.6 Overview of Hydrodynamic Models
The study of changes of the surface morphology of titanium and its alloy Ti-6Al-4V under
heavy high-energy ion implantation have been performed experimentally. The particular
experimental conditions performed in this work highlight other important pattern forming
mechanisms. Similarities of low and medium energies phenomena to those at high energies is
117
Chapter 7. Discussion
observed. The possible validation of known theories of pattern formation at these particular
experimental conditions has been discussed.
However, other difficulties in assessing the behavior of ion sputtered surfaces has recently
been highlighted [26,124]. These for example are related to long time behavior, impurities of the
target material and the lack of patterns at low angles of incidence. The use of hydrodynamic
models is a promising technique for modeling the behavior of patterns even in the existence
of these issues. These in particular are overcame by coarse-grained approximations that
often result after modeling the ion-atom interactions as a continuum medium.
In our case, Au ion implantation at a few MeV of energies generate surface ripples in
the nano to micrometer range. The emergence of surface patterns resembling ripples were
obtained with similarities to those generated at low energies of semiconducting materials
even in the presence of irregularities. The study of these pattern has been pushed forward
due to possible applicationsin the medical industry.
Analogies with macroscopic phenomena exist; formation of sand dunes by particle trans-
port is a common comparison to ion beam sputtering (IBS) experiments. In the case of IBS
experiments, the interplay between the bombarding ion and the ion-atom induced effects
is believed to be the main growth mechanism. The rippling structure length scaling with
energies is seen to occur, confirming the possible control of the pattern by the ion and energy.
Continuum models successful approach to IBS experiments has emphasize this natural
choice for time and length scales. Additionally, the formation of surface structures from
Bradley-Harper type and/or Bradley-Shipman type theories considerations can account for
the generation of surface structures independent of substrate type. Explicitly, ripple forma-
tion in Ti and Ti-6Al-4V in the nano and micrometer length scale relate processes during
ion bombardment being of the order with respect to the penetration depth. In consequence,
the mention models conform to approximation due to the damage region produced with the
Au-ion beam.
118
Chapter 7. Discussion
Presently, experimental assessments of the damage region provides information that may
elucidate the actual physical process that occurs during ion-atom collisions. The macroscopic
observed phenomena accounts for large space-time dimensions. This study is believed to
encourage further work of ion-induced effects of metallic samples at high energies. Either
from experiments or from theoretical studies, there still a long way before a full understanding
of pattern formation in metallic samples, specially at high energies.
7.7 Applications of Surface Structures in the Medical
Industry
Nano and micrometer sized surface structures obtained after ion implantation may be utilized
as future technological devices [13]. Our studies focus on ion induced surface morphologies of
titanium and its alloy. These structures are expected to play important roles in orthopedic
implant devices [10,15]. Making use of this method can improve mechanical and chemical
surface properties of biomedical implants. Well-controlled surface morphologies are known
to require additional post surface chemical treatment so that successful use in orthopedic
devices is achieved. Hydroxyapatite (HA), for example, is a bone mineral that may be used
with surface coatings for osseointegration of the metallic implant with the bone tissue.
Many studies have already performed these types of experiments [5,16], with interest in
developing devices that use ion implantation techniques along with surface chemical treat-
ments for biomedical applications. The formation of surface roughness in the range of a few
nanometers has the potential for anchoring and proliferation of associated bone cells onto the
metal implant [8]. In the study of Braceras and collaborators [5], noble gas ion implantation of
titanium surfaces have been performed. Subsequent ion-implanted analysis included adhe-
sion, proliferation and wettability tests. WST-1 [175] assestments of hFOB 1.19 [176] cells were
119
Chapter 7. Discussion
performed revealing an early growth and proliferation on ion-implanted samples following
ATTC recommended protocols.
Moreover, bio-devices based of micrometer-size channels are also of interest. These can
indeed allow better control of fluids in the relevant dimensions for molecules like DNA and/or
proteins. A number of pattern forming methods have been used for microfluidic applications
including laser photolithography [178], reactive ion [179,180] and focused ion beam (FIB) [181]
etching techniques. Microfluids in a channel are controlled by electroosmosis, by applying
voltages in reservoirs, specific fluids are driven within a channel along a prepared pattern
structure. Explicitly, in the work of Han and Craighead [179] their constructed device can
effectivey drive a diluted solution of DNA while holding and separating different DNA size
chains becoming a method for filtration of long polymers.
In our work [147], we have speculated that the obtained micrometer-sized surface struc-
tures can succesfully be used in biomedical applications. Biomedical devices such as or-
thopedic implants based of ion-implantation methods are able to include bio-compatibility,
high strength-to-weight ratio enhancing longevity in hip and knee replacements applications.
These modulated structures in the nano and micrometer size can promote growth and prolif-
eration of bone cells onto the metal implant [8]. Post chemical coatings must also be performed
on ion-implanted titanium samples in order to cope with biocompatibility issues [6,7,10,16] and
thus successfully utilize them as biomedical devices.
120
Chapter 8
Conclusions
Titanium based biomaterials are frequently used in orthopedic implants. Surfaces of Ti-based
biomaterials determine the adherence and proliferation of biomolecules, and are important
for clinical studies. Surface modification by noble ion implantation has proved to be an im-
portant technique for controlling the behavior of the near-surface region. One possibility has
been the implantation of Au ions near the surface of Ti and its alloy Ti-6Al-4V. Additionally,
Au ion implantation modifies the top surface layers of both materials while at the same time
maintaining their biocompatibility properties.
In particular, in the present work 1.0-MeV Au+ ions are bombarded into Ti and its alloy
Ti-6Al-4V at various angles of incidence. Many different shapes are observed that depend
on the experimental conditions. Changes of the surface morphology are characterized with
the help of optical microscopy (OM), scanning electron microscopy (SEM), atomic force
microscopy (AFM), along with Rutherford backscattering spectroscopy (RBS) and X-ray
photoelectron spectroscopy (XPS) techniques.
Our main experimental results are as follows:
� The formation of ripples on ion sputtered Ti and Ti-6Al-4V surfaces evolves from an
initial incubation fluence of Φ = 6.0× 1016 ions cm−2.
121
Chapter 8. Conclusions
� Initial symmetric (small) ripple structures develop that evolve into asymmetric (larger)
structures as the ion fluence increases. Small ripples of wavelength ∼ 1µm develop into
bigger structures (λ ∼ 3µm) that merge laterally to form longer structures.
� Rippling structures grow in height with respect to the ion fluence. The development
of flat tops is seen to occur for Ti in contrast to the alloy.
� A non-zero threshold angle (0°< θ <45°) of incidence exists for ripple formation on
both materials.
� Ion implantation at other angles of incidence produce similar smaller features even at
high fluences.
� Studies involving XPS, revealed the formation of the intermetallic compounds Au2Ti
and AuTi3.
� Ripple-like features have also been studied using Vickers� micro-hardness indentation
which corroborate single angle experiments of Ti and its alloy.
� Surface transport is seen to occur confirming a viscous flow in the case of a continuum
model approach due to the atomic damage.
� Surface shapes obtained in this work are similar to those of low and medium energy
experiments performed in semiconducting materials.
Before ending, continuum type equations are commonly utilized for the description of ion
bombardment of materials. Initial studies from Bradley-Harper type models have improved
over time given by experimental insights. Low and medium energy theories may be utilized
in order to compare with experiments performed at high energies. The observation of surface
ripples and other structures on ion-implanted Ti and Ti-6Al-4V at high energy is likely to
122
Chapter 8. Conclusions
be described by extension of current known theories. The study of intermetallic compounds
can account for the formation of surface structures in metallic target materials.
Finally, the generated micrometer-sized periodic height modulations, often labeled as
ripples areof interest in the medical industry. These surface ripples may be utilized as
biomedical template for biomolecules’ adhesion and growth in orthopedic implants while for
microfluidic applications in the separation and characterization of long polymeric chains like
DNA and/or proteins. Post surface treatments would be required in accordance to ATTC
protocols for biomedical applications.
123
Chapter 9
Outlook
Collective effects of single-ion interactions are approximated by continuum models. Impor-
tant experiments at low and medium energies have motivated theoretical studies of surface
evolution of ion sputtered materials. This description has been successful in providing un-
derstanding of the general atomic processes although a general model is still lacking. In this
sense, experimental work along with theoretical descriptions are continuously been performed
in order to reach a better agreement even at those at high energies. Our understanding is
slowly advancing within the scope of out-of-equilibrium phenomena in condensed matter
physics.
In the best of our interests, gaining a better understanding of the physics require further
work on the system at hand. The most appropiate approach is to carried out careful X-
ray Photoelectron Spectroscopy (XPS) analysis of implanted samples with respect to the
development of surface structures. This may be utilized in order to couple the density of top
surface layers with respect to a continuum model approach as given by Bradley-Shipman
type models. In this approach, the knowledge of the top layers can shed light on the atomic
process that occur during ion implantation. This method can test the vialibility of low-energy
theories at high energy experiments especially in metallic materials.
124
Outlook
The understanding of interface phenomena is of great interest in the materials science
community; motivated technologically by possible near-future applications in the semicon-
ducting, biomedical and environmental industries. The availability of high-energy ion im-
planters has directed out our interest into other possible applications outside of the usual
silicon-based technologies. Many areas of current research are known to affect our everyday
life; a better understanding of surface modification of materials would allow novel uses. For
example, this work has proven to be of interest in the medical field for orthopedic implants
and of microfluidic applications, where their utilization has been held back by uncontrolled
surface modification.
The surface modification by ion implantation along with post chemical treatments may
render a bio-active surface for biomedical applications. These must meet strict medical
protocols based for example by ATTC recommendations. Further studies are required, in
particular for our experimental conditions relating metallic target at high energies. Hopefully
this piece of work can motivate further biomedical studies of Au ion-implanted Ti and Ti-
6Al-4V samples.
125
Appendix A: Sputtering yield
The semi-empirical Yamamura-Tawara [99] formula predicts the sputtering yield of single
element targets; given an incident particle of energy E at normal (θ = 0°) incidence angles.
An angular dependence is resolved considering the following geometrical representation (see
Figure 1) and represented by a cosine function relation:
ion
beam
q
surface
Y(0)Y( )q
Figure 1: Schematic representation of the sputtering yield at an angle. The sputtering yield
maybe characterized by the yield from that given by its normal incidence yield.
Y (θ) =
Y (0)
cos(θ)
(1)
where Y (θ) is the sputtering yield at an angle θ, Y (0) is the yield at normal incidence. Then
for instance at θ = 45 is given by:
Y (θ = 45) =
Y (0)
cos(45)
=
Y (0)√
2/2
=
2√
2
Y (0) ≈ 1.414 · Y (0) (2)
126
Sputtering yield
Now recall that the Yamamura result for 1.0-MeV Au at normal incidence on Ti is given by
Y (0) = 6.1 atoms/ion, then:
Y (θ = 45) ≈ 1.414Y (0) ≈ 1.141(6.1) ≈ 8.61 (3)
In this case, approximately 8 Ti atoms are expell per incident Au ion into the material.
This geometrical representation falls short to approximate the sputtering yield at a given
angle θ, this is due to its rough geometrical approximation. Nastasi [1], Yamamura et al. [98]
suggest that the related expression would follow an inverse cosine power relation:
Y (θ)
Y (0)
= cos(θ)−fs (4)
where the exponent fs is a function of M2/M1 and Y (θ) and Y (0) are the yield produced
from an angle θ and from normal incidence respectively. In the approximation employed
by Yamamura-Itoh [98], where the ration M2/M1 is smaller than 10, the sputtering yield is
dominated by collision cascades near the surface. This approximation is given as:
fs ≈ 1 +
(
< Y 2 >D
< ∆X2 >D
< X >2D
< ∆X2 >D
)
(5)
where < X >D is the average depth of damage,
√
< Y 2 >D and
√
< X2 >D are the damage
straggling of the ion horizontal and perpendicular to the ion trajectory, respectively. These
values are called the moments of the distribution, and may be obtained from a function of
M2/M1.
For simplicity, Nastasi suggests that for M2/M1 < 1, for example in the case of Au
(M1 = 196.966) into Ti (M2 = 47.867), the mass ratio is M2/M1 = 47.867/196.966 ≈ 0.243,
then this follows that the ratio < ∆X2 >D / < X >
2
D and < Y
2 >D / < X >
2
D are nearly
constant on average with values of 0.4 and 0.15 [1] respectively. The rough calculation follows
127
Sputtering yield
like:
fs ≈ 1 +
(
< Y 2 >D
< ∆X2 >D
< X >2D
< ∆X2 >D
)
, but
< ∆X2 >D
< X >2D
= 0.4
≈ 1 +
(
< Y 2 >D
< ∆X2 >D
1
0.4
)
, but
< Y 2 >D
< X >2D
= 0.15
≈ 1 +
(
0.15
0.4
< X >2D
< ∆X2 >D
)
, but
< ∆X2 >D
< X >2D
= 0.4
≈ 1 +
(
0.15
0.4
1
0.4
)
≈ 1 +
(
0.15
0.16
)
≈ 1 + 0.9375 ≈ 1.9375 (6)
The angular sputtering yield is then given by:
Y (θ)
Y (0)
= [cos(θ)]−fs
= [cos(θ)]−1.9375 → Y (θ) = [cos(θ)]−1.9375Y (0) = Y (0)
[cos(θ)]1.9375
(7)
Consider, the previous problem of determining the sputtering yield of 1.0 MeV Au ions into
Ti, where the incident angle is θ = 45 and Y (0) = 6.1 atoms/ion, thus:
Y (θ = 45) =
6.1
[cos(45)]1.9375
, but cos(45) =
√
2
2
=
6.1
(
√
2/2)1.9375
=
(
2√
2
)1.9375
× 6.1 = 1.957× 6.1 = 11.938 ≈ 12.0 (8)
In the implantation of 1.0 MeV Au ions into Ti, it is computed that at an angle of
θ = 45, the sputtering yield will be equal to 11.93 Ti atoms/ion. This value is consistent to
simulation values from TRIM [91].
128
Appendix B: Linear stability analysis
The Bradley-Harper (BH) linear model describes the surface growth of ion-beam sputtering
(IBS) experiments at early times. In this model, a linear stability analysis is performed in
accord to a long-wave disturbance of the surface thus obtaining the growth rate and length
scale. Consider the linearized model (up to 4th order) for IBS experiments, in its isotropic
version:
∂h
∂t
= −v0 + γ∇h− ν∇2h + Ω∇3h−K∇4h (1)
where ∂h/∂t describes the growth of a surface with respect to an initially flat surface; v0,
γ, ν, Ω and K are constants and taken as positive values relating the mean surface erosion
velocity, the transport velocity, the surface-induced tension, an anisotropic surface velocity
transport and the relaxation mechanism either thermal or ionic, respectively. Meanwhile the
anisotropic version in accordance with an ion-beam direction in the x-direction is given by:
∂h
∂t
= −v0 + γ
∂h
∂x
− νx
∂2h
∂x2
− νy
∂2h
∂y2
+ Ω1
∂3h
∂x3
+ Ω2
∂
∂x
(
∂2h
∂y2
)
−K∇4h (2)
Assume a small perturbation function, h(x, y, t):
h(x, y, t) = −v0t + A exp [i(qxx+ qyy) + ω(qx, qy)t] (3)
129
Linear stability analysis
The growth rate (dispersion relation) is given by:
ω(qx, qy) = −iγqx − νxq2x − νyq2y + iΩ1q3x + iΩ2qxq2y −K(q2x + q2y)2 (4)
The real part associates the growth rate of a particular mode qx,y, depending on the ion
beam direction while the imaginary part describes its anisotropic surface transport mode:
Re ω(qx, qy) = νxq
2
x + νyq
2
y −K(q2x + q2y)2 = νxq2x+ νyq2y −K(q4x + q4y + 2q2xq2y) (5)
Im ω(qx, qy) = −γqx + Ω1q3x + Ω2qxq2y (6)
If one considers the x-direction (qx, qy = 0) for the real part, then the most pronounced
surface structure is located at the greatest value of the growth rate.
Re ω(qx, qy = 0) = νxq
2
x −Kq4x (7)
Thus the maximum unstable Fourier mode is given by:
∂Re ω(qx, 0)
∂qx
∣
∣
∣
∣
qmax
x
= 0 ⇒ 2νxqmaxx − 4K(qmaxx )3 = 0 ⇒ qmaxx =
√
νx/2K (8)
This relates a length scale for the pattern forming system of:
lc =
2π
qmaxx,y
= 2π
√
2K
min(|νx,y|)
(9)
In an isotropic geometry, the dispersion relation is ω(q) = νq2 − Bq4 and its length scale
also given by equation (17) by removing the x or y dependence.
The growth rate is the competition between surface erosion (q < qmax) and surface relax-
130
Linear stability analysis
ation mechanisms (q > qmax). Being erosion the mechanism dominant at short-wavelength
instability which compete with the long-wavelength relaxation mechanism.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
-1.5
-1.0
- 0.5
0.0
q
Ω
H
q
L
Figure 2: Isotropic dispersion relation for surface ripples. Red-solid line for a linear Bradley-
Harper model and blue-dashed line for the damped linear Bradley-Harper model.
131
Appendix C: Linear and nonlinear
corrections
The noisy Kuramoto-Sivashinsky (nKS) equation is written as:
∂h
∂t
= ν∇2h− B∇4h+ λ
2
(∇h)2 + η (1)
Higher-order linear and nonlinear terms maybe obtained by performing a higher-order Taylor
expansion of the geometric factor. These corrections apply to the first and third terms of the
nKS equation; as the second equation being a fourth-order term that considers temperature-
dependent relaxation of the surface which has been added as an ad-hoc term in order to
induce smoothing.
The corrections to the linear term are given by:
L0 = ν
∫
dd~x
√
1 + (∇h)2 = ν
∫
dd~x
[
1 +
1
2
(∇h)2 − 1
8
(∇h)4 + 1
16
(∇h)6 + · · ·
]
(2)
Substituing into the time-dependent Ginzburg-Landau free energy functional (2.4) and ap-
plying functional derivatives. The first-three higher-order linear corrections terms are:
∂h
∂t
= ν(1)∇2h− 1
2
ν(2)∇(∇h)3 + 3
8
ν(3)∇(∇h)5 + · · ·+ η (3)
132
Linear and nonlinear corrections
Similarly in the case of the nonlinear term; the geometroc erosion velocity factor is also
Taylor expanded (recall Figure 2.3):
δh
δt
= v
√
1 + (∇h)2 = v
[
1 +
1
2
(∇h)2 − 1
8
(∇h)4 + 1
16
(∇h)6 + · · ·
]
= v +
1
2
λ(1)(∇h)2 − 1
8
λ(2)(∇h)4 + 1
16
λ(3)(∇h)6 + · · · (4)
Finally taking hold of the first two terms in the linear and nonlinear corrections and com-
bining:
∂h
∂t
= v + ν(1)∇2h− 1
2
ν(2)∇(∇h)3 −B∇4h+ 1
2
λ(1)(∇h)2 − 1
8
v(2)(∇h)4 + η (5)
In a similar fashion as given in section §2.3, transforming to a comoving reference frame.
The corrected noisy Kuramoto-Sivashinsky equation to second order is given by:
∂h
∂t
= ν(1)∇2h− 1
2
ν(2)∇(∇h)3 − B∇4h+ 1
2
λ(1)(∇h)2 − 1
8
λ(2)(∇h)4 + η (6)
In summary, this growth equation describes the time evolution of a surface that undergoes
diffusion and lateral growth with second-order corrections. These are represented by the
second and fifth terms on this equation of motion.
133
Epilogue
In the vast number of natural phenomena, the emergence of critical phenomena during ion
bombardment of materials resulted in a practical study of a system found far away from
equilibrium. The evolution of surfaces and interfaces has become an interesting starting
point into the general framework of study of growth phenomena. Additionally, material sci-
ence posses a unique opportunity into accessing physical processes occurring in our natural
environment and to understand macroscopic phenomena as large as desert dunes up to entire
galaxies and possibly extend it to parallel universes.
M.A. Garcia
Ciudad Universitaria, January 2017
134
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	Portada 
	Contents
	Abstract 
	Chapter 1. Introduction 
	Chapter 2. Surface and Interface Growth 
	Chapter 3. Experimental Approaches: a Review 
	Chapter 4. Theories of ion Induced Surface Growth 
	Chapter 5. Experimental Techniques 
	Chapter 6. Results 
	Chapter 7. Discussion 
	Chapter 8. Conclusions
	Chapter 9. Outlook 
	Appendixes
	Epilogue
	Bibliography