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 Model