Ion Beam Induced Pattern Formation on Si and Ge Surfaces
Von der Fakultät für Physik und Geowissenschaften
der Universität Leipzig
genehmigte
DISSERTATION
zur Erlangung des akademischen Grades
Doctor rerum naturalium
Dr. rer. nat.
vorgelegt
von M.Sc. Bashkim Ziberi
geboren am 11. September 1974 in Radiovce (Mazedonien)
Gutachter:
Prof. Dr. B. Rauschenbach (IOM Leipzig und Universität Leipzig)
Prof. Dr. Michael J. Aziz (Harvard University)
Prof. Dr. S. Linz (Westfälische Wilhelms-Universität Münster)
Tag der Verleihung 19.12.2006
Diese Arbeit wurde im Zeitraum Mai 2002 bis Juni 2006 am Leibniz-Institut für
Oberflächenmodifizierung e. V. Leipzig angefertigt.
This work was prepared from May 2002 until June 2006 at the Leibniz-Institut für
Oberflächenmodifizierung e. V. Leipzig.
Betreuer / Supervisors:
Prof. Dr. B. Rauschenbach / Dr. F. Frost
Bibliographische Beschreibung
Ziberi, Bashkim
Ion Beam Induced Pattern Formation on Si and Ge Surfaces
Universität Leipzig, Dissertation
135 S., 148 Lit., 79 Abb., 9 Tab.
Referat:
Die vorliegende Arbeit beschäftigt sich mit systematischen Untersuchungen zur
Ionenstrahlerosion von Si- und Ge-Oberflächen. Im Mittelpunkt standen dabei die
Aufklärung von Selbstorganisationsprozessen, die beim Beschuss mit Ne+-, Ar+-, Kr+-, und
Xe+-Ionen mit eine Energie zwischen 500 eV bis 2000 eV auftreten. Besonderes Interesse
galt dabei den entstehenden hoch-geordneten Ripple- und Punktmustern, mit Dimensionen
der Einzelstrukturen deutlich kleiner als 100 nm. In diesem Zusammenhang wurde der
Einfluss verschiedener Prozessparameter, wie Ionenenergie, Ioneneinfallswinkel, und
Ionenfluenz und -fluss, auf die Strukturbildung im Detail untersucht. Ein weiteres
Hauptaugenmerk lag auf dem Einfluss zusätzlicher sekundärer Ionenstrahlparameter, und
dabei speziell der Winkelverteilung der Ionen im Breitstrahl. Weiterhin wurde der
Zusammenhang zwischen verschiedenen Prozessparametern und der Größe sowie dem
Ordnungsverhalten untersucht. Dazu wurde u. a. das Skalierungsverhalten
unterschiedlicher Rauhigkeitskenngrößen mit den oben genannten Parameter bestimmt.
Bei Einstellung geeigneter Prozessparameter können die entstehenden Strukturen einen
sehr hohen Ordnungsgrad aufweisen. Bei Verwendung geeigneter Breitstrahlionenquellen
bietet das hier untersuchte Verfahren einen alternativen und kosteneffizienten Ansatz zur
Realisierung großflächig nanostrukturierter Oberflächen für unterschiedlichste
Anwendungen.
Zur Charakterisierung der Oberflächentopographie und der oberflächennahen Bereiche
wurden die Rasterkraftmikroskopie (AFM), die Rasterelektronenmikroskopie (REM),
Transmissions-Elektronenmikroskopie (TEM) sowie die Röntgen-Kleinwinkelstreuung
(GISAXS) und Röntgenbeugung unter streifendem Einfall (GID) eingesetzt.
Table of Contents
1
Introduction
3
2
Basics of Ion Beam Sputtering
7
2.1
Ion-Target Interaction.................................................................................... 7
2.2
Ion Range and Energy Distribution ............................................................... 9
2.3
Sigmund's Sputtering Theory ........................................................................12
3 Continuum Theory of Pattern Formation
15
3.1
Linear Continuum Model ..............................................................................16
3.2
Nonlinearities in the Continuum Model ........................................................22
3.3
Damped Kuramoto-Sivashinsky Equation ....................................................23
4 Experimental Setup and Analysis Methods
4.1
25
Ion Beam equipment......................................................................................25
4.1.1 Design of the Ion Beam Equipment ........................................................25
4.1.2 Characterization of the Broad Beam Ion Source .....................................26
4.2
Analysis Methods .......................................................................................... 31
4.2.1 Scanning Force Microscopy (AFM)........................................................31
4.2.2 X-ray Scattering Techniques ................................................................... 35
5
6
4.2.2.1
Grazing Incidence Small Angle X-Ray Scattering (GISAXS).......37
4.2.2.2
Grazing Incidence Diffraction (GID) .............................................38
General Properties of the Surface Topography on Si and Ge
41
5.1
Overview of Emerging Topographies ...........................................................41
5.2
Influence of Ion Species ................................................................................45
Ripple and Dot Patterns on Si and Ge Surfaces
1
49
Table of Contents
6.1
Influence of Ion Energy................................................................................. 50
6.2
Ion Fluence and Flux..................................................................................... 61
6.3
Geometrical Shape ........................................................................................ 68
6.4
GISAXS and GID.......................................................................................... 72
6.4.1 Ge ............................................................................................................ 73
6.4.2 Si.............................................................................................................. 80
7
Pattern Transitions on Si and Ge Surfaces
7.1
87
Role of Ion Incidence Angle ......................................................................... 87
7.1.1 Influence of Ion Incidence Angle on Pattern Transition on Si................ 88
7.1.2 Influence of Ion Incidence Angle on Pattern Transition on Ge .............. 90
7.1.3 Discussion ............................................................................................... 92
7.2
Role of Secondary Ion Beam Parameters on the Surface Topography ......... 94
7.2.1 Secondary Ion Beam Parameters vs. Ion Incidence Angle ..................... 95
7.2.2 Secondary Ion Beam Parameters vs. Ion Energy .................................... 98
7.2.3 Summary ................................................................................................. 98
8
9
Comparison of Experimental Results with Theory
101
8.1
Bradley-Harper Model and the Nonlinear Extension.................................... 101
8.2
Surface Relaxation Mechanisms ................................................................... 103
8.3
Other Models................................................................................................. 105
Conclusions
109
Appendices
111
A1
Details of the Continuum Equation............................................................... 111
A2
The model for GISAXS and GID Simulations.............................................. 113
List of Acronyms
115
Bibliography
116
2
Chapter 1
Introduction
The fabrication of regular nanostructures on the nanometer length scale builds the basis
for many technological applications in variety of fields. These applications range from
optics to optoelectronics, to biological optics, to templates for the deposition of functional
thin films, and to heteroepitaxial growth of quantum dots or wires [1-3]. Further
applications are in the field of data storage industry, for creating high density magnetic
media [4,5]. In general there are two approaches for the fabrication of nanostructures. One
is top-down technique that involves lithographic patterning using optical sources with a
wavelength much smaller than the visible light. This technique is time consuming and costintensive. In contrary to this technique is the bottom-up approach. This technique relies on
self-assembly or self-organization processes. There are various methods like
semiconductor heteroepitaxy [6] and block copolymer lithography [7,8]. Another method
for the generation of self-organized nanostructures is the low-energy ion beam erosion of
solid surfaces. This erosion process usually known as sputtering is a widespread technique
used in many surface processing applications. It is the main tool applied in depth profiling,
surface analysis, sputter cleaning and deposition. Ion beam sputtering is also used to
modify the surface of the solid by producing different topographies. For particular
sputtering conditions, due to self-organization processes, these topographies can evolve in
well ordered nanostructures on the surface like ripples or dots. Nanostructure formation is
observed on different materials such as metals [9-16], crystalline and amorphous
semiconductors [17-32], and other materials [33,34]. During the last years, it has been
reported about the formation of dot nanostructures on III/V (InP, GaSb, InAS, InSb)
semiconductors under normal ion incidence or oblique ion incidence with sample rotation
[27,28,35]. The evolving structures revealed particular domains with hexagonal ordering.
These investigations paved the way for further intensive studies in this field, by also
including other materials for nanostructuring.
The main advantage of ion erosion method is the possibility to produce large-area
nanostructured surfaces in a one-step process. Another advantage is the easy control of
different process parameters. However, the main disadvantage is the long range lateral
ordering of nanostructures and the less control of their shape. The basics behind the ion
beam erosion method is the interaction of the incoming ions with target atoms. During this
interaction, ions transfer energy and momentum to target atoms, eventually leading to
sputtering of the surface. The process of nanostructure formation itself is a complex
interplay between sputtering that roughness the surface and different relaxation
mechanisms, that act to smooth the surface. Indeed, this interplay depends on different
sputtering parameters. Therefore, from the above discussion it is important to study the
3
Chapter 1: Introduction
influence of different process parameters on the evolution of the surface topography. This,
not only to identify the dominant mechanisms, but also to find the process parameters that
can influence evolution, size, shape, and lateral ordering of nanostructures.
This will be the task of the work presented here. Namely, to study the evolution of the
surface topography on Si and Ge surfaces and the possibility to form nanostructures with
large scale ordering, in particular the formation of ripple and dot patterns.
During studies on III/V semiconductors, it came out that it is difficult to understand the
process of pattern formation and the influence of different sputtering parameters. One
reason was that these systems were made of two components, leading to preferential
sputtering, and the enrichment of the dot nanostructures with one component, for example
with Indium or Gallium. Therefore the idea was to use simpler one component systems for
the process of pattern formation, like Si and Ge. Further, to verify if the formation of dot
nanostructures is characteristic only of III/V semiconductors, or it can be applied also on
other materials. Another reason for choosing these two systems is their importance for
applications in different fields of technology. Last but not least, the lack of a systematic
study on the evolution of the surface topography during low-energy ion beam erosion at
room temperature.
For Si, according to earlier reports on ripple formation under noble gas ion beam
erosion, two cases can be distinguished: I) The formation of ripple patterns at relatively
high ion energies (typically above 20 keV) at room temperature or below. Under these
conditions, ripples form at ion incidence angles ranging from 35 deg and 65 deg with
respect to the surface normal, with ripple wavelength above 300 nm [36-38]. Upon ion
bombardment, the target surface becomes amorphous. II) Ripple patterns are also reported
at lower ion energies (e. g., at 750 eV) but at temperatures above 400 °C [26]. In the later
case, ripples form at ion incidence angles near 67° with wavelength larger than 200 nm.
Under these erosion conditions the target surface remains crystalline. In general, it can be
stated that, independent of ion energy, ripple patterns are observed at ion incidence angles
from 40° up to 70° and with ripple wavelength > 100 nm. Dot patterns have been observed
for Si at normal ion incidence for an ion energy of 1200 eV [30], but with a low degree of
ordering.
Concerning Ge, only few reports about pattern formation during low-energy ion beam
erosion exist. Up to now ripple patterns are observed for 1000 eV Xe+ ion beam sputtering
at high temperatures (~ 300 °C) under an ion incidence angle of 55 deg [21]. The surface
of the forming ripples remains crystalline after ion bombardment with a ripple wavelength
of 200 nm.
In this work, a detailed study of the surface topography evolution for Si and Ge during
noble gas ion beam erosion, for low ion energies between 500 eV and 2000 eV will be
presented. Chapter 2 contains a short introduction of the ion beam sputtering theory. In
Chapter 3, the theory of ripple formation by presenting different relaxation mechanisms
will be given. The description of the ion beam setup used for the experiments will be given
4
in Chapter 4. In this chapter a new parameter for controlling the evolution of the surface
topography will be introduced, by discussing its influence on ion beam properties. Atomic
force microscopy and small angle X-ray scattering techniques used to characterize the
surface topography will be presented in Chapter 4.2. Chapter 5 will deal in general, with
possible topographies evolving on Si and Ge surfaces by discussing the role of ion
incidence angle and the ion mass. The influence of ion energy, erosion time and ion flux on
the size and ordering of ripple and dot nanostructures will be discussed in Chapter 6. Also
the geometrical shape of nanostructures will be addressed in this chapter. Results will be
discussed by using both characterizing methods. In Chapter 7 completely new phenomena
not observed up to now, will be addressed. Namely the transition from dots to ripples and
again to dots. During these transitions the evolving dots show an almost perfect array of
dots covering the whole sample area. In this context a new parameter for controlling the
large scale ordering of dot nanostructures will be presented. The last chapter is devoted to
the discussion of the current theory of pattern formation and its consistency with the
experimental results.
5
Chapter 2
Basics of Ion Beam Sputtering
The bombardment of solid targets with energetic particles (ions) gives rise to different
processes. For example the backscattering of incident ions, implantation of ions in to the
target, and emission of electrons and photons. Additionally, due to collision processes, an
ion penetrating the target surface will slow down by transferring its kinetic energy and
momentum to the target atoms. Depending on the transferred energy this may lead to the
displacement of atoms (creating vacancies and interstitials) causing lattice defects. If these
atoms (primary knock-on atoms) receive enough kinetic energy they will induce additional
collisions with other target atoms and hence additional atomic displacements. This
situation is referred to as a collision cascade [39,40]. If a small number of atoms is set in
motion and for an isotropic distribution of collision density one speaks of a linear collision
cascade. Such a cascade can be described by binary collisions between moving ions and
stationary atoms. On the other hand due to momentum reversal, target atoms may travel
towards the surface. If these atoms posses enough kinetic energy they can overcome the
surface binding energy barrier and will be ejected away from the surface. Under
continuous bombardment of the surface this will lead to a material erosion. A process
known as sputtering. Furthermore, the increasing number of ions will induce additional
defects in the crystalline structure, and for a high defect density this may lead to a surface
amorphisation.
For the kinetic energy loss of the incoming ion, elastic (nuclear collisions) and inelastic
processes (electronic excitation) must be considered. However, for the low-energy range
(up to 2000 eV ion energy) the energy loss of ions happens mainly through nuclear
collision processes. This nuclear collision processes determine the energy deposition and
the range of ions in the target. The amount of energy deposited by an ion and the ion range
depend on the material properties (density structure, atomic mass etc.), and the ion mass
and energy. These two sets of parameters describe completely the collision cascade
processes and are the basis for the continuum theory of pattern formation that will be
presented in Chapter 3.
2.1 Ion-Target Interactions
Ions penetrating the surface undergo different collision processes with target atoms.
During this process they transfer part of their kinetic energy to target atoms [39]. If only
the nuclear collision processes are considered, then the differential energy loss of an ion
per unit length dx is given by
7
Chapter 2: Basics of Ion Beam Sputtering
dE
= − NS n (E )
dx
(2.1)
where N is the atomic density in the solid, and Sn(E) is the nuclear stopping cross section
(energy loss rate), E is the initial energy of an incoming ion.
The nuclear stopping cross section gives the average energy dissipated during the
collision processes. The expression for Sn(E) depends on the form of the screening
(potential) function used to describe the interaction between the ions and atoms. For low
ion energies, where the screening of the coulomb interaction is essential, the Sn(E)
according to Sigmund is given by [39,41,42]:
Sn ( E ) =
Tmax
Tmax
− m −1− m
∫ Tdσ ( E , T ) = ∫ TCm E1 T dT =
0
0
Cmγ 1− m E 1− 2 m
1− m
(2.2)
with
⎛M ⎞
C m = λm a ⎜⎜ 1 ⎟⎟
2
⎝ M2 ⎠
π
2
s
m
⎛ 2 Z1 Z 2 e 2 ⎞
⎜⎜
⎟⎟
⎝ as ⎠
2m
4 M 1M 2
γ =
.
(M 1 + M 2 )2
(2.3)
Here dσ is the interaction cross section, T the energy transferred from an ion with
Tmax = γE being the maximum energy transmitted by an elastic collision (head on
collision). The parameter m characterizes the power law potential used to describe the
Coulomb interaction between atoms, and λm is a dimensionless function of m that can take
values from λ1 = 0.5 up to λ0 ≈ 24, and as is the screening length [39]. M1, Z1 and M2, Z2 are
the mass and atomic number of the ion and target atoms, respectively.
To simplify calculations the reduced notations for the nuclear cross section and the
energy are introduced. In this way the problem of different ion-target combinations is
reduced into a two body collision process [43].
A rather successful approach for calculating the energy loss and cross sections is the
Ziegler, Biersack and Littmark (ZBL) approximation. ZBL proposed a universal screening
function for the interaction potential by presenting numerical solutions for Sn(E) and sn(ε)
[44]. The analytical expressions for Sn(E) and sn(ε) used for the fitting procedure are
Sn ( E ) = 8.462 × 10−15 ×
M 1Z1Z 2 sn (ε )
.
( M 1 + M 2 )( Z10.23 + Z 20.23 )
(2.4)
For ε ≤ 30 that includes the case for low ion energy:
sn (ε ) =
0.5 ln(1 + 1.1383ε )
(ε + 0.01321ε 0.21226 + 0.19593ε 0.5 )
8
(2.5)
2.2: Ion Range and Energy Distribution
and the reduced energy ε is given through
ε=
0.03253E ( eV ) M 2
.
Z1Z 2 ( M 1 + M 2 )( Z10.23 + Z 20.23 )
(2.6)
Here sn(ε) is the reduced nuclear stopping cross section independent of the ion-target
combination. The ε is the dimensionless reduced energy parameter, and it describes how
energetic a collision is. The parameter m presented above depends on the value of ε and
according to Winterbon et al.
m = 1 / 3 for ε ≤ 0.2
(2.7)
m = 1 / 2 for 0.08 ≤ ε ≤ 2.
The above model for the energy loss is also the basis for the Monte Carlo simulations
(SRIM 2003) that will be used later for some ion-target combinations [44]. In Fig. 2.1 the
stopping cross sections Sn(E) for Si and Ge calculated for an ion energy Eion = 100 eV –
2000 eV are plotted. The plots for different ion species show that heavier ions loose more
energy per collision compared to lighter ions (see Table 2.1), i. e. they undergo fewer
collisions with target atoms until they come at rest. This means the mean penetration path
(that will be discussed in the next section) for heavier ions is shorter compared to lighter
ions.
2.2 Ion Range and Energy Distribution
Due to elastic and inelastic collision processes an ion penetrating the target will loose its
energy until it comes at rest after a certain range. The traveling distance of the ion in the
target depends on the mass ratio of the colliding particles and on the ion energy. The range
20
20
+
Sn(E) [eVnm /atom]
15
+
+
Ar
5
+
Ne
0
500
1000
1500
2000
+
Kr
15
2
Xe
+
Kr
10
0
Xe
Ge
2
Sn(E) [eVnm /atom]
Si
10
5
0
2500
ion energy Eion [eV]
+
Ar
+
Ne
0
500
1000
1500
2000
2500
ion energy Eion [eV]
Figure 2.1: Nuclear stopping cross section of Ne+, Ar+, Kr+ and Xe+ ions in Si and Ge targets, calculated
according to Eq. (2.4).
9
Chapter 2: Basics of Ion Beam Sputtering
Table 2.1: The energy loss per unit length in Si and Ge calculated using Eq. (2.1) for different ion species for
Eion = 2000 eV.
energy loss dE/dx (eV/nm)
Ne+
Ar+
Kr+
Xe+
Si
211
389
577
631
Ge
195
420
762
942
of an ion can be calculated by integrating the stopping cross section Sn(E) using Eq. (2.2)
[45]
1
R( E ) = −
N
0
∫
Eion
dE
1
=
Sn (E) N
Eion
∫
0
dE
1 − m m−1 E 2 m
γ
.
≅
Sn (E)
NCm
2m
(2.8)
By introducing the reduced length ρ, it follows:
2
⎛ 0.8854a0 ⎞
M 2 M1
⎟
ρ (ε ) = 4πNR⎜⎜ .23
.23 ⎟
2
⎝ Z1 + Z 2 ⎠ ( M 1 + M 2 )
(2.9)
with a0 = 0.0529 nm being the Bohr radius.
In the experiments usually the projected range Rp is accessible. Rp is defined as the
distance between the hitting point of the ion on the surface and the point where this ion
comes at rest along the direction of incidence and Rp < R. For amorphous and
polycrystalline targets and low ion energies Schiøtt [46] introduced a formula for the
projected range that depends on the ion-target combination
2m
⎡⎛ Z12 / 3 + Z 22 / 3 ⎞ ⎤
⎟⎟ E ⎥ .
R p ~ M 2 ⎢⎜⎜
Z
Z
1 2
⎠ ⎦
⎣⎝
(2.10)
Due to the stochastic nature of ion penetration depths (for ions of the same type) one
considers the range of many ions. In this case a broad statistical distribution of ion ranges
and ion range straggling is obtained.
Winterbon et al. [47] showed that the amount of energy deposited by an ion in a
primary collision over a range dx, for an elastic scattering, is given by
dE = NS n ( E ( x))dx = FD ( x)dx
(2.11)
with FD(x) being the energy depth distribution function. Sigmund in the context of the
linear collision cascade theory [42] derived the expression for FD(x). By introducing a set
of transport equations and using the Edgeworth expansion, it could be shown that the
average energy deposited at a point r(x,y,z) in the target by an ion traveling along the z axis
10
2.2: Ion Range and Energy Distribution
is given by
FD (r ) =
⎛ ( z − h0 + a ) 2 x 2 + y 2 ⎞
ν (E)
⎟
⎜⎜ −
exp
−
2α 2
2 β 2 ⎟⎠
( 2π )3 / 2αβ 2
⎝
(2.12)
with ν(E) being the total energy deposited. Expression (2.12) has a Gaussian form and is
valid for amorphous targets. The parameter a is the average depth of the deposited energy,
α and β are the width of the distribution (straggling) parallel and perpendicular to the ion
beam projection, respectively (Fig. 2.2). The parameters a, α and β depend on the ion
energy and on the ion-atom mass ratio, i. e. on the kinematics of the collision process. As
shown from Eq. (2.8) and Eq. (2.12) the range and the mean distribution function depend
on the stopping cross section function. This means the distribution of the deposited energy
can be calculated by studying the energy loss process using, for example Eq. (2.4), Eq.
(2.5) and Eq. (2.6). Usually simulation programs for calculating the collision processes are
applied. A successful software code with a very broad field of applications is the Monte
Carlo based SRIM 2003 (Stopping and Range of Ions in Matter) simulation program
developed by Ziegler et al. [44]. It is based on two body collision processes in amorphous
targets (binary collision approximation BCA). Through a detailed calculation, the history
of every collision between the ion and target atoms as well as the collision between recoil
atoms with other target atoms can be followed. These processes are followed until all the
participating particles come at rest. In this work the SRIM 2003 simulation program is
used to calculate the ion range and the distribution of the deposited energy in the target.
If the energy transferred to the target atom during the collision process is higher than the
displacement energy (~ 15 eV for Si and Ge, taken from Ref. [44]) the atom will live its
lattice site creating a vacancy. If the energy of this recoil atom is smaller than the
displacement energy it will stay in the lattice, by releasing the rest of the energy as
phonons and remaining as an interstitial atom. If a moving atom hits another atom
transferring its energy by knocking it out of lattice site, and taking its place in the lattice,
one speaks of a replacement collision. In general it is not possible to deduce FD(x) directly
Ion beam
h
z
x
a
β α
Figure 2.2: Schematic drawing of the distribution of the deposited energy of an ion in the target. Also given
are the parameters characterizing the distribution.
11
Chapter 2: Basics of Ion Beam Sputtering
from SRIM simulations, but can be calculated from the number of displaced (vacancies
and replacements) atoms. According to Bolse [48]
FD ( x ) = nv ( x)
ν (E)
Nv
(2.13)
with nv(x) being the vacancy density and Nv the total number of vacancies, and ν(E) is
defined in Eq. (2.12).
In the low-energy regime, the collision processes are confined at the near surface
region. If the amount of ions per area and time (ion flux) hitting the target is increased, also
the number of defects will increase up to a critical point that the upper surface layer
becomes amorphous. The formation of this layer depends also on the target material and
temperature, and on the ion mass and energy. However, the dominant influence on
amorphization, at room temperature, comes from the ion fluence (ions per area). Detailed
studies about the surface damage accumulation of Ge and Si materials, at ion energies up
to 3000 eV, are performed by different groups [40,49-52]. By bombarding the Ge surface
with 3000 eV Ar+ ions Kido et al. [50] showed that at a fluence of ~ 1 × 1014 ions/cm2 an
amorphous layer is formed. Similar results were reported by Bock et al. [51] for studies on
Si and Ge surfaces using Ar+ ions (ion energy from 100 eV up to 3000 eV). Both these
studies showed that the layer thickness increases with ion fluence and it saturates above ~
1 × 1015 ions/cm2. Further increases on the layer thickness, for a given temperature, are
expected with increasing ion energy. They also showed that the amorphization and
saturation values for Si and Ge are very close to each other. The lowest ion fluence used in
this work is 1.87 × 1016 ions/cm2 well above the amorphization threshold. For this reason a
surface amorphization on all samples is expected. This correlates with results presented in
Section 6.3 using HRTEM (high resolution transmission electron microscopy) where an
amorphous layer covering the sample surface is observed.
2.3 Sigmund's Sputtering Theory
If an atom moving toward the target surface gains enough energy to overcome the
surface binding energy it will be ejected from the target. This process is characterized by
the sputter yield, Y, and gives the mean number of emitted atoms per incident ion. Most
contribution to the sputter yield comes from the recoil atoms traveling in backward
direction. According to Sigmund [39,41,42], the sputtering yield depends on the target
material, on the ion beam parameters and on the experimental (geometrical) conditions,
and is is defined as (by making use of Eq. (2.11))
Cmγ 1−m E 1−2 m
Y ( E , α ion ) = ΛFD ( E , α ion ) = Λα I NS n ( E ) = Λα I N
.
1− m
(2.14)
The sputter yield has a linear dependence from the distribution of the deposited energy,
12
2.3: Sigmund's Sputtering Theory
i. e. the number of displaced atoms. FD(E,αion) corresponds to the depth distribution
function FD(x) and can be determined from Eq. (2.12). In Eq. (2.14) αI (for low ion
energies) is a dimensionless function of the mass ratio, and of the ion beam incidence angle
αion. Λ is the material dependent parameter given by
Λ=
3
1
4π NC0 Esb
2
(2.15)
2
with C0 = (πλ0 aBM
) / 2 from Eq. (2.3) and Esb being the surface binding energy.
The final relation of the sputter yield given by Sigmund [41], for normal ion incidence,
has the form
α S (E)
(2.16)
Y ( E ,0) = 0.042 I n
.
Esb
Expression (2.16) is deduced for low ion energies, near the sputtering threshold (m = 0).
However, as discussed by Sigmund in the same work, with a low uncertainty Eq. (2.16)
can be well applied also at higher ion energies where m ≥ 1/3. For oblique ion incidence
the sputter yield is a function of the cosine of αion
Y ( E ,α ion ) = Y ( E ,0)(cosα ion ) −b
(2.17)
with the exponent b being a function of the mass ratio. However, for M2/M1 ≤ 3, b ≈ 5/3 is
independent of the mass ratio [41]. Equation (2.17) is valid for not too oblique incidence
angles (between 50 deg and 60 deg). With further increase of the incidence angle the
sputter yield will decrease due to the increased amount of reflected ions from the target
surface. For all calculations in the context of this work the sputter yield value at normal
incidence will be applied. In order to be compatible with experimental studies, for the rest
of the work the energy of incoming ions will be denoted by Eion.
13
Chapter 3
Continuum Theory of Pattern Formation
For describing the formation and evolution of surfaces under ion beam bombardment,
different models are proposed. These models mainly can be divided into continuum models
and microscopic models. In microscopic models, the atomic and crystalline structure of the
evolving topography on the surface is taken in consideration [53-56]. The microscopic
models are based on Monte Carlo and Molecular dynamic simulations. In contrary, in the
continuum models the surface topography is described by a continuous function, without
taking in consideration the atomic and crystalline structure of the surface. Therefore these
models are valid for amorphous materials. For the time evolution of the surface topography
at a mesoscopic scale, differential partial equations are used [57-61]. These models were
originally developed to describe rough interfaces and fractal topographies, and they posses
some specific scaling properties in the spatial and temporal evolution. Furthermore, the
continuum models have prevailed over microscopic models, for describing the evolution of
periodicities on the surface due to ion beam sputtering. The continuum models provide
quantitative predictions about the temporal evolution of the surface topography, and about
the scaling properties of the evolving structures.
The basis of these continuum equations, is Sigmund's theory of sputtering of amoprhous
targets, presented in the previous Chapter. It was shown that an ion hitting the target will
transfer it’s energy to the target atoms. Due to this energy transfer a removal of atoms from
the surface takes place. This material removal can lead to modifications on the surface
topography, usually producing rough surfaces. However, due to self-organization processes
this surface erosion process sometimes can produce well ordered nanostructures on the
surface [9,12,17,21,26-28,30-32,62-64]. The distribution of the deposited energy equation
(2.12) is given for one ion. For a flux of incoming ions that penetrate the sample
simultaneously at different points one has to integrate Eq. (2.12) over all individual events.
The erosion rate v(A) at a given point A, depends from the energy deposition of all ions in
a region around A, with a width comparable to a. The total erosion rate is given by
v( A) = Λ ∫Φ (r ) FD (r )dr
(3.1)
A
with Λ given by Eq. (2.15), and Φ(r) is the ion flux J corrected for the local curvature
variations on the surface [42]. The integral is performed over the region of all points that
contribute to the energy deposited at A.
In absence of a perfectly smooth surface the distribution of the deposited energy
depends on the local spatial shape of the surface. This will result in local variations of the
sputter yield and the erosion velocity.
15
Chapter 3: Continuum Theory of Pattern Formation
The evolution of the surface topography during the sputtering process is a result of
complex processes taking place on the surface and near surface region. In the continuum
models usually there are two competing mechanisms: i) curvature dependent sputtering
that leads to a rough surface with time, and ii) relaxation processes acting to smooth the
surface. These relaxation processes can be of different origin like thermally activated
surface diffusion, viscous flow, or erosion related smoothing mechanisms. There are
additional processes that can influence the topography evolution like re-deposition of the
material and the re-emission of particles.
It is the competition between roughening and the smoothing mechanisms, that can result
in laterally ordered structures on the surface. This is also the basis of the continuum models
that will be presented in the next sections, together with their main properties. In general,
in the continuum model two cases can be distinguished. The evolution of the surface
topography at oblique ion incidence without sample rotation, whereupon ripple structures
evolve on the surface. In this case, there is an anisotropy present, given by the direction of
the ion incidence angle. This situation is described by the anisotropic continuum equation.
Second, the topography evolution at normal ion incidence or oblique ion incidence with
sample rotation. Due to rotational symmetry structures having isotropic distribution, like
dots, form on the surface. Hence, the isotropic version of the continuum equation is used.
3.1 Linear Continuum Model
Based on Sigmund theory of sputtering of amorphous targets Bradley and Harper (BH)
developed a model to explain the formation of ripple structures on the surface [61,65].
They showed that the variation of the sputter yield with the local surface curvature induces
a roughness on the surface, that leads to ripple structures. This surface roughness is caused
ion beam
ion beam
B
A
C
C
b) crest
a) trough
Figure 3.1: Schematic drawing of different erosion rates during ion beam sputtering. The surface at point A
(trough) is eroded faster than in B (crest). This because the average energy deposited at point A by ions
hitting the surface at point C is greater than the energy deposited at point B (the thick solid line has the same
length). The doted arrows have the same length and indicate that the ion hitting the surface has the same
distance from point A, respectively B.
16
3.1: Linear Continuum Model
by the different erosion rates in troughs (Fig. 31.(a)) compared with crests (Fig. 3.1(b)). By
combining the curvature dependent sputtering, with surface smoothing due to thermally
activated surface diffusion, they developed a linear continuum equation for the evolution
of the surface topography:
4
∂h
∂ 2h
∂ 2h
∂ 2h
∂ 4h ⎞
th ⎛ ∂ h
⎜
S
S
D
= −v0 + v0'
+
+
−
+
x
y
⎜ ∂x 4 ∂y 4 ⎟⎟.
∂t
∂x 2
∂x 2
∂y 2
⎠
⎝
(3.2)
Equation (3.2) describes the temporal evolution of the surface height h(x,y,t) for an
angle αion with respect to the normal of an initial flat surface. The coordinate system axis x
and y lie parallel and perpendicular to the projection of the ion beam on to the surface,
respectively. Eq. (3.2) is valid for small slope approximation, i. e. the radius of curvature is
much larger than the mean depth a (Fig. 2.2). The first term, on the right hand side of Eq.
(3.2), represents the angle dependent erosion velocity of a flat surface. The second term,
describes the lateral movement of structures on the surface. As argued by Makeev et al.
[66], these two terms do not affect the characteristics of structures (wavelength and
amplitude), therefore they can be omitted from the surface evolution equation. The third
term describes the curvature dependent erosion rate, and the final term describes the
surface relaxation due to material transport on the surface [67,68].
The BH coefficient S depends on the ion energy, ion incidence angle and the material
properties
S x, y =
Ja
Y0 (α ion ) Γ x , y (α ion ).
N
(3.3)
Here J is the flux of incoming ions, and Y0(αion) is the angle dependent sputter yield of
an initial flat surface. The Γx(αion) and Γy(αion) coefficients account for the local variations
of the sputter yield, expressed as [61]
Γ x (α ion ) =
B ⎛
A
A2 ⎞
AC ⎛
A2 ⎞
⎟⎟ cos α ion − 2 ⎜⎜ 3 +
⎟ cos α ion
sin α ion − 2 ⎜⎜1 +
2 B1 ⎝ 2 B1 ⎠
B1
B1 ⎝
B1 ⎟⎠
Γ y (α ion ) = −
β2 ⎛1
AC ⎞
⎜ B2 +
⎟ cos α ion .
2 ⎜
a ⎝2
B1 ⎟⎠
(3.4)
These coefficients depend on the ion incidence angle and on the parameters a, α and β
characterizing the distribution of the deposited energy, and can be calculated using the
simulation code SRIM 2003 introduced in Section 2.2. The coefficients A, B1, B2, C are
given in Appendix A1. In Fig. 3.2, Γx(αion) and Γy(αion) as function of ion incidence angle,
for Si using Xe+ ions with 2000 eV ion energy are plotted. The plots show that Γx(αion) can
have both positive and negative values, while Γy(αion) has always negative values. For
symmetry reasons at αion = 0 deg, Γx = Γy in this case no ripple structures are expected on
the surface.
17
Chapter 3: Continuum Theory of Pattern Formation
Γx
Γy
Γx, Γy
0.3
a = 2.85 nm
α = 1.83 nm
β = 1.13 nm
0.0
-0.3
0
20
40
60
ion incidence angle αion [deg]
80
Figure 3.2: Coefficients Γx(αion), Γy(αion) calculated for 2000 eV Xe+ ion beam erosion of Si, according to Eq.
(3.4). The parameters a, α and β are deduced from SRIM simulations.
The coefficient Dth in Eq. (3.2), describes the thermally activated surface diffusion
[67,68] related with material transport on the surface, having the form
D th =
⎛ − ∆E ⎞
Ds γ Ω 2 N
⎟⎟
exp⎜⎜
k BT
k
T
⎝ B ⎠
(3.5)
with Ds being the surface diffusion constant, γ is the surface free energy per unit area, Ω is
the atomic volume, ∆E the activation energy for surface diffusion, kB the Boltzmann
constant, and T the surface temperature.
As stated at the introduction of this Section, it is the simultaneous acting of the third and
fourth term in Eq. (3.2) that can lead to the formation of ripple structures on the surface.
A simpler way to explain Eq. (3.2) is by taking the Fourier transform of it [69]. By
defining h(qx,qy) as the Fourier transform of the surface height, and q ≡ (qx,qy) the wave
vector, Eq. (3.2) can be written as
d h(q x , q y , t )
dt
[
(
= − S x q x2 − S y q y2 − D th q x4 + q y4
)] h(q , q , t ).
x
y
The solution of Eq. (3.6) is
h(q x , q y , t ) = h(q x , q y ,0) exp[R(q)t ]
(3.6)
(3.7)
with h(qx,qy,0) being the initial amplitude spectrum of the Fourier component and the
growth factor
(
)
R ( q x , q y ) = − S x q x2 − S y q y2 − D th q x4 + q y4 .
18
(3.8)
3.1: Linear Continuum Model
Equation (3.7), represents the time evolution of the amplitude of the Fourier
components, and it increases exponentially for positive R(q) values. In this case the surface
roughens. While, for negative values of R(q) the surface smoothens. The factor R(q) has a
maximum at q* = (max|Sx,y| / 2Dth)1/2 with |Smax| being the larger in absolute value of -Sx, or
-Sy, respectively. While in Eq. (3.8) the diffusion term is always positive, the sign of R(q)
will depend on the coefficient Sx,y given in Eq. (3.3), i. e. on the parameters Γx(αion) and
Γy(αion). From Eq. (3.7) and Eq. (3.8) the amplitude with the wavenumber q* will grow
faster than all others. This will result in a periodicity with the wave number q*, that will
dominate the surface topography. The alignment of the wave vector of the periodicities
depends on the values of Sx, and Sy.
According to Fig. 3.2 for αion < 45 deg Γx(αion) < Γy(αion) < 0, the coefficient
S x = ( Ja / N )Y0 (α ion )Γ x (α ion ) . The wave vector qx = (|Sx| / 2Dth)1/2 is aligned along the xaxis and is parallel to the projection of the ion beam on to the surface.1 For αion > 45 deg
Γy(αion) < Γx(αion) and Γy(αion) < 0, S y = ( Ja / N )Y0 (α ion ) Γy (α ion ) , in this case qy =
(|Sy| / 2Dth)1/2 is aligned along the y-axis, and perpendicular to the ion beam projection.
From the above discussion, the characteristic wavelength of ripples can be written as
⎛ 2 D th
2π
= 2π ⎜
λ=
⎜ max S x , y
q x, y
⎝
1/ 2
⎞
⎟ .
⎟
⎠
(3.9)
In following the behavior of λ for certain sputtering parameters will be analyzed.
i)
Concerning the ion energy Eion by using Eq. (2.8) and Eq. (2.14) it follows that
λ~
ii)
iii)
iv)
1
.
E 1/ 2
(3.10)
This means the wavelength of ripples decreases with increasing ion energy.
Making use of Eq. (3.3) the coefficient S ~ J, i. e. λ ~ (1 / J)1/2 is a decreasing
function of the ion flux.
Taking into account that Dth is independent of αion and from Fig. 3.2 and Eq.
(2.17) the coefficients Γx(αion) and Γy(αion) and Y0(αion) are increasing functions of
αion, then λ decreases with αion (at least for not to oblique incidence).
In Eq. (3.9) λ does not depend on the ion fluence, i. e. no changes of the ripple
wavelength with ion fluence are expected.
While for high temperatures thermal diffusion can be regarded as dominating relaxation
mechanism, with decreasing temperature its effectiveness decreases. Makeev et al. [70],
introduced the ion-induced effective surface diffusion ESD as the main relaxation
mechanism at low temperatures. This mechanism does not imply a real mass transport
along the surface, but is generated by preferential erosion of the target during the ion beam
1
The wave vectors qx,y are deduced from Eq. (3.8) by differentiating with respect to q.
19
Chapter 3: Continuum Theory of Pattern Formation
sputtering. By including the ESD term, and neglecting thermal diffusion, Eq. (3.2) can be
written in the form
∂ 4h
∂ 4h
∂ 4h
∂ 2h
∂ 2h
∂h
= S x 2 + S y 2 − DxxESD 4 − DxyESD 2 2 − D yyESD 4 .
∂y
∂x ∂y
∂x
∂y
∂x
∂t
(3.11)
The last three terms of Eq. (3.11) are equivalent to the relaxation term in Eq. (3.2). The
coefficients DxxESD , DxyESD , and D yyESD can be fully determined from the parameters for the
distribution of the deposited energy, from the ion flux, and the ion incidence angle. For the
symmetric case where α = β the coefficients in Eq. (3.11) are expressed as:
DxxESD =
2
2
⎡⎛ a ⎞ 4 4
⎛
⎞⎤
⎞
2
4⎜⎛ a ⎞ ⎛ c
⎟⎥,
⎜
⎟
s
c
s
+
+
−
−
3
6
4
12
⎟
⎜
⎟
2 ⎢⎜
2
⎜ s
⎟
⎜
⎟⎥
α
α
⎝
⎠
⎝
⎠
a
⎛ ⎞ ⎢
⎝
⎠
⎝
⎠⎦
24⎜ ⎟ ⎣
⎝α ⎠
(
Fa 3
D
ESD
xy
)
2
⎤
Fa 3 ⎡⎛ a ⎞ 2 2
2
2
+
−
2
=
s
c
c
s
⎜
⎟
⎥,
⎢
2
⎛ a ⎞ ⎣⎢⎝ α ⎠
⎦⎥
4⎜ ⎟
⎝α ⎠
D yyESD =
Sx =
Fa 3c 2
⎛a⎞
8⎜ ⎟
⎝α ⎠
2
(3.12)
,
Fa ⎛ 2
a 2 2⎞
2
⎜ 2 s − c − s c ⎟,
2 ⎝
α
⎠
(3.13)
Sy = −
with
F≡
⎛
JEion Λa
a 4c 2
exp⎜⎜ −
2 2
αβ 2πf
⎝ 2α β f
Fa 2
c ,
2
⎞
a2
a2
⎟⎟ ; s = sin α ion ; c ≡ cos s = sin α ion and f = 2 c 2 + 2 s 2 .
α
β
⎠
The general form of coefficients in Eq. (3.11) will be given in Appendix A1.2.
If ESD is the main relaxation mechanism then the ripple wavelength can be calculated
1/ 2
as follows
⎛ 2 DxxESD
⎞
, yy ⎟
⎜
(3.14)
λESD
.
x , y = 2π
⎜ max S x , y ⎟
⎝
⎠
By making use of Eq. (2.8), Eq. (3.12) and Eq. (3.13) the scaling behavior of λESD as a
function of Eion is given by
2m
λESD ~ a ~ Eion
,
(3.15)
i. e. the wavelength is increasing with ion energy.
20
3.1: Linear Continuum Model
Also from Eq. (3.14), it follows that λESD is independent of the ion flux and ion fluence
used. The dependence of λESD from the ion incidence angle is given by the relation between
DESD and Sx,y, that depend on ion incidence angle.
Another relaxation mechanism that can contribute to surface smoothening is the ioninduced viscous flow (IVF) [67,68]. The viscous flow term is introduced by Mullins [68]
and Orchard [71] using Navier-Stokes equations. IVF is related with material transport
along the surface and depends on the concentration of defects created within a cascade
[72]. Concerning IVF two cases can be distinguished. In the first case proposed by Chason
et al. the bulk viscous flow is extended to the range of λ. The term contributing to
smoothening is Fb q with the coefficient Fb = γ / ηb. Here γ is the surface energy and ηb is
the bulk viscosity coefficient. Umbach et al. [29], using the model of Orchard, discussed
the other case, when viscous flow is restricted to a surface layer of thickness d. This
thickness is comparable to the ion range in the solid a, and the amplitude of structures, but
much smaller than the wavelength of structures. For this case, in the growth rate equation
the smoothening term –Fs d3 q4 is added. The surface relaxation rate coefficient Fs = γ / ηs,
with ηs being the surface viscosity coefficient. Mayer et al. following the suggestions by
Volkert et al. proposed a more meaningful measure of viscosity by taking ηr = ηb,s J. ηr is
κ
the flux-independent viscous relaxation per ion, and 1 / ηr ~ Eion
. The exponent κ describes
how strong the relaxation rate depends on the ion energy. The growth rate factor in Eq.
(3.8) for the two cases has the form
Rb (q ) = − S x q x2 − S y q y2 − Fb (q x + q y ) ,
(
(3.16)
)
Rs (q ) = − S x q x2 − S y q y2 − Fs d 3 q x4 + q y4 .
(3.17)
Differentiation of Eq. (3.16) does not deliver a characteristic wave vector for the
ripples. While from Eq. (3.17) q = [max|Sx,y| / (2 d3 Fs)]1/2 and for the ripple wavelength, by
taking d = a, it follows that
λIVF
⎛ 2dF
s
= 2πd ⎜
⎜ max S x , y
⎝
⎞
⎟
⎟
⎠
1/ 2
⎛
⎞
2γN
⎟.
= 2πd ⎜
⎜ Y0 (α ion )η r max Γ x (α ion ), Γ y (α ion ) ⎟
⎝
⎠
(3.18)
From Eq. (3.18) it can be concluded that λ is independent from the ion flux and ion
fluence. Due to increasing values of Y0(αion) and Γx(αion) and Γy(αion) with αion, the
wavelength will decrease with ion incidence angle. For the ion energy, by approximating
κ ≈ 1 from Ref. [72] the wavelength
3m
λ ~ Eion
.
21
(3.19)
Chapter 3: Continuum Theory of Pattern Formation
3.2 Nonlinearities in the Continuum Model
The linear BH model predicts an exponential increase of the ripple amplitude in Eq.
(3.7). However, to account for amplitude saturation with erosion time observed
experimentally (see Section 6.2), and for the stochastic nature of the sputtering process,
Cuerno et. al. [59,60] proposed the anisotropic noisy Kuramoto-Sivashinsky equation
∂ t h = S x ∂ 2x h + S y ∂ 2y h +
λx
2
(∂ x h )2 +
λy
2
(∂ h )
2
y
− D (∂ 2x h + ∂ 2y h ) + η ( x , y , t ).
2
(3.20)
This is valid for Sx, < 0, Sy < 0. For positive S values the scaling properties are described
using the Kardar-Parisi-Zhang equation [73].
The relaxation term D may be of thermal nature, ion-induced ESD, or viscous flow. The
last term in Eq. (3.20) represents the stochastic nature of ions arriving on the surface. The
third and fourth term account for the angle dependence of the erosion velocity. The
coefficients λx and λy (not to be confused with the wavelength) can be calculated using
sputter parameters and the energy distribution parameters a, α and β. For the symmetric
case α = β it follows [66]
F cos α ion a 2
λx =
2α 2
⎡
a2
2
2
2
2⎤
⎢3(sin α ion ) − (cos α ion ) − α 2 (sin α ion ) (cos α ion ) ⎥
⎦
⎣
(3.21)
F (cos α ion ) 3 a 2
.
λy = −
2α 2
From Eq. (3.21) it is evident that λy is negative for all incidence angles, while λx can
take both positive and negative values [66]. Due to the nonlinearity of Eq. (3.20) general
analytical solutions are not possible. Through numerical integration, Park et al. have
shown that Eq. (3.20) can be divided in to parts (regimes). For short times the linear
regime is dominating the surface topography up to a crossover time tc, and (3.20) takes the
form of (3.2). In this regime the amplitude (represented through the surface roughness) of
y
x
a)
b)
c)
Figure 3.3: Evolution of the surface topography after Park et al. [74] for λx × λy > 0. Ripple patterns in a)
and b) evolve in the linear regime. a) Sy < Sx < 0, b) Sx < Sy < 0. c) Long time behaviour when non-linear
terms dominate the process. The incident beam is oriented along the y axis.
22
3.3: Damped Kuramoto-Sivashinsky Equation
ripple structures increases exponentially. After the crossover time tc, the nonlinear terms
dominate the evolution of the surface, and the amplitude saturates. While the coefficients
λx and λy do not influence the wavelength of ripples, their sign plays an important role in
the evolution of the surface topography in the non-linear regime. Theoretical simulations
have shown that, for λx × λy > 0 the surface roughness saturates and the ripple structures,
formed at short sputter times, during which the linear regime is dominating the sputter
process, disappear (Fig. 3.3). With further sputtering, the surface roughness exhibits kinetic
roughening [74,75]. Also for λx × λy < 0, the surface roughness saturates and the ripples
disappear. However for prolonged sputtering a new type of rotated ripples (with respect to
linear regime ripples) is observed, with a rotation angle θ c = tan −1 ( − λ y / λx )1 / 2 or
θ c = tan −1 ( − λx / λ y )1 / 2 (Fig. 3.4).
For normal incidence the coefficients in Eq. (3.20) are isotropic with S ≡ Sx = Sy,
λ ≡ λx = λy, D ≡ Dxx = Dyy. Numerical simulations by Kahng et al. showed that for λ > 0
dot structures evolve on the surface, while for λ < 0 holes evolve on the surface [75].
For off-normal ion incidence with sample rotation, due to rotational symmetry, Eq.
(3.20) also becomes isotropic and the coefficients are expressed as S = Sx + Sy, λ = λx + λy
[65,76]. As shown by Bradley [65] this is true for fast rotating substrates with angular
velocity much larger than the ripple amplitude growth rate ω >> Sav2 / D , and for Sav < 0.
3.3 Damped Kuramoto-Sivashinsky Equation
The main disadvantage of the KS equation is the failure to account for dot or ripple
stabilization for long sputter times observed experimentally (see Section 6.2). Furthermore,
the model can not account for the hexagonal ordering of particular domains of dot
structures. Recently, through simulations, Facsko et al. [77] using the damped KS equation
for the sputtering process, showed hexagonal ordered dots evolving on the surface similar
to those in III/V semiconductors [27,28]. The damped KS equation is based on Eq. (3.20)
with an additional damping term of the form –χ h introduced by Chaté et al. [78]. Although
the physical meaning of the term is not yet understood, this term has an important role in
y
x
a)
b)
c)
Figure 3.4: Evolution of the surface topography after Park et al. [74] for λx × λy < 0, Sx < Sy < 0 and
different times. a) t = 104, b) t = 2 × 105 and c) t = 107.
23
Chapter 3: Continuum Theory of Pattern Formation
the surface evolution process. It suppresses the transition to kinetic roughening during the
temporal evolution of the surface in the nonlinear regime. For normal ion incidence the
damped KS equation is given by
∂ t h = − χh + S ∇ 2 h − D ∇ 4 h +
λ
2
(∇h )2 + η ( x, y, t ).
(3.22)
By considering only the damping term, Eq. (3.22) it gives an exponential decrease of
the surface height, during the temporal evolution. This decrease, counteracts the
exponential increase of the surface height due to the linear term [79]. Numerical
simulations showed that depending on the value of χ also the surface topography varies
accordingly [77,80] (Fig. 3.5). For values of χ larger than a critical value χc, hexagonally
ordered patterns evolve on the surface. With decreasing χ, (χ < χc), the hexagonal ordering
decreases until for even lower χ values structures with no ordering are present on the
surface. Long time simulations of Eq. (3.22) show that the hexagonal ordering of patterns
is maintained [79]. These simulations also demonstrated that the wavelength of structures
remains constant with erosion time in agreement with experimental results in Section 6.2.
a)
c)
b)
Figure 3.5: Surface topography calculated using the DKS equation (3.22) after [77]. a) Early time regime
for χ = 0.24, b) Late time regime for χ = 0.24. c) Structures with no apparent ordering for χ = 0.15. Inset:
Corresponding Fourier images showing the characteristic spatial frequencies dominating the surface (see
Section 4.2.1).
24
Chapter 4
Experimental Setup and Analysis Methods
In this chapter a description of the ion beam equipment and the methods used to
characterize the evolution of the surface topography will be presented. Especially the setup
of the ion source employed for sputtering experiments and the characteristics of the
extracted beam, will be discussed in Section 4.1. In the second part atomic force
microscopy as the main method applied in this work to characterize the surface topography
will be presented. In this context, a summary of the statistical quantities for characterizing
the surface topography will be given. Further, the grazing incidence small angle X-ray
scattering methods used for ex-situ studies of structures will be shortly described.
4.1 Ion Beam Equipment
4.1.1 Design of the Ion Beam Equipment
The samples investigated in this work where all treated in a home built ion beam
equipment (ISA 150). A simplified overview of the equipment is given in Fig. 4.1. The
main parts are: a) pumping system; b) gas system for supplying sputter gases; c) the load
lock for sample handling; d) Faraday cup arrays; e) sample holder, and f) the ion source.
The base pressure in the chamber is about 2 × 10-6 mbar. Depending on the gas species,
the working pressure was varying between 5 × 10-5 mbar and 1 × 10-4 mbar. In this work
four different inert gases were used Ne+, Ar+, Kr+, and Xe+. The variations on the working
pressure were necessary in order to maintain the stable operation of the beam source.
However, it is observed that pressure variations (achieved by varying the amount of gas
supplied in the chamber) do not influence the evolving topography.
The distance between the aluminum sample holder and the ion source (acceleration
grid) amounts around 400 mm, and is smaller than the mean free path length of ions
amounting around 1 m, for the working pressure given above. Therefore the extracted ions
will reach the sample without collisions that could effect their kinetic energy and lead to a
broad beam. The sample holder offers the possibility of rotating around its axis with about
12 rotations per minute. Additionally, it can be tilted from 0 deg up to 90 deg with respect
to the axis of the ion beam source. Further, to avoid thermal effects on the sample the back
side of the sample holder is water cooled. However, variations of the temperature from
room up to 60 °C did not have any influence on the evolution of the surface topography on
Si and Ge (this has been performed by measuring the water temperature on the back side of
the sample holder).
25
Chapter 4: Experimental Setup and Analysis Methods
load lock for sample handling
6.5e-3 mbar
gas system
Faraday cup array
m
sa
ion
source
e
pl
l
ho
r
de
2.0e-6 mbar
2.0e-6 mbar
1.0e-2 mbar
sample rotator
vacuum pumping system
Figure 4.1: Schematic view of main parts of the ion beam equipment ISA 150. The distance between the ion
source and the sample holder is about 40 cm.
The Faraday cup array is used to determine the ion current density distribution. Five
probes are mounted, one in the middle and four in each edge covering a radius of ~
120 mm. A deviation of up to 15 % in ion current density value between the middle and
edge probes is present, and it depends on the ion source parameters. However, as it will be
shown later the ion current density has no influence on the topography evolution.
4.1.2 Characterization of the Broad Beam Ion Source
The ion source, is a home built broad beam source of Kaufman-type [81,82] with a two
grid ion optics system (Fig. 4.2). All inner parts of the ion source are made of purified
graphite.
The ion source is equipped with a hot filament (tungsten wire) that emits electrons, by
applying an appropriate current. The electrons are then accelerated toward the anode rings
lying under positive voltage called discharge voltage Udis (in this case Udis = 100 V). The
Udis voltage controls the acceleration of emitted electrons in the filament sheath. With
increasing Udis the energy of electrons will increase also, thus increasing the number of
double charged ions. The variation of Udis between 40 V and 100 V showed no influence
on the evolution of the surface topography at least for hole structures (see Chapter 5). Due
26
4.1: Ion Beam Equipment
Permanent Magnet
Discharge Vessel
Screen Grid
Anode
Hot
Filament
Udis
Ub
Uacc
Gas Supply
Acceleration Grid
Figure 4.2: Schematic drawing of the main components of the, Kaufman type, broad-beam ion source (ISQ
150) of the ion beam equipment ISA 150.
to collisions with gas atoms present in the discharge vessel an ionization takes place, i. e.
the plasma is created. There is a discharge current flowing to the anode rings, determined
by filament heating current. Additionally, the discharge vessel is equipped with permanent
magnets that makes the electrons perform spiral trajectories, performing larger travel
lengths in the vessel. In this way, the ionization efficiency is increased, before they reach
the vessel walls. The ion optical system is made of two multi-aperture grids having a
180 mm diameter. The multi-aperture plane parallel grids are made of holes (for the given
diameter around 3000 holes) with a cylindrical form covering the whole grid surface. In
order to have higher transparency holes are hexagonally arranged [83]. The grid optics,
consists of the screen grid and the acceleration grid that are used to extract the ions from
Figure 4.3: Potential diagram across the different parts of the ion source and outside, up to the sample
holder.
27
Chapter 4: Experimental Setup and Analysis Methods
the plasma (see the potential distribution in Fig. 4.3). The characteristics of the grid system
are: a) hole diameter of 2.5 mm each; b) The grid opening is 180 mm; c) The screen grid
has a thickness of 1 mm, while the acceleration grid is 2 mm thick. d) The distance
between grids is 2 mm.
After the plasma is created the potential of the discharge anode is determined by the
voltage applied in the screen grid Uscr. It is the anode voltage (Udis + Uscr) that determines
the ion beam energy, thereafter called beam voltage Ub. By applying an appropriate
negative voltage at the acceleration grid Uacc ions will be extracted from the plasma and
accelerated toward the second grid.
The total extraction voltage is given by the absolute values of Ub and Uacc, Uextr = Ub –
Uacc [84]. Under experimental conditions the beam and accelerator grid can take values
that vary between 100 V ≤ Ub ≤ 2000 V and -10 V ≤ Uacc ≤ -1000 V.
The ion beam is also characterized by the total current Ib transported in the beam.
Furthermore, in case of improper values of the grid voltages, part of the extracted ion beam
directly hits the accelerator grid resulting in a direct grid current Iacc (see plot in
Fig. 4.4(a)). Because of the resulting fast grid destruction and beam pollution, such an
operation mode is usually avoided (refer to other plots in Fig. 4.4). Beside of the extracted
ions also neutrals can diffuse from the plasma chamber leading to a density of neutrals
within the grid system. In a charge-exchange collision between an extracted beamlet ion
and a neutral, a secondary ion is created which could be accelerated towards the
accelerator grid. This is the origin of the unavoidable accelerator grid current Iacc which
typically amounts a few percent of the beam current [34]. It is the difference between the
total beam current Ib and Iacc that gives the real beam current density at the sample position.
Between the ion source and the sample holder an electron emitting tungsten wire is
mounted to prevent the charging of the sample.
Before discussing some characteristics of the beam it is useful to give several
a)
Plasma
sheath
boundary
500 V
-100 V
500 V
-400 V
500 V
-900 V
1000 V
-100 V
1000 V
-400 V
1000 V
-900 V
d)
e)
b)
f)
c)
Figure 4.4: Beamlet plots for Uacc = -100 V, -400 V, and -900 V at corresponding experimental conditions.
(a-c): Ub = 500 V and a high plasma density np = 5 × 1010 cm-3; (d-f): Ub = 1000 V and medium plasma
density np = 2 × 1010 cm-3. The plots are calculated using the simulation code IGUN [83,85].
28
4.1: Ion Beam Equipment
definitions. The beam extracted from one hole is defined as a beamlet. The superposition
of all beamlets yields the broad-beam, therefore the broad-beam properties are mainly
defined by the beamlet properties. Due to the diverging ion trajectories the beam broadens
with the distance from the ion source. This is usually described by the beam divergence.
The divergence angle, is defined as the half opening angle of the beamlet cone that
contains a certain amount of the overall current (for example 75 % or 95 %) [83,86-89].
Additionally, the angular distribution of ions leaving the hole should be considered. These
beam properties will be referred as secondary ion beam parameters.
In Fig. 4.4 are plotted the ion beam trajectories extracted from one hole for different
Uacc.1 The simulations are performed with the computer code IGUN [90] using the
geometrical dimensions of the grid system given above. The simulations are plotted for
two different plasma density values. This is performed to point out that the beam
broadening depends on the plasma density, that influences the plasma sheath boundary.
The ion trajectories plotted in Fig. 4.4(a-c), for Ub = 500 V, show that the beam broadens
with decreasing acceleration voltage. This is the case for high plasma density np =
5 × 1010 cm-3. For medium plasma density np = 2 × 1010 cm-3 the beamlets (Fig. 4.4(d-f))
plotted for Ub = 1000 V show a broadening of the beam with increasing Uacc. This means
with increasing Uacc the beamlet divergence, i. e. broad beam divergence increases. The
last extraction conditions are valid for the experimental studies presented in this work,
when varying the Uacc.
The angular distribution of ions in the beamlets for different accelerator voltages is
quantitatively presented in Fig. 4.5. The angular distribution for Ub = 500 V and Uacc = 100 V is affected by the direct impingement of the beamlet on the accelerator grid. The
a) Ub= 500 V
1.0
Uacc= -100V
0.5
0.5
0.0
1.0
0.0
1.0
Uacc= -400V
0.5
0.0
1.0
Uacc= -900V
0.5
0.0
intensity [a.u.]
intensity [a.u.]
1.0
0
5
10
15
20
25
b) Ub= 1000 V
Uacc= -100V
Uacc= -400V
0.5
0.0
1.0
Uacc= -900V
0.5
30
angle [deg]
0.0
0
5
10
15
angle [deg]
20
25
30
Figure 4.5: Angular distribution obtained from one beamlet for different acceleration voltages for two
different Ub values, deduced from Fig. 4.4.
1
In fact the simulations yield the beam for a half hole diameter with the supposition that the beam is
symmetric to the other half.
29
Chapter 4: Experimental Setup and Analysis Methods
plots for Ub = 1000 V show that the beamlet with the highest angular distribution, i. e. with
the highest beam divergence, is obtained for Uacc = -900 V. In this case most of the ions
leave the hole with an angle between 4 deg and 6 deg. With decreasing Uacc this angle
range shifts toward smaller values. The superposition of these beamlet configurations to
the number of holes, making up the grid opening, defines the broad beam properties.
Additionally to Uacc also the Ub, i. e. the ion energy, influences the secondary ion beam
parameters [83]. For the particular case the increase of Ub leads to a decrease of the beam
divergence. However, the influence is not so pronounced as for Uacc [85].
Beside the plasma conditions and the extraction voltages, the geometrical parameters of
the two grid optical system also influence the beamlet divergence (see Fig. 4.2). For
example, with increasing distance between grids the beamlet divergence decreases. An
increase of the screen grid thickness or decrease of the hole diameter leads to a decrease of
the beamlet divergence. A more detailed description of the relations between the grid
parameters and the beam characterizing properties is given by Tartz et al. [83,91,92].
These beam properties are valid for the particular ion source and ion beam conditions. The
variation of one of the above parameters could lead to results different from those
presented here.
30
4.2: Analysis Methods
4.2 Analysis Methods
4.2.1. Atomic Force Microscopy
The investigation of the surface topography in the nanometer range, requires a
characterization method with a very high resolution by still yielding a good statistics.
Especially, for structures with a low aspect ratio (ratio height to length) and dimensions
below 50 nm, presented in this work, there are only few methods that can be applied. One
of these methods is the Atomic Force Microscopy AFM [93,94]. The developments of last
years made the AFM to one of the mostly used methods for characterizing the surface
topography on different materials. Beginning from conductive to non-conductive materials
up to biological samples [95,96]. The main advantage of AFM is its sub-nanometer
resolution and the possibility to give a direct real space image of the surface. Today there
are hand-full of books that explain the working principle of scanning probe microscopy
[97,98]. With the AFM method, one scans the surface with very sharp tip mounted on a
piezo-scanner that can be set in motion by applying a voltage on it. Usually there are three
modes used for measurements with AFM: i) contact mode, ii) non-contact mode and iii)
TappingMode™ or dynamic mode. The last mode was developed as a method to achieve
high resolution, without inducing destructive frictional forces between the sample and the
tip. With this technique, the cantilever is oscillated near its resonant frequency with
constant amplitude as it is scanned over the sample surface. As the tip is brought close to
the sample surface its amplitude will reduce. A feedback positioning unit takes care that
the distance between the tip and the sample, i. e. the oscillation amplitude, remains
constant during scanning, without getting in contact with each other.
The measurements presented in this work, were all performed using Dimension 3000
with a Nanoscope IIIa controller, in TappingMode™ from Veeco Instruments [96], and
MFP-3D ,AFM, in dynamic mode from Asylum Research [95]. Compared to Nanoscope
IIIa controller that can record up to 512 points per line, the MFP-3D offers the possibility
to perform large area scans with high resolution up to 4096 points per line. The
measurements were performed in air, using silicon tips with nominal radius smaller than
10 nm, and sidewall angles < 18 deg [99].2 During the measurements, a special attention
was paid to the tip artifacts [100-102]. Sometimes the tip-sample interaction may lead to a
deterioration of the tip. If the tip radius is too large to access the structure inter-distance,
then the measured topography will be determined by the tip shape. Such a situation is
visualized in Fig. 4.6. Therefore, for most of the data presented in this work, several
measurements using different tips for every sample were performed. In this way the tip
artifacts are kept at minimum. Other important factors to be considered are the image size,
2
Olympus cantilevers with tetrahedral tip for standard applications where used (Typ: OMCL-AC160TS).
Additionally, ultra sharp tips (Typ: OMCL-AC160BN) were used to measure some samples, for comparison.
But, no differences on the measured surface topography were observed.
31
Chapter 4: Experimental setup and analysis methods
height [nm]
10
5
0
0
50
100
150
200
length [nm]
250
300
Figure 4.6: Height profile of ripple structures, and the schematic drawing of the geometrical tip shape. If the
tip radius is larger than radius of the valley between ripples, then the tip artefacts are present in the AFM
image.
the number of points one image has, and the spacing between points [103,104]. If L is the
image size, N the number of points, and d the data point spacing then L = N d. Therefore,
in order to have an accurate estimation and a better statistics, it is important that
measurements with different size and high enough resolution are performed. For example,
for a measurement with a scan size 4 µm × 4 µm with 1024 × 1024 data points, gives a
point spacing of ~ 4 nm. For structures with a mean size of 40 nm it means around 10 data
points per structure.
The most important statistical quantity used to characterize the height fluctuations of the
surface is the rms-root mean square roughness. It describes the fluctuations of the surface
heights around an average height, and is given by
rms = w =
1
N
N
∑ ( h ( x, y ) − h )
2
.
(4.1)
i =1
Here h(x,y) represents the surface height at a given point, and h - the mean height of
the surface. For quantitative analysis of the surface roughness samples with scan size of
2 µm × 2 µm and 4 µm × 4 µm were used with 512 × 512 and 1024 × 1024 data points,
respectively. This means a spacing d < 4 nm. However, a comparison with measurements
having data point spacing d < 2 nm between points, showed no difference on the rms
roughness value. This indicates that the largest contribution to the rms value comes from
structures (ripples, dots) present on the surface.
The rms value alone can not fully characterize the surface because it gives information
only along the vertical direction [105,106]. However, sometimes it is very useful to know
the positional correlation between different points on the surface. Especially on structured
surfaces quantities like the wavelength, homogeneity and lateral correlation of structures
are of great importance. This is best achieved by analyzing the height profile in the
reciprocal or Fourier space. The main statistical quantity used to characterize the surface
32
4.2: Analysis Methods
topography in the reciprocal space is the Power Spectral Density (PSD) function.
Performing the discrete two dimensional Fast Fourier Transformation 2D-FFT, of the
height profile of AFM images, it is possible to deduce the dominating spatial frequencies
present on the surface and the amplitude of the roughness. The FFT for the discrete case is
given by
FFT ( f x , f y ) =
1
N2
N
N
∑ ∑ h ( x, y )e
− i 2πL / N ( xf x + yf y )
(4.2)
x =1 y =1
for a square image L × L with N and d equal in x and y directions, fx, and fy are the spatiel
frequency coordinates along the x-axis and y-axis, respectively.
Two representative examples for ripple and dot structures are given in Fig. 4.7. The
FFT images give information about spatial frequencies (fx,fy) ranging from –128 µm-1 to
128 µm-1. The spatial frequency has a minimum value at the center of the image and it
increases by moving away from the center. Due to the preferred orientation of ripples
(Fig. 4.7(a)), in the FFT image (Fig. 4.7(c)) spots are visible. The direction of spots gives
the direction of the wave vector of ripples.
The AFM image in Fig. 4.7(b), shows dot structures emerging on the surface. The
image reveals the short range ordering of dots building domains having hexagonal
ordering. However, these domains are randomly distributed in between them, therefore, a
ring is visible in the corresponding FFT spectra (Fig. 4.7(d)).
The position of the spot (ring) determines the characteristic spatial frequency of ripples
(dots), i. e. the wavelength of structures in the real space. From the width of the spot (ring)
information about the homogeneity, and spatial correlation of periodicities can be deduced.
Additional spots (rings), are multiples of the first one, and are related to the high lateral
ordering of structures. The higher the number of spots (rings), the better is the lateral
ordering. In fact the position of the first spot (ring) gives the mean separation between the
structures. For the ripples, it is assumed that the separation between ripples is equal to the
ripple size. Due to the wavelike form of ripples, the term ripple wavelength will be used to
describe their size. This will be done with the supposition, that the separation in between
ripples is equal to their size. For dots, the mean separation is equal to the mean lateral size
by supposing that the dots are close packed to each other.
However, from the FFT images is difficult to make quantitative estimation about these
quantities. For this reason, the area or two dimensional PSD(fx,fy) function of the surface
height is introduced, by taking the square of FFT [105,107,108]. A more practical way of
evaluating the data is by introducing the angular averaged PSD(f), especially for structures
with an isotropic distribution [106]. This is done by performing angular averaging over all
spatial frequencies with constant distance f 2 = f x2 + f y2 . Summarizing it follows that,
PSD ( f ) =
1
Nf
Nf
∑ 2 D − PSD =
n =0
33
1
Nf
Nf
∑ ( FFT ( f
n =0
x
, f y )) 2
(4.3)
Chapter 4: Experimental setup and analysis methods
z = 7 nm
z = 20 nm
a)
b)
z
0 nm
500 nm
500 nm
c)
d)
FFT
f = - 128 µm-1 … 128 µm-1
FFT
5
10
ζ ~ 1/FWHM
5
10
e)
λ
3
3
PSD [nm ]
10
4
4
PSD [nm ]
f)
λ
1
10
10
1
10
ζ ~ 1/FWHM
-1
10
-1
-3
-2
10
-1
10
10
10
-1
spatial frequency f [nm ]
-3
10
-2
-1
10
10
-1
spatial frequency f [nm ]
Figure 4.7: Typical AFM images of ripple and dot structures on, a) Si and b) Ge surfaces. (c,d) The
corresponding FFT spectra. (e,f) Angular averaged PSD spectra.
where Nf is the number of points at constant distance f [107]. The range of f values depends
on L and N and is limited by 1/L < f < N/2L.
The unit of angular averaged PSD spectra is (length)4. For the rest of the work the
notation PSD will be used, instead. It is obvious that in the case of ripples there is an
asymmetry in the distribution of the spatial frequencies. However, by performing angular
averaging of the FFT image, the dominating spatial frequencies (the spots) will contribute
mostly to the PSD spectra compared to the rest of the Fourier spectrum. The PSD spectra
are plotted in Fig. 4.7. The position of the first peak gives a quantitative information about
the wavelength of structures.
There are different models for describing the lateral ordering of structures. Applying the
model of Zhao et al. [106] the system correlation length ζ can be used as quantity to
describe the lateral ordering of ripples and dots. The ζ gives the length scale up to which
34
4.2: Analysis Methods
spatial correlation is present i. e. the mean domain size. It is deduced from the FWHM of
the first order PSD peak, and is inverse proportional to the FWHM, ζ ~ 1/ FWHM.
At the end it is worth to mention that from the PSD function, by integrating over the
spatial frequency range under consideration, the square of the rms surface roughness can
be deduced which is equivalent to Eq. (4.1).
4.2.2 X-Ray Scattering Techniques
The X-ray scattering measurements were performed at the European Synchrotron
Radiation Facility (ESRF) in Grenoble, France. The ESRF is a third generation
synchrotron facility with a storage ring circumference of 844.4 m, working at an energy of
6 GeV [109]. The experiments were carried out at the ID01 beam line. The ID01 is
equipped with devices, a wiggler and an undulator, that produce synchrotron radiation of a
high brilliance [110]. Additional optical devices are used for tuning, focusing and
alignment of the X-ray beam (Fig. 4.8). A detailed description of the ESRF facility, ID01
beam line, and the experimental setup is given at the ESRF home page [110].
The samples were mounted vertically on the sample holder, fixed on a 6 axes
diffractometer. To avoid air scattering of the beam the sample was covered with a Kapton
foil filled with Helium. The scattered intensity is detected using a linear position sensitive
detector (PSD) mounted at the end of the detector arm (Fig. 4.9). All the experiments
presented in this work were performed at an energy of 8 keV corresponding to a
wavelength of 1.55 Å.
undulator
Figure 4.8: Picture view of the optics used at the ID01 for aligning focusing and tuning of the X-ray beam
source [110].
35
Chapter 4: Experimental setup and analysis methods
x-ray
a)
Sample holder
b)
Figure 4.9: a) The setup used for measurements showing the outcome of the X-ray and the PSD detector with
the detector arm; b) Sample holder with the Helium filled Kapton foil [110].
For sample investigation grazing incidence X-ray techniques are used. The idea behind
this geometry is to probe near surface regions, i. e. very suitable for the investigation of
surface structures. However, the rather widely distributed intensity in the reciprocal space
and the weak signal coming from the surface structures, requires very bright and
collimated beam. This can be achieved in third generation synchrotron radiation facilities
[111]. In grazing incidence techniques the X-ray beam impinges on the substrate under a
small angle, typically several tenths of degree, and is partly specularly and partly diffusely
reflected. The small impingement angle is chosen in order that the penetration of the X-ray
beam on the substrate is only few nanometer. This ensures an enhancement of the ratio of
αc
0
Ge
Si
nomralized intensity
10
-2
10
-4
10
-6
10
-8
10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
αi [deg]
Figure 4.10: Reflectivity curves for Si and Ge. The critical angle αc = 0.22 deg for Si and αc = 0.25 deg for
Ge is deduced from the position at which the intensity has its maxium.
36
4.2: Analysis Methods
the scattered intensity from nanostructures compared to that of the bulk. Usually the
incidence and the scattering angle are chosen in such a way, that the condition for a total
reflection of the scattered intensity is fulfilled. The critical angle for total reflection αc =
π/2 – arcsin(n), depends on the refraction index n. Typically the αc is between 0.1 deg –
0.5 deg depending on the X-ray energy, and on the material used for studying. For the
experiments presented in this work the condition for total reflection is fulfilled for αc <
0.22 deg for Si, and αc < 0.25 deg for Ge. It is deduced from the reflectivity curves
(Fig. 4.10), by performing a scan for different incoming angles αi and scattering angles αf
of the beam, i.e. by varying the depth of the impingement of the X-ray beam on the
substrate. The position where the reflectivity curve has its maximum gives the total
reflection angle. With further increase of αi and αf the scattered intensity decreases rapidly.
Below, a short description of the grazing incidence methods and their geometries used
for the sample investigation will be given.
4.2.2.1 Grazing Incidence Small Angle X-Ray Scattering
Grazing incidence small angle X-ray scattering (GISAXS) is a technique used for
scattering experiments under grazing incidence and exit angles [111-116]. By probing the
scattered intensity close to the (000) reciprocal lattice point, GISAXS is not sensitive to the
crystalline structure of the material, but only on the index of refraction i. e. electron density
variations. In Fig. 4.11 the scattering geometry for the GISAXS is given. The incidence
and the scattering angle are chosen in such a way that the condition for a total reflection of
the scattered intensity is fulfilled, i. e. αi = αf = 0.2 deg. In the GISAXS geometry 2θ is the
in plane scattering angle, and ω is the azimuthal angle. The rotation of ω, allows to study
the distribution of nanostructures (isotropic or not), and the angular distribution of ripples.
a)
ω
b)
PS
D
αf
αi
qx
y
x
2θ
2θ
qII qy
ki
kf
2θ
Figure 4.11: Schematic presentation of the GISAXS geometry. a) The incident X-ray is along the ripples,
and the scattered intensity is collected in the PSD detector set parallel to the sample surface. b) The wave
vectors of the incident and the scattered beam together with the scattering vector in the x-y plane are
presented.
37
Chapter 4: Experimental setup and analysis methods
In this geometry the sample is mounted in such a way that the ripples are aligned along the
X-ray beam.3 Usually the scattered intensity is plotted in the reciprocal space, and is
characterized by the momentum transfer q (scattering vector), between the incident and the
scattered X-ray beam with wave vectors ki and kf, respectively. From the geometrical
sketch in Fig. 4.11(b), the momentum vector components of the reciprocal space can be
expressed by the scattering experimental angles using the relations
qx = k (cos αi − cos α f cos 2θ )
(4.4)
q y = k (cos α f sin 2θ )
(4.5)
where k i = k f = k = 2π / λ .
4.2.2.2 Grazing Incidence Diffraction
In comparison to GISAXS, Grazing Incidence Diffraction (GID) is usually used to
study the crystalline properties of nanostructures [111,112,115,116]. A schematic drawing
is given in Fig. 4.12. The sample is rotated around the surface normal until the Bragg
condition of a particular lattice plane is fulfilled. In the GID setup, the PSD detector is set
perpendicular to the sample surface in order to measure the scattered intensity for different
αf (Fig. 4.12(a)). The GID geometry of the incident and the scattered wave vector is given
in Fig. 4.12(b). For the present studies, two scan modes in the reciprocal space were used:
ω -2θ-scan
a)
PSD
b)
αi
ω
ω-scan
y
αf
qang
x
2θ
qrad
2θ
qII
ki
ω
kf
2θ
lattice plane
Figure 4.12: Schematic presentation of the GID geometry. a) The PSD is set vertically to the sample surface
to collect the scattered intensity for different αf, b) The wave vectors of the incident and scattered beam
projected in the x-y plane are given. Also given are the two components of the scattered vector qrad and qang
and the two scan modes used.
3
Ripples form independent of the crystalline orientation of the sample. Therefore, no attention is paid to
the alignment of ripples with respect to the crystalline direction during the sputtering experiments.
38
4.2: Analysis Methods
i) The angular scan (ω-scan), in this case the length of the scattering vector is kept
constant, by varying his direction. ii) Radial scan performed along the scattering vector by
varying his length (ω-2θ-scan). This makes the radial scan strain sensitive. Due to these
scan modes the scattering vector q is divided into two components usually notated with
qang and qrad. This reciprocal space coordinates can be derived from the experimental
scattering angles using the relations
qang = 2k sin θ * sin(ω − θ )
(4.6)
qrad = 2k sin θ * cos(ω − θ ).
39
(4.7)
Chapter 5
General Properties of the Surface Topography on Si and Ge
As already mentioned in the introduction, the aim of the work is to study the evolution
of the surface topography on Si and Ge surfaces during low-energy ion beam erosion.
Especially, the capability to form large-scale ripple and dot nanostructures on the surface is
of pronounced interest. There are many parameters which play a crucial role for the
formation of nanostructures on the surface. Beginning with the geometrical parameters of
the ion-optical system, continuing with the extraction voltages applied on the grid system,
and ending with the parameters that influence the ion-target interactions. For the rest of the
work, the experimental results concerning the influence of these parameters on the
evolution of ripple and dot patterns on Si and Ge surfaces will be discussed.
At the beginning, an overview of topographies emerging on Si and Ge surfaces will be
given. In general, the role of ion incidence angle1 will be discussed. Section 5.2 will deal
with the influence of ion species on the evolution of the surface topography on both
materials.
5.1 Overview of Emerging Topographies
During low-energy ion beam erosion of Si and Ge surfaces, different topographies can
evolve on the surface. Features like holes, bumps, ripples, and dots are common. As
mentioned above, the evolution of features strongly depends on the conditions under which
the experiments are performed. An important role on the evolution of the surface
topography, at oblique ion incidence angles, exerts the rotation respectively non-rotation of
the target holder around its surface normal, Fig. 5.1 (see description of the ion beam
equipment in Section 4.1.1). From here on, “SR” denotes sample rotation and “NSR” no
sample rotation, respectively. In cases with sample rotation, due to rotational symmetry,
there is an isotropic evolution of the surface topography. This is true under the supposition
that the substrate is rotating fast enough [65]. Usually SR leads to roughness suppression,
i.e. the surface smoothens. However, as shown recently, sample rotation at oblique
incidence can lead to the formation of well-ordered dot nanostructures on the surface
[28,31,35]. For the case with NSR, there is an anisotropy present on the surface given by
the ion beam direction. In general, this results in the formation of structures (usually
ripples) with preferred spatial orientation. There are plenty of examples showing the
formation of ripples at off-normal incidence in many materials (for an overview see
1
During the experiments as ion incidence angle is taken the angle between the surface normal of the ion
source grid system and the surface normal of the sample holder.
41
Chapter 5: General Properties of the Surface Topography on Si and Ge
SR
NSR
Ion beam
Ion beam
αion
αion
b)
a)
Target
Target
Figure 5.1: Schematic presentation of the experimental setup for the case: a) with sample rotation, and b)
without sample rotation.
Chapter 1 and Ref. [117,118]). However, as it will be shown in Chapter 7, this is not
necessarily the case. For example, although there is an anisotropy present, the dot
structures show an isotropic spatial distribution, emerging on the surface.
The samples used in this work were Si(100) and Ge(100). However, as discussed in
Chapter 2 the surface evolution processes take place inside an amorphous layer covering
the sample surface.2 The layer thickness is in the range of the ion penetration depth.
Therefore the crystallographic orientation of the sample does not influence the surface
topography. The Si substrates were epi-polished, p-type with a conductivity of 0.01 –
0.02 Ω cm, while the Ge substrates were undoped with a conductivity of > 30 Ω cm. AFM
measurements of the initial surfaces of both materials revealed a root mean square
roughness ~ 0.2 nm. All investigated samples were sputtered at room temperature.
Examples of possible topographies emerging on Si surfaces after low-energy ion beam
erosion are presented in Fig. 5.2. Si substrates were bombarded with Ar+ ions, at ion
energies Eion ≤ 2000 eV, with an ion flux of J = 1.87 × 1015 cm-2 s-1 for 3600 s,
corresponding to a total ion fluence of Φ = 6.7 × 1018 cm-2. The AFM images reveal a
complexity of different topographies on the surface by varying the ion incidence angle αion.
In the case of SR for Eion = 500 eV and αion = 0 deg, hole structures evolve on the surface
(Fig. 5.2(a)). With increasing the ion incidence angle the height of structures decreases
until at αion = 45 deg the surface smoothens (Fig. 5.2(b)). By further increase of αion at
75 deg, dot structures with isotropic distribution evolve on the surface (Fig. 5.2(c)). For
NSR, structures showing preferential orientation form on the surface (Fig. 5(d-f)). Figure
5.2(d) reveals ripple structures on the Si surface at Eion = 1500 eV for αion = 15 deg.
Ripples are aligned perpendicular to the ion beam projection. At αion = 45 deg the surface
smoothens, and at αion = 75 deg columnar structures aligned along the ion beam projection
form on the surface. The same topography is observed by using Kr+ and Xe+ ions to
bombard the Si surface. Similar results are also found on Ge surfaces using Kr+ and Xe+
2
Investigations on amorphous Si and Si(111) samples showed no difference on the evolution of the
surface topography compared to results in Si(100).
42
5.1: Overview of Emerging Topographies
(a)
8 nm
(d)
0 nm
500 nm
0 nm
500 nm
Eion = 500 eV, αion = 0°, SR
(b)
Eion = 1500 eV, αion = 15°, NSR
2 nm
(e)
0 nm
500 nm
2 nm
0 nm
500 nm
Eion = 1500 eV, αion = 45°, NSR
Eion = 500 eV, αion = 45°, SR
(c)
7 nm
10 nm
(f)
0 nm
250 nm
200 nm
0 nm
500 nm
Eion = 500 eV, αion = 75°, SR
Eion = 1500 eV, αion = 75°, NSR
Figure 5.2: AFM images of different topographies on Si surfaces after Ar+ ion beam erosion. The black
arrow indicates the ion beam direction.
ions, especially with NSR.3 The topography is analyzed in terms of rms surface roughness,
that can also be taken as a measure for the height fluctuations on the surface and the
amplitude of structures [106]. These results are summarized in Fig. 5.3 where the surface
roughness w is plotted as a function of αion for Si, using Ar+ ions. The graph shows that the
roughness decreases up to a minimum value with ion incidence angle. By further
increasing of the αion the w increases again. Fig. 5.3 reveals that the evolution of w with αion
is independent of the ion energy used and if there is sample rotation or not. In general,
three regions with regard to αion can be distinguished. Region I: the surface is rough for αion
between 0 deg and ~ 40 deg, and features like dots, holes, and ripples form on the surface.
Region II: smooth surfaces for αion from ~ 40 deg up to ~ 60 deg. Region III: the surface
roughens again at grazing incidence above 60 deg and features like dots or columnar
structures emerge. Analogous results are obtained using different ion species to
3
The role of ion species on the evolution of the surface topography will be discussed in Section 5.2.
43
Chapter 5: General Properties of the Surface Topography on Si and Ge
rms roughness w [nm]
100
+
Ar , SR
+
Ar , NSR
+
Ar , SR
Eion = 500 eV
10
1
I
II
III
Eion = 1500 eV
Eion = 1800 eV
0.1
0
15
30
45
60
75
ion incidence angle αion [deg]
Figure 5.3: Development of rms surface roughness with ion incidence angle for Si using Ar+ ions at different
ion energies (Φ = 6.7 × 1018 cm-2). The results are plotted for the case with and without sample rotation.
bombard the Si surface. Results for Kr+ and Xe+ ions are plotted in Fig. 5.4(a), exemplary
for Eion = 2000 eV. Figure 5.4(b) shows a similar behavior for the evolution of the surface
roughness with ion incidence angle on Ge surfaces using Kr+ and Xe+ ions.
There is a parameter region for both materials with its center at 45 deg that is always
valid independent from all the sputtering parameters treated in this work. Namely, at this
ion incidence angle the surface remains smooth with a roughness w < 0.2 nm. Hence, this
sputtering condition is very well suited for large area surface smoothing, and finds a broad
application in the field of optic manufacturing [119-121].
In this section, the evolution of the surface topography in terms of surface roughness as
a function ion incidence angle was discussed without a detailed treatment of the particular
+
10
III
II
I
1
rms roughness w [nm]
rms roughness w [nm]
100
Kr
+
Xe
Eion = 2000 eV
+
10
I
0
15
30
45
60
0.1
75
ion incidence angle αion [deg]
II
III
1
a)
0.1
Xe
+
Kr
Eion = 2000 eV
b)
0
15
30
45
60
75
ion incidence angle αion [deg]
Figure 5.4: Rms surface roughness w as a function of αion using Kr+ and Xe+ ions to bombard the surface (Φ
= 6.7 × 1018 cm-2, Eion = 2000 eV, without sample rotation). a) Si, b) Ge.
44
5.2: Influence of Ion Species
structures. It was shown that there is a general behaviour of the surface roughness with ion
incidence angle for different sputtering parameters. A detailed discussion especially of
region I will be given in Chapter 7. Moreover, the discussion concerning the influence of
other process parameters will be given in the next Chapters. Obviously, it is difficult and
beyond the scope of this work to study the influence of different process parameters on all
topographies presented in Fig. 5.2. Especially, as it will be shown in the topography
diagrams in Section 7.2, by varying the sputtering conditions additional structures evolve
on the surface. As discussed in Chapter 1, particular interest will be paid to ripple and dot
structures, and hence to the conditions under which these structures evolve.
5.2 Influence of Ion Species
When an ion penetrates the target surface it transfers its energy and momentum due to
collision processes to the target atoms until it comes at rest. This process of slowing down
of ions gives rise to different phenomena on the surface and near-surface region. The most
important process parameters are the range and straggling of the distribution of the
deposited energy of incoming ions. This distribution depends on the energy of incoming
ions, ion incidence angle, and the properties of the target material. Additionally, the
distribution depends on the mass of incoming ions.
Experimental results show that the evolution of the surface topography on Si and Ge
surfaces is ion species dependent. During the bombardment of Si surfaces with Ne+ ions at
ion energies 300 eV ≤ Eion ≤ 1000 eV structures evolve on the surface. For 1000 eV ≤ Eion
≤ 2000 eV the surface remains smooth.4 Using Ar+, Kr+, and Xe+ as bombarding ions the
surface roughens i. e. structures evolve. Their formation depends also on other sputtering
conditions as it will be shown later in this work. Similar dependence of the surface
topography on Si with different ion species, was observed previously by Carter for
intermediate ion energies (above 20 keV) [122].
In the case of Ge, no structures are commonly observed (the surface remains smooth)
when Ne+ and Ar+ ions are used to bombard the surface. An example of topography
evolution on Ge surfaces for different ion species is given in Fig. 5.5. The AFM images
show that in the case of Ar+ ions, the surface remains smooth, while for Kr+ and Xe+ ions a
dot like structure evolves on the surface. However, there is a region (Eion = 1300 eV –
2000 eV and αion = 0 deg – 20 deg) where dot like structures are also observed using Ar+
ions, but with no ordering and an amplitude below 1 nm.
Due to lack of experimental studies, up to now, on the influence of ion species on the
surface topography on Si and Ge, it is difficult to give an exact explanation for the above
observations. Nevertheless, two possible explanations can be propsed for the emerging
4
For ion energies 300 eV ≤ Eion ≤ 1000 eV the formation of structures depends on other sputtering
parameters, like ion incidence angle similar to the discussion in Section 5.1. Contrary to this, for 1000 eV ≤
Eion ≤ 2000 the surface remains smooth independent of ion incidence angle.
45
Chapter 5: General Properties of the Surface Topography on Si and Ge
3 nm
(a)
3 nm
(b)
0 nm
500 nm
0 nm
0 nm
500 nm
500 nm
Ar+
3 nm
(c)
Kr+
Xe+
Figure 5.5: Surface topographies on Ge after ion beam erosion with different ion species at Eion = 1200 eV,
αion = 15 deg without sample rotation.
surface topography using different ion species:
(i) It is known from the theory and experiments that the main contribution to the sputter
yield originates from atoms ejected from the uppermost surface layers [40]. This
ejection process is related to the distribution of the deposited energy just below this
surface layer, that depends on the mass of the incoming ions. In Fig. 5.6 the depth
profiles of the deposited energy FD in Si using Ne+, Ar+, Kr+, and Xe+ ions at
Eion = 2000 eV are plotted using the SRIM simulation code [123]. The profiles show
that with increasing ion mass, the mean penetration depth (Fig. 5.4(b)) and the width of
the distribution decrease. This means that for heavier ions the energy distribution
maximum is located closer to the surface region than for lighter ions, i.e. more recoils
are created in the upper surface layer for heavier ions (for comparison see energy loss
values in Table 2.1). A similar behavior of the energy distribution and the mean depth
is also observed for simulations using Ge as a target material.
(ii) In addition to the energy deposition also highly energetic sputtered target atoms as well
as backscattered projectile ions become more important for primary ions with lower ion
mass. These sputtered particles contribute to additional sputtering of peaks compared to
valleys, hence prohibiting the evolution of structures and leading to smooth surfaces
[124]. This is supported by TRIM.SP [125,126] calculations using Ne+, Ar+, and Kr+
ions. Some conclusions of these simulations are: a) For lighter projectile ions, the
number of highly energetic particles emitted at an emission angle between 0 deg and
15 deg with respect to the surface plane, is higher when Ne+ ions are used compared to
Ar+, and Kr+ ions. b) Additionally, the number of highly energetic particles emitted
under these angles increases with ion energy, for a given ion species. c) The energy of
sputtered particles increases with decreasing ion mass. d) This number increases also
with increasing ion incidence angle, that would explain for example the larger surface
roughness observed at small incidence angles and its decrease with increasing angle up
to 45 deg.
From the above discussion it is obvious that the evolution of the surface topography
depends on the ion species used. For ion species with lighter mass than the target, usually
46
5.2: Influence of Ion Species
160
+
80
40
mean depth a [nm]
FD (z) [a.u.]
120
0
5
+
Ne
+
Ar
+
Kr
+
Xe
Fits using Eq. (2.12)
a)
0
5
10
15
mean depth a [nm]
20
4
3
Ne
+
Ar
+
Kr
+
Xe
power fit
2
b)
1
800
1200
1600
2000
ion energy Eion [eV]
Figure 5.6: a) Depth distribution of the deposited energy on Si for different ion species (Eion = 2000 eV, at
normal incidence). The solid line is a fit using the Eq. (2.12). b) Mean depth a of the deposited energy for
different ion energies and different ion species. A power law fit of data for Ar+ ions is performed with m =
0.31 (see discussion for Fig. 6.3 in Section 6.1). All simulations were performed with SRIM code.
no structures are observed. On the other side, once these structures form, their
characteristics like mean size, lateral ordering, homogeneity, and height do not depend on
the ion species used as it will be shown in Chapter 6.
47
Chapter 6
Ripple and Dot Patterns on Si and Ge surfaces
In Section 5.1, a short introduction of different topographies evolving on Si and Ge
surfaces during low-energy ion beam erosion was given. Depending on sputtering
conditions, features like holes, ripples, dots, and smooth surfaces can evolve. However,
from particular interest, and the main subject of this work is the formation of ripple and dot
patterns. Theoretically, the process of pattern formation is related to the competition
between the curvature dependent sputtering and different relaxation processes that results
in the formation of nanostructures on the surface. The question that arises is, it is possible
to control the evolution, lateral ordering and size of structures? If yes, which parameters
are more relevant? To answer this question the influence of different process parameters
has to be addressed. Some of these parameters include the ion energy Eion, ion fluence Φ,
and ion flux J which will be discussed in detail in the next sections. Section 6.1 will show
that with increasing ion energy the wavelength of structures increases. This increase
correlates with theoretical predictions, by considering ion induced effective surface
diffusion ESD or ion-induced viscous flow IVF as a dominant relaxation mechanism.
However, there are conditions under which a completely new behavior is observed, namely
a change in orientation of the wave vector of ripples in Si, or a transition from ripples to
dots on Ge, with increasing ion energy. In Section 6.2, the role of ion fluence and ion flux
on the evolution of the surface topography and the lateral ordering of structures will be
discussed. While the wavelength of structures remains constant with ion fluence the lateral
ordering increases. The geometrical shape of ripples and dots investigated with highresolution transmission electron microscopy is given in Section 6.3. Additional to the AFM
method also small angle X-ray scattering techniques are used to characterize the ripple and
dot structures. These results will be discussed in Section 6.4.
Results for both materials, Si and Ge, are presented. In the case of Si, results will be
given for Ar+, Kr+, and Xe+ ions. Also, for Si two cases are distinguished, a) SR and b)
NSR. With SR the results for a grazing incidence angle of 75 deg, where dot structures
evolve on the surface, will be given [31]. For Ge only the case with NSR will be discussed.
Dot structures form also on Ge surfaces with SR. However, under experimental conditions
used in this work, they show only a vague lateral ordering, and a large size distribution
making it very difficult to deduce general statements.
49
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
6.1 Influence of Ion Energy
Silicon
This section is devoted to the role of ion energy Eion on the formation of patterns on Si
surfaces. The Eion is varied between 500 eV and 2000 eV, which is limited by the power
supply of the ion source. The experiments were performed at room temperature with an
ion-current density jion ~ 300 µA cm-2 (corresponding to an ion flux J =
1.87 × 1015 cm-2 s-1) and a total ion fluence of Φ = 6.7 × 1018 cm-2. The ion fluence used
ensures that the evolving patterns are well above the saturation regime concerning the
surface roughness (see Section 6.2).
In Fig. 6.1, the AFM images of ripple patterns emerging on Si surfaces, at an ion
incidence angle αion = 15 deg from the surface normal, without sample rotation are given.
The Si surface is bombarded with Ar+ ions with three different Eion: a) 800 eV, b) 1500 eV,
and c) 2000 eV. In order to study the characteristic wavelength (i. e. spatial frequency) of
ripples, Fast Fourier transformation (FFT) of the AFM images is performed using Eq.
(4.2). The FFT images show clear spots in the spatial frequency spectra that correspond to
the dominating ripple wavelength in the real space. Additional spots in the FFT image
indicate the high lateral ordering of ripples. The FFT image shows that the wave vector of
ripples is parallel to the projection of the ion beam onto the surface plane. Compared to
high spatial-frequency ripples, AFM images reveal additional low spatial-frequency
corrugations on the surface with a wave vector perpendicular to the ion beam projection.
Due to a rather broad size distribution, it is difficult to determine their wavelength.
However, the amplitude of these corrugations is very small compared to that of short
wavelength ripples.1 For quantitative determination of the characteristic wavelength of
ripples, the power spectral density (PSD) function is obtained from FFT images by angular
averaging using Eq. (4.3). The corresponding PSD graphs for three different ion energies
are given in Fig. 6.2. The position of the first peak on the PSD graph gives the
characteristic spatial frequency of ripples, i. e. the ripple wavelength λ. From the PSD
graphs a shift of the first peak towards smaller spatial frequencies with increasing ion
energy is observed. Further, the integral over the PSD function, which can be taken also as
a measure for the rms surface roughness w, indicates an increase of w with Eion. This
means also that the mean height (amplitude) of ripples increases with ion energy. Results
about the change of the ripple wavelength λ, and the system correlation length ζ with ion
energy are quantitatively summarized for Ar+ and Kr+ ions in Fig. 6.3. The graphs show an
increasing λ from 40 nm up to 70 nm by varying the ion energy from 500 eV up to
2000 eV. In Fig. 6.3, the system correlation length is normalized to the ripple wavelength
1
For example, by excluding the short wavelength contributions, applying a Fourier filtering, to an AFM
image with 16 × 16 µm2 scan size for Eion = 1200 eV at αion = 15 deg, a mean amplitude of the long
wavelength corrugations of ~ 0.6 nm was deduced.
50
6.1: Influence of Ion Energy
4 nm
FFT
d)
FFT
e)
FFT
f)
0 nm
500 nm
a) Eion = 800 eV, αion = 15°, NSR
7 nm
0 nm
500 nm
b) Eion = 1500 eV, αion = 15°, NSR
10 nm
0 nm
500 nm
c) Eion = 2000 eV, αion = 15°, NSR
f = - 128 µm-1 … 128 µm-1
Figure 6.1: AFM images of self-organized ripple patterns on Si surfaces after Ar+ ion beam erosion at Φ =
6.7 × 1018 cm-2, for different ion energies. (d-f) Corresponding Fourier images (image size ± 128 µm-1). The
arrows indicate the direction of the incoming ion beam.
ζ/λ. The ratio ζ/λ, shows in how many periods a perfect lateral ordering of ripples is
present. While for Kr+ ions the ordering of ripples is independent of Eion, for Ar+ ions the
best ordering is achieved at Eion = 1200 eV. For the given sputtering conditions in the case
of Kr+ at Eion = 500 eV no ripple structures are observed.
Theoretically the scaling behavior of λ with Eion depends on the particular relaxation
mechanism under consideration. Under given sputtering conditions, two relaxation
mechanisms can be considered from relevance for the process of ripple formation.
i) Ion-induced ESD as presented in Section 3.1. From Eq. (3.8), (3.12), and (3.13) and by
making use Eq. (2.8) for the ion range it follows
2m
λ ~ a ~ Eion
.
(6.1)
51
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
6
st
800 eV
1500 eV
2000 eV
1 peak
4
power spectral density PSD [nm ]
10
4
10
2
10
0
10
-2
10
-3
-2
10
-1
10
10
-1
spatial frequency f [nm ]
Figure 6.2: Angular averaged PSD functions obtained from FFT images in Fig. 6.1. The first peak
corresponds to the characteristic ripple wavelength.
The wavelength is proportional to the mean penetration depth a, which scales with Eion
2m
as a ~ Eion
. In order to compare the evolution of λ with Eion with the theory, a linear
(λ ∝ Eion )
(
)
2m
and power-law λ ∝ Eion
fit of the experimental results for the case of Ar+
ions is performed. The power-law fit gives an exponent m = 0.22. This is in a reasonable
80
60
50
+
Ar
+
Kr
linear fit
power fit
40
30
ζ/λ
ripple wavelength λ [nm]
/
/
20
40
10
20
400
800
1200
1600
2000
0
ion energy Eion [eV]
Figure 6.3: The dependence of the ripple wavelength λ and the system correlation length ζ (normalized to λ)
on ion energy Eion (Φ = 6.7 × 1018 cm-2, αion = 15 deg, Ar+ and Kr+ ions) for Si. The linear fit and power fit
(with an exponent m = 0.22) of the data for Ar+ ions are also given.
52
6.1: Influence of Ion Energy
agreement with the power-law fit of the mean depth SRIM data plotted in Fig. 5.4 for
Ar+ ions, resulting in an exponent m = 0.31.
ii) Another relaxation mechanism that can influence the process of ripple formation is the
ion-enhanced viscous flow (IVF) [29]. Here smoothing occurs by ion-induced viscous
relaxation confined to a near surface region of depth d, with d being of the order of the
ion range a, and much smaller than the ripple wavelength λ (a ~ d << λ). Then from Eq.
3m
, i. e. the ripple wavelength increases with ion energy, and m = 0.15 is
(3.19) a ~ Eion
deduced.
From the above discussion it seems that both mechanisms predict an increase of λ with
Eion similar to experimental results. However, the value of the parameter m calculated from
the experimental data should be taken with caution. In order to better distinguish between a
linear and power-law dependency the energy range would have to be extended. However,
this was not possible due to the limited range of ion energy feasible in the experiments.
A ripple wavelength increasing with ion energy is observed also by using Xe+ ions to
sputter the Si surface for Eion >1000 eV (Fig. 6.4). The results are given for three different
ion incidence angles. The graph shows also that the wavelength of ripples decreases with
increasing ion incidence angle, a fact that will be elaborated in more detail in Chapter 7.
However, there is a more complex behavior of the surface topography dependent on ion
energy in the case of Xe+ ions, which is not predicted by the continuum theory. In Fig. 6.5
AFM images of Si surfaces after sputtering with Xe+ ions, at αion = 20 deg, for different ion
energies are given. For Eion = 1200 eV ripple patterns with the wave vector parallel to the
ion beam projection form on the surface with wavelength λ = 46 nm (Fig. 6.5(a)). By
decreasing the ion energy to Eion = 800 eV, ripples vanish and the surface remains smooth
5°
15°
20°
ripple wavelength λ [nm]
75
60
45
1200
1400
1600
1800
2000
ion energy Eion [eV]
Figure 6.4: The dependence of wavelength on ion energy for ripples on Si, using Xe+ ions, for different ion
incidence angles.
53
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
(Fig. 6.5(b)). Further decrease of ion energy results in a new type of ripples evolving on
the surface (Fig. 6.5(c)). Now, the wave vector is oriented perpendicular to the ion beam
projection. These ripples have a mean height similar to ripples with parallel wave vector,
but with wavelength λ = 111 nm, which is more than two times larger. The spots in the
corresponding FFT image show a larger radial distribution of these ripples compared to
ripples with parallel wave vector.2 The parameter region of a smooth surface between these
two mode ripples depends on sputtering conditions. For example, it can be shifted toward
smaller Eion values with increasing αion. For a better presentation, the results for the
5 nm
FFT
d)
FFT
e)
FFT
f)
0 nm
500 nm
a) Eion = 1200 eV, αion = 20°, NSR
2 nm
0 nm
500 nm
b) Eion = 800 eV, αion = 20°, NSR
5 nm
0 nm
1000 nm
f = - 128 µm-1 … 128 µm-1
c) Eion = 500 eV, αion = 20°, NSR
Figure 6.5: (a-c) Surface topography on Si after Xe+ ion beam sputtering at Φ = 6.7 × 1018 cm-2 for different
Eion. (d-f) Calculated FFT images with image size ± 128 µm-1. Please note the different scale in AFM images.
2
The larger radial distribution for ripples with perpendicular wave vector is mainly due to sticking
together of ripples at some positions that contributes to the characteristic spatial frequency on the FFT.
54
6.1: Influence of Ion Energy
ion energy Eion [eV]
B
parallel
mode
ripples
1500
ripples + dots
2000
pillar
structures
1000
A
smooth
surfaces
C
500
0
15
30
45
60
75
ion incidence angle αion [deg]
Figure 6.6: Topography diagram giving the surface topography on Si due to Xe+ ion beam erosion (Φ =
6.7 × 1018 cm-2) for different ion energies and ion incidence angles, without sample rotation. A – represents
hole structures, B – represents hillock structures, and C – the normal mode ripples. The symbols indicate the
experimental data. - smooth surfaces,& - hole structures, ^ - hillock structures, 1 - perpendicular-mode
ripples, - parallel-mode ripples + dots, u - parallel-mode ripples, = - columnar structures.
evolution of the surface topography for different Eion and αion are plotted using a so-called
topography diagram TD. Such a TD is presented in Fig. 6.6. It reveals a complexity of
different topographies that depend on Eion, and αion. Obviously, the surface topography
reliance on Eion will also depend on the value of αion. The boundaries (doted lines) on the
TD are used as a guide to the eye, to distinguish between different topography regions. In
most of the cases their position is taken as a middle point between the experimental data
(symbols in Fig. 6.6) representing two different topographies.3 For explanation, as parallelmode ripples are meant ripples with the wave vector parallel to the projection of the ion
beam on to the surface. With ripples plus dots are meant surfaces where parallel-mode
ripples and dots coexist. Ripples with the wave vector oriented perpendicular to the ion
beam projection, are named perpendicular-mode ripples.
In the SR case (ensuring an isotropic evolution of the surface topography) at oblique ion
incidence, depending on sputtering parameters dot patterns can form on the surface. Dot
patterns form also at normal ion incidence [31], similar to those reported by other research
groups [30]. The focus here will lie on dot patterns at oblique ion incidence, because of the
larger amplitude and the much better lateral ordering compared to the dots at normal
incidence, making them more appropriate for other applications. In Fig. 6.7 an example of
3
This is done with the supposition that topographical transitions are continuous and that the transition
point lies in the middle between the experimental data.
55
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
20 nm
FFT
d)
FFT
e)
FFT
f)
0 nm
500 nm
a) Eion = 500 eV, αion = 75°, SR
20 nm
0 nm
500 nm
b) Eion = 1000 eV, αion = 75°, SR
30 nm
0 nm
500 nm
c) Eion = 2000 eV, αion = 75°, SR
f = - 128 µm-1 … 128 µm-1
Figure 6.7: (a-c) Dot structures on Si after Kr+ ion beam erosion with SR at Φ = 6.7 × 1018 cm-2 for different
Eion. (d-f) Corresponding FFT images with size ± 128 µm-1.
dot patterns evolving on the Si surface during sputtering with Kr+ ions (αion = 75 deg, Φ =
6.7 × 1018 cm-2) is given. The AFM images show domains of close-packed, hexagonally
ordered dot structures evolving on the surface. These domains are randomly ordered with
respect to each other. The position of the first ring on the FFT specifies the mean lateral
size of dots λ. Additionally, FFT images reveal that the size of dots increases with Eion,
giving the possibility to control their size in a certain range. Similar results are observed by
using Ar+ and Xe+ ions. Quantitatively, the dependence of λ and ζ (equal to the mean
domain size for dots) on Eion is summarized in Fig. 6.8. The results show an increase of the
mean dot size, from 25 nm up to 50 nm, with increasing ion energy. In the case of Ar+
ions, the mean dot size increases up to Eion = 1000 eV. For further increasing Eion, dots
overlap with each other creating conglomerates of dots with no ordering, until at Eion =
56
6.1: Influence of Ion Energy
(a)
λ
ζ/λ
40
mean dot size/periodicity λ [nm]
9
+
Ar
6
3
30
50
(b)
0
9
+
Kr
6
40
3
30
50
(c)
ζ/λ
50
0
9
+
Xe
6
40
3
30
400
800
1200
1600
ion energy Eion [eV]
2000
0
Figure 6.8: The variation of the mean dot size and the normalized correlation length on Si with ion energy
for different ion species (αion = 75 deg, and Φ = 6.7 × 1018 cm-2).
2000 eV the surface smoothens. For Kr+ and Xe+ ions, dots form in the range 500 eV ≤ Eion
≤ 2000 eV and their size increases with Eion. The determination of the mean dot size for
Xe+ ions at Eion = 500 eV is associated with a large uncertainty due to the marginal lateral
ordering of dots. The ratio ζ/λ shows that for Eion > 750 eV the lateral ordering of dot
structures remains constant. For Ar+ ions the best lateral ordering of dots is obtained at Eion
= 500 eV. A behavior similar to that of Ar+ is observed for Ne+ ions under the given
sputtering conditions. However, the uniformity and ordering of dots is much less
pronounced than for Ar+ ions.
Germanium
Analogous investigations for the surface topography with ion energy were performed on
Ge. The parameters used for the sputtering process are identical to those for Si. A
representative example of ripple patterns on Ge is given in Fig. 6.9 after Xe+ ion beam
sputtering with Eion = 2000 eV, at αion = 5 deg, and ion fluence Φ = 6.7 × 1018 cm-2. The
FFT image reveals that the wave vector of ripples is parallel with respect to the ion beam
projection. The findings on the dependence of λ on Eion are summarized in Fig. 6.10. The
ripple wavelength increases with ion energy, similar to Si. For the given sputter conditions,
57
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
10 nm
FFT
0 nm
500 nm
f = - 128 µm-1 … 128 µm-1
Eion = 2000 eV, αion = 5°, NSR
Figure 6.9: AFM image of ripple patterns on Ge surfaces after Xe+ ion beam erosion without sample
rotation, for Φ = 6.7 × 1018 cm-2.
75
ζ/λ
60
power fit, m = 0.21
SRIM: a, m = 0.21
λ
15
10
45
ζ/λ
mean wavelength λ & mean depth a [nm]
no ripples were observed at Eion = 500 eV. The ratio ζ/λ shows that for Eion > 1000 eV the
lateral ordering of ripples is improved significantly compared to Eion = 800 eV. By
considering the ESD term as the main relaxation mechanism, a power-law fit to the
experimental data yields an exponent m = 0.21. This is in a very good agreement with the
exponent m = 0.21 deduced by a power-law fit of the mean depth a from the SRIM
simulations data (Fig. 6.10). However, similar to Si, a larger Eion range would be necessary
to enable a better determination of the scaling behavior of λ with Eion.
By studying the surface topography with ion energy at different ion incidence angles, an
interesting effect is observed. Namely, by increasing Eion for αion = 20 deg a transition from
5
2
1
400
800
1200
1600
2000
0
ion energy Eion [eV]
Figure 6.10: The dependence of λ and ζ/λ on Eion for Xe+ ion beam erosion of Ge (αion = 5 deg, Φ =
6.7 × 1018 cm-2). Also plotted is the mean depth a for Xe+ ions on Ge for different Eion calculated with SRIM.
The power-law fits performed for λ and a, give a scaling exponent m = 0.21.
58
6.1: Influence of Ion Energy
5 nm
FFT
d)
FFT
e)
FFT
f)
0 nm
500 nm
a) Eion = 500 eV, αion = 20°, NSR
5 nm
0 nm
500 nm
b) Eion = 800 eV, αion = 20°, NSR
5 nm
0 nm
500 nm
c) Eion = 1200 eV, αion = 20°, NSR
f = - 128 µm-1 … 128 µm-1
Figure 6.11: Surface topography on Ge after Xe+ ion beam erosion with no sample rotation, for different Eion
(Φ = 6.7 × 1018 cm-2) (d-f) Corresponding FFT images. The arrows indicate the ion beam direction.
ripple to dot structures occurs. Such an example is presented in Fig. 6.11, where AFM
images of the Ge surface after sputtering with Xe+ ions for: a) Eion = 500 eV, b) Eion =
800 eV, and c) Eion = 1200 eV are depicted. At 500 eV, parallel-mode ripples, as shown in
the corresponding FFT image, form on the surface. The large angular distribution of spots
is related to the less pronounced lateral ordering of ripples. By increasing Eion, ripples and
dots evolve simultaneously on the surface. In the FFT image the spots have a more curved
form indicative of dots, but the preferential direction of ripples is still observable. At
1200 eV, the surface consists only of dot structures with particular domains showing
59
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
500
dots
hillock structures
1000
parallel
mode ripples
dots
s
dot
1500
s+
ple
rip
ion energy Eion [eV]
2000
smooth
A
0
5
10
15
20
25
ion incidence angle αion [deg]
30
35
Figure 6.12: Topography diagram for Ge after Xe+ ion beam sputtering (Φ = 6.7 × 1018 cm-2) for different
ion energies and ion incidence angles. Region A represents perpendicular-mode ripples. The symbols
represent the experimental data: ^ - hillock structures, - smooth surfaces, 1 - perpendicular-mode
ripples, - parallel-mode ripples + dots, u - parallel-mode ripples, = - columnar structures, - dots.
hexagonal ordering. The evolution of the surface topography with ion energy for different
incidence angles is summarized in the topography diagram in Fig. 6.12. The TD gives a
complex picture of evolving topographies similar to Si. The topography depends not only
on Eion but also on the incidence angle. For αion = 5 deg a transition from perpendicularmode to parallel-mode ripples is observed with increasing ion energy. The same discussion
as for Si can be made concerning the boundaries between different parameter regions.
Summary
The above results for Si and Ge show that ion energy, is a key-parameter for the
formation of ripple and dot structures and for determining their wavelength. Generally, the
results evidence an increase of the wavelength of nanostructures with ion energy. This
correlates with the theoretical models that also predict an increase of λ with Eion by
considering ESD or IVF as relaxation mechanisms (at least for certain ion energy range).
However, there are experimental conditions under which completely new phenomena are
observed. One is the formation of a new type of ripples with the wave vector perpendicular
to the projection of the ion beam. The wavelength of these ripples is approximately two
times larger compared to the wavelength of parallel-mode ripples. Moreover, there is a
transition from ripples to dots with increasing ion energy on Ge surfaces. Both
observations are not consistent with the theoretical model presented in Chapter 3. It should
be mentioned that the role of ion incidence angle on the topographical transitions will be
discussed in Section 7.1.
60
6.2: Ion Fluence and Flux
6.2 Ion Fluence and Flux
In this chapter the temporal evolution of ripple and dot patterns on Si and Ge will be
discussed in detail. Explicitly, results about the evolution of the characteristic wavelength λ
of nanostructures and the surface roughness w with ion fluence Φ will be presented. The
results for different ion species will be given. The ion fluence equals the total number of
ions hitting the surface per unit area. For a given ion flux the ion fluence Φ is equivalent
with the sputter time, or with the thickness of the removed layer.
All experiments were conducted under conditions, under which well ordered ripple and
dot structures are formed.
Silicon
A representative example of evolving ripple patterns, with increasing ion fluence on Si
is given in Fig. 6.13. The surface is sputtered with Kr+ ions at Eion = 1200 eV, αion = 15 deg
and ion current density jion = 300 µA cm-2 for different ion fluences: a) Φ = 3.3 × 1017 cm-2
(sputter time 180 s), b) Φ = 2.2 × 1018 cm-2 (1200 s), c) Φ = 1.3 × 1019 cm-2 (7200 s). The
AFM image in Fig. 6.13(a) reveals ripple topography from the beginning of the sputtering
process. The first spot on the corresponding FFT image clearly indicates the characteristic
wavelength of ripples in Fig. 6.13(d). The wave vector of ripples is oriented parallel to the
ion beam projection. However, the rather broad radial and angular distribution of the first
spot, reveal that ripples have a rather poor lateral ordering (alignment) and size
homogeneity. With increasing ion fluence the ordering of ripples increases (Fig. 6.13(b,e)),
as seen from the FFT image where the radial and angular distribution of the spots
decreases. Also more high order peaks become visible (Fig. 6.13(c,f)).
The AFM images show that ripples are interrupted by defects (denoted by the circle in
Fig. 6.13(c)), producing two new ripples or coalescence of two ripples in one. The number
of defects decreases with Φ leading to almost perfectly ordered ripples with approximately
2 defects per 1 µm2, for the highest fluence shown in Fig. 6.13(c).
In Fig. 6.14 the PSD functions of the corresponding FFT images from Fig. 6.13 are
plotted. For comparison the PSD of an untreated substrate is included. The PSD graphs
show a clear distinct peak growing at a given spatial frequency. The position of the peak
does not change with ion fluence indicating independent wavelength values with Φ. In
contrast, the width of the peak decreases. These results are quantitatively summarized in
Fig. 6.15 by plotting the evolution of the ripple wavelength λ and the system correlation
length ζ with ion fluence for Ar+, Kr+, and Xe+ ion species. The results are presented for
Eion = 1200 eV, and αion = 15 deg. The graphs show a ripple wavelength λ ~ 50 nm, that is
not varying with ion fluence. At the same time ζ increases with ion fluence. At the
beginning (up to an ion fluence Φ = 2 × 1018 cm -2) there is a steeper increase than for
larger fluences. For a total fluence of Φ = 4 × 1019 cm-2, the ζ extends up to 1 µm. This
61
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
7 nm
FFT
d)
FFT
e)
FFT
f)
0 nm
500 nm
a) Φ = 3.4 x 1017 ions/cm2, NSR
7 nm
0 nm
500 nm
b) Φ = 2.2 x 1018 ions/cm2, NSR
7 nm
0 nm
500 nm
f = - 128 µm-1 … 128 µm-1
c) Φ = 1.3 x 1019 ions/cm2, NSR
Figure 6.13: (a-c) Surface topography on Si after Kr+ ion beam erosion with Eion = 1200 eV, αion = 15 deg,
for different ion fluences. The solid circle in c) indicates an existing defect between ripples. (d-f)
corresponding Fourier images.
value of the system correlation length for ripples can be interpreted as the mean distance
between the defects on the AFM image.
Concerning dot structures forming at oblique ion incidence, with sample rotation,
similar behavior of λ and ζ, like for ripples is observed. These results are plotted in
Fig. 6.16 for: a) Φ = 1.1 × 1017 cm-2 (sputter time 60 s), b) Φ = 2.2 × 1018 cm-2 (1200 s), c)
Φ = 1.3 × 1019 cm-2 (7200 s). The surface is bombarded with Kr+ ions at Eion = 1000 eV,
αion = 75 deg and ion current density jion = 300 µA cm-2. The images show an increase of
the lateral ordering and size homogeneity of dots with ion fluence. For prolonged
sputtering (Fig. 6.16(c)) large domains of close packed dots showing hexagonal order form
on the surface. In the corresponding FFT images the radial width of the ring decreases and
62
6.2: Ion Fluence and Flux
10
2
10
0
10
600
40
17
2
Φ = 3.4 x 10 ions/cm
18
(b)
2
Φ = 2.2 x 10 ions/cm
(c)
300
20
80
0
900
+
Kr
60
600
40
300
20
80
0
900
+
Xe
60
(d)
600
40
19
2
-3
-2
0
-1
2
a) Φ = 1.1 x 10 ions/cm , SR
d)
10
0
40
18
-2
Figure 6.15: Ion fluence dependence of wavelength
λ and the system correlation length ζ for ripples on
Si with Eion = 1200 eV, αion = 15 deg for different
ion species.
5 nm
20 nm
20 nm
0 nm
0 nm
0 nm
500 nm
500 nm
17
5
300
ion fluence Φ [10 cm ]
Figure 6.14: Angular averaged PSD functions
obtained from FFT images in Fig. 6.13(d-f).
For comparison the PSD function of an
untreated (not eroded) surface is given (a).
500 nm
λ
ζ
20
Φ = 1.3 x 10 ions/cm
10
10
10
-1
spatial frequency f [nm ]
FFT
900
60
4
10
2
10
0
10
+
Ar
system correlation length ζ [nm]
4
80
(a)
18
2
b) Φ = 2.2 x 10 ions/cm , SR
e)
FFT
c) Φ = 1.3 x 1019 ions/cm2, SR
FFT
f)
f = - 128 µm-1 … 128 µm-1
4
10
2
10
0
10
Φ=0
ripple wavelength λ [nm]
4
power spectral density PSD [nm ]
4
10
2
10
0
10
Figure 6.16: (a-c) AFM images of dot structures on Si surfaces after Kr+ ion beam sputtering, with sample
rotation, at Eion = 1000 eV, αion = 75 deg for different ion fluences. (d-f) The corresponding FFT images.
63
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
the number of multiple rings increases. Fig. 6.17 shows the evolution of the mean size of
dots λ and the system correlation length ζ with Φ. The results are given for Ar+ at Eion =
500 eV and Kr+ and Xe+ ions at Eion = 1000 eV.4 From the graphs, the mean size of dots
does not change while ζ increases with ion fluence. For prolonged sputtering a system
correlation length ζ up to 150 nm is observed. Also the results indicate a mean size of dots
that is independent from the ion species used.
Germanium
The evolution of λ and ζ with ion fluence for structures on Ge, displays similar result
like for Si. This is shown for the case of dot patterns evolving on the Ge surface, after Xe+
ion beam sputtering with NSR, for Eion = 2000 eV and αion = 20 deg (see Fig. 6.11(c) and
Section 7.1). Results for the dependence of λ and ζ/λ on ion fluence are plotted in Fig. 6.18.
Similar to the case of dots in Si also in Ge the mean size of dots remains constant with ion
fluence while the ordering increases with ion fluence until it saturates. The mean size
fluctuations for low ion fluences are related to the large size distribution of dots.
(a)
150
40
100
mean dot size/periodicity λ [nm]
20
50
+
0
60
Ar , αion = 75°, Eion = 500 eV
0
(b)
150
40
100
20
50
+
0
60
Kr , αion = 75°, Eion = 1000 eV
λ
ζ
(c)
40
150
100
20
0
0
system correlation length ζ [nm]
60
+
Xe , αion = 75°, Eion = 1000 eV
0
4
8
12
18
-2
ion fluence Φ [10 cm ]
50
0
Figure 6.17: Dependence of the mean dot size and the system correlation length on ion fluence for dots on
Si. The results are given for different ion species.
4
As shown in Fig. 6.8 for Ar+ ions dots with the best lateral ordering evolve for Eion = 500 eV.
64
6.2: Ion Fluence and Flux
4
80
ζ/λ
λ
mean dot size λ [nm]
60
3
ζ/λ
40
20
0
+
Xe , αion = 20°, Eion = 2000 eV
0
4
8
2
12
18
-2
ion fluence Φ [10 cm ]
Figure 6.18: Evolution of mean dot size and the normalized system correlation length with ion fluence, for
dots on Ge.
Evolution of the Structure Height
Next, the evolution of the amplitude of ripple and dot structures, with ion fluence for Si
and Ge will be addressed. This will be done in terms of rms surface roughness w using Eq.
(4.1). It is known that the surface roughness depends on the length scale of the image used
to deduce w. For this reason images with scan size 2 µm × 2 µm were used for all samples.
Fig. 6.19 gives the evolution of the rms surface roughness w with ion fluence for ripples
and dots on Si. The experimental results show that for small sputtering fluences, up to
5.6 × 1017 cm-2, w seems to grow exponentially (dotted line). For Φ ~ 1 × 1018 cm-2, the
1
+
Ar αion = 15°, Eion = 1200 eV
+
Kr αion = 15°, Eion = 1200 eV
+
Xe αion = 15°, Eion = 1200 eV
exp. growth
0.1
0
4
8
12
18
b)
rms roughness w [nm]
rms roughness w [nm]
a)
1
+
Ar , αion = 75°, Eion = 500 eV
+
Kr , αion = 75°, Eion = 1000 eV
+
Xe , αion = 75°, Eion = 1000 eV
exp. growth
0.1
40
-2
ion fluence Φ [10 cm ]
0
4
8
12
18
-2
ion fluence Φ [10 cm ]
Figure 6.19: Rms surface roughness evolution with ion fluence in Si for different ion species. The dotted line
illustrates the exponential grow for the initial stage of sputtering. a) ripples, b) dots.
65
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
roughness, i. e. the ripple and dot amplitude, saturates and remains constant upon further
sputtering. The results in Fig. 6.19 prove also that the evolution of the amplitude with ion
fluence behaves similar for Ar+, Kr+, and Xe+ ions. The results show a similar behavior of
w with Φ for ripples and dots.
The evolution of w with Φ is investigated also for Ge. A representative example is given
in Fig. 6.20 using the same sputtering conditions like in Fig. 6.18. In this case the surface
roughness saturates at an ion fluence of Φ = 8.4 × 1016 cm-2, corresponding to an erosion
time of only 45 s.
The results for w (i. e. structure amplitude) with ion fluence for Si and Ge can be
summarized as following: (i) For small ion fluences the surface roughness seems to have
an exponential increase, (ii) with increasing ion fluence the roughness saturates and
remains constant even for prolonged sputtering (up to Φ = 4 × 1019 ions / cm-2).
Another point to be addressed in the context of ion fluence is the evolution of the
surface topography for conditions where ripples and dots evolve simultaneously (see also
discussion in Chapter 7) on Si and Ge. This is done in order to observe how stable are
these regions, and if one type of structures is dominating the other or they simply coexist.
Figure 6.21 gives an example of surface evolution on Si during Xe+ ion beam erosion at
Eion = 2000 eV, αion = 34 deg, with no sample rotation and for different ion fluences. The
AFM image in Fig. 6.21(a) shows a simultaneous formation of ripples and dots for low ion
fluences at Φ = 3.4 × 1017 ions / cm-2 (sputter time 180 s). The characteristic ring
representative for dots and additionally the peaks for ripples are present on the 120 min by
an ion flux of jion = 300 µA cm-2). The only difference is that the alignment of ripples
+
rms roughness w [nm]
Xe , αion = 20°, Eion = 2000 eV
exp. growth
1
0
4
8
12
18
-2
ion fluence Φ [10 cm ]
Figure 6.20: Evolution of rms surface roughness w as function of Φ for Xe+ ion beam erosion of Ge surfaces.
The results are plotted for dots with Eion = 2000 eV, at αion = 20 deg and without sample rotation.
66
6.2: Ion Fluence and Flux
3 nm
FFT
d)
FFT
e)
FFT
f)
0 nm
500 nm
a) Φ = 3.4 x 1017 ions/cm2
4 nm
0 nm
500 nm
b) Φ = 2.2 x 1018 ions/cm2
4 nm
0 nm
500 nm
f = - 128 µm-1 … 128 µm-1
c) Φ = 1.3 x 1019 ions/cm2
Figure 6.21: (a-c) Surface topography on Si after Xe+ ion beam sputtering without sample rotation for
different ion fluences at Eion = 2000 eV and αion = 34 deg. (d-f) Corresponding FFT images. The solid circle
visualizes the equal radius of the rings in the FFT spectra.
increases, which influences also the ordering of dots that form along the ripples. The FFT
(Fig. 6.21(f)) reveals the quadratic ordering of dots, but still a preferred orientation is
observed along the wave vector of ripples.
Ion Flux
Here the influence of the ion flux on the evolution of ripples and dots is studied.
Experimental results for the evolution of the wavelength and surface roughness with ion
flux J are summarized in Fig. 6.22. The results are presented for ripples on Si and dots on
Ge, sputtered with 2000 eV Xe+ ions at αion = 20° and a total fluence Φ = 6.7 × 1018 cm-2.
67
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
/
/
Si/Ge
Si/Ge
4
wavelength λ [nm]
60
3
40
2
20
0
0.0
1
0.5
1.0
1.5
15
rms roughness w [nm]
5
80
0
2.0
2
ion flux J [10 cm /s]
Figure 6.22: Structure wavelength and rms surface roughness with ion flux for ripples on Si and dots on Ge
surfaces after Xe+ ion beam erosion with Eion = 2000 eV, αion = 20 deg.
The erosion time was varied so that the total amount of ions hitting the surface Φ remains
constant. In the ion flux range used for the experiments (jion was varied from 80 µA cm-2 up
to 300 µA cm-2), no variation of the wavelength λ is observed. Also the surface roughness
w remains constant with ion flux.
6.3 Geometrical shape
The geometrical shape of ripples and dots is studied using the cross-section profile
taken from AFM images. A better method to investigate the geometrical shape and the
surface damage on the atomic scale is the High Resolution Transmission Electron
Microscopy (HRTEM). HRTEM was performed in a 400 keV microscope possessing a
point resolution of 0.155 nm. Cross-sectional samples were prepared by gluing samples
face to face, embedding resulting sandwiches in alumina tubes, wire-saw cutting, planparallel grinding, one-sided polishing, other-sided dimpling followed by polishing to a
residual thickness of about 15 µm, and Ar+-ion beam etching at 2.8 keV.
In Fig. 6.23 a magnified image of Fig. 6.13(c) for the ripple topography together with
the cross-section profile is given. The cross-section profile, taken from the solid line of the
AFM image, gives a separation of ~ 50 nm between ripples, and a height of ~ 4 nm. Also
from the profile, the site facing the ion beam is less steep than the other side with sidewall
angles varying between 8 deg and 14 deg, respectively. This asymmetry is present for
whole ion incidence angle range where ripples evolve (from 5 deg up to 25 deg). The
amount of asymmetry, i. e. the ratio of ~ 1 : 2 between the site facing the ion beam and the
opposite site, remains the same. Fig. 6.24 shows a cross section HRTEM image of the
68
6.3: Geometrical Shape
(a)
1 µm
(b)
2.5 nm
1 µm
0.8 µm
0
0.6 µm
50
~ 14°
100
~ 8°
150
200
Length [nm]
250
300
0.4 µm
0.2 µm
Figure 6.23: AFM image of ripple nanostructures on Si after Kr+ ion beam erosion (Eion = 1200 eV, αion =
15 deg). (b) Height profile along the line drawn in the AFM image showing the asymmetric form of ripples.
The arrows indicate the ion beam direction.
same sample. The HRTEM reveals very well aligned ripples having a homogenous height
of ~ 5 nm, and a separation of 50 nm in between. Also the asymmetric form of ripples
similar to the AFM cross section profile is clearly observed. Additionally the HRTEM
image reveals an amorphous layer covering the ripple surface. To this amorphous layer
contributes also an oxide layer forming due to the delay between sputtering and HRTEM
measurements. The thickness of the amorphous layer is ~ 6 nm and it depends on the ion
energy, ion incidence angle, and the ion species used. The surface amorphization is
expected because the total ion fluence used to sputter the surface is some orders of
magnitude higher than the amorphisation threshold. As reported by Gnaser for 2 keV Xe+
ions on Si, the threshold value is Φ ~ 1015 ions/cm2 [40] (details are discussed in Section
2.2).
Beneath the amorphous layer the surface is single crystalline with the corresponding
FFT
FFT
50 nm
Figure 6.24: Cross-sectional HRTEM image of ripple patterns. The arrows give the direction of the ion
beam. Inset: The amorphous covering layer has a thickness of 6 nm. Also given are the FFT images
calculated from the amorphous respectively crystalline part of ripples.
69
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
(b)
Height [nm]
200 nm
(a)
10 nm
0
50
100
150
Length [nm]
200
250
150 nm
100 nm
50 nm
Figure 6.25: A magnified image of dot nanostructures on Si after Kr+ ion beam erosion with Eion = 1000 eV
and αion = 75 deg. (b) Height profile along the line drawn in the AFM image showing the sinusoidal-like form
of dots. The AFM image and the cross-section profile graph have an one to one aspect ratio.
lattice parameter for Si (0.543 nm). The FFT images deduced from the inset (Fig. 6.24)
clearly show that the upper layer is amorphous, while the layer beneath reveals a
crystalline structure. Further, the HRTEM image indicates a strong correlation on the shape
of ripples between the amorphous layer a-Si and the crystalline interface c-Si.
Fig. 6.25 shows a magnified AFM image of dot nanostructures evolving on Si with
similar conditions like in Fig. 6.16(c). The cross section (solid line) displays the sinusoidal
like form of dots. The dots have an average height of up to 10 nm and a 40 nm separation
in between. The sinusoidal form of dots can be supplementary verified from the HRTEM
cross-section measurements (Fig. 6.26). The HRTEM analysis gives a separation between
dots ~ 37 nm and a dot height of ~ 9 nm. Similar to ripples the dot surface is covered with
a thin amorphous plus oxide layer. The thickness of the amorphous layer it varies from
2 nm on the valleys up to 3.5 nm on the top of dots. This depends on the variations of the
local surface angle.
HRTEM investigations on Ge structures were also performed. In Fig. 6.27 an example
for the dot structures evolving at 20 deg ion incidence (see Section 7.1.2) is given.
3.5 nm
2 nm
Figure 6.26: Cross-sectional HRTEM image of dot structures with a height of ~ 9 nm and a separation of ~
37 nm. Inset: The anisotropic distribution of the amorphous layer is highlighted.
70
6.3: Geometrical Shape
Figure 6.27: A cross-sectional HRTEM image showing the crystalline structure of a dot on Ge covered with
an amorphous layer.
Similar to Si the surface is covered with ~ 6 nm amorphous layer, and the dots have a
height of ~ 4 nm.
Additionally, the HRTEM image of ripples (shown in Fig. 6.9) reveals a homogeneous
structure with ~ 5nm height and an amorphous layer of ~ 8 nm (Fig. 6.28). The distance
between ripples is ~ 60 nm. The asymmetry of ripples is less pronounced than in the case
of Si.
Figure 6.28: Cross-sectional view of ripple structures on Ge samples. The surface is covered with an
amorphous layer.
71
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
6.4 GISAXS and GID
The main aim of GISAXS and GID investigations was to study the periodicity, ordering
and lateral correlation of nanostructures. Furthermore, the investigation of nanostructures
with GISAXS and GID provides a much better statistics compared to AFM. This because
the scattered intensity is an average over the beam spot illuminating the surface that can be
up to some millimeters in size (spot size in the experiments was 200 µm × 100 µm in width
and few millimeters along the beam spot on the sample surface). As given in Section 4.2.3,
GISAXS gives information about the size and correlation of nanostructures on the surface,
and is not sensitive to the crystalline structure. As it is known from the HRTEM
investigations, the nanostructures are composed of an amorphous layer covering the
crystalline part of nanostructures. Therefore, GID is used to deduce information about the
crystalline part of nanostructures by setting up the in plane angle to fulfill the Bragg
condition. While only the horizontal ordering of nanostructures is of interest the qy
component of the scattering vector is used for GISAXS plots. The qz component gives no
contribution for αi = αf = constant [111]. For discussing the experimental results, and
especially the lateral correlation length of nanostructures from the scattered intensity
spectra, certain theoretical models should be applied to the experimental data. Usually, in
small angle X-ray scattering the scattered intensity is given as a product (in the reciprocal
space) of the square of a form factor F (q) for a given object, and the correlation function
C (q) that describes the position of structures (i. e. the lateral ordering) [127,128].
I (q) = F (q) 2 C (q )
(6.2)
For the fitting procedure the form factor of a cone is used (the expression is given in
Appendix A2). The shape of the cone can be influenced by varying the tilt angle and by
choosing a reasonable radius value comparable to the radius of structures obtained from
the HRTEM images. The ordering of ripples is described using the one-dimensional linear
paracrystal model (Eq. (A2.2)). While for dots, the hexagonal paracrystal model is used to
account also for the hexagonal arrangement of dots and for peaks having different
distances between them (Eq. (A2.4)) [127].
Theoretically, the ordering of nanostructures is generally discussed in the frame of two
models: a) The long range order (LRO) model that assumes perfect arrangement of
nanostructures inside a given domain independent of distance; b) The short range order
model (SRO) assuming a decreasing order with distance [111,116,129]. According to
Schmidbauer [111] by considering the auto-correlation function in real space the LRO
model gives equidistant intensity peaks with a constant width. For the SRO model, the
peaks are equidistant but the width increases in contrast to the intensity of peaks, which
decreases, with increasing number of peaks. In the reciprocal space the intensity profiles
72
6.4: GISAXS and GID
show equidistant peaks decreasing with increasing q for both models. However in the
case of LRO the peak width remains constant and can be expressed as:
δq =
2π
(6.3)
ξ
where ξ is the correlation length and it shows up to which length scale positional
correlation of nanostructures is present. In the SRO model the peak width decreases and is
given by the relation deduced from Stangl et al. [130]
δq =
(σ q ) 2
d
(6.4)
with σ being the standard deviation of the mean distance d between two neighboring
structures, i. e. the width of the first peak of the correlation function in real space. For
convenience with AFM studies, d
will be substituted with λ.5 In this model for the
correlation length it follows
λ3
ξ=
.
2σ 2
(6.5)
From (6.4) and (6.5) a relation between δq and ξ can be deduced
δq =
2π 2
ξ
.
(6.6)
As will be shown from the experimental scattered intensity plots below, the peak width
increases with peak number suggesting that the SRO model can be applied, i. e. the
ordering of structures disappears gradually with distance.
The samples used for investigation with GISAXS and GID were previously treated in
the ion beam equipment and characterized with the AFM. Detailed description of the
samples analyzed with AFM was already given in the previous sections. It should be
mentioned that, due to the time limit restrictions at the ESRF facility and well in advance
planning of the measurements, only few samples were investigated.
6.4.1 Ge
In Fig. 6.29 an example of the GISAXS spectra by varying the in-plane angle 2θ for
ripples in Ge is given. The sample was sputtered with Xe+ ions using Eion = 2000 eV, at αion
= 5 deg and ion fluence Φ = 6.7 × 1018 cm-2 (Fig. 6.9). The scans are performed for two
azimuthal angles ω = 0° and 180°. The intensity spectra show multiple equidistant peaks
5
The mean separation of structures deduced from the AFM studies is denoted with λ and is equivalent
with d .
73
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
10
10
-q1 +q
1
∆q
intensity [a.u.]
8
10
ω=0
ω = 180°
6
10
4
10
2
10
-0.16
-0.08
0.00
-1
qy [Å ]
0.08
0.16
Figure 6.29: GISAXS scan of a Ge ripple sample (Eion = 2000 eV) for two azimuth angles (ω = 0° and 180°).
(up to 8th order), appearing due to the high lateral ordering of ripples. The distance between
peaks ∆q = ± 0.0113 Å-1, equivalent with the position of the first order peak q1,6
corresponds to a ripple wavelength λ = 2π / ∆q = 55 nm. The slight asymmetry in the
intensity profile between the scans at 0° and 180° is due to the slight asymmetric shape of
ripples. The high intensity of the central peak is a contribution of the specular beam and
the diffuse scattering coming from the lateral uncorrelated roughness, including short
spatial frequency corrugations observed on AFM images.
A comparison of GISAXS spectra, for ripples formed under different ion energies,
reveals a decreasing distance between peaks, i. e. an increase of the ripple wavelength with
increasing Eion (Fig. 6.30). However, the spectra show a slightly better ordering of ripples
with increasing Eion. A GID angular scan of samples at the vicinity of the ( 2 20) Bragg
reflection (along the [ 1 10] crystallographic plane) displays equivalent results compared
with GISAXS (Fig. 6.30). This indicates that surface ripples correlate quite well with the
crystalline part of ripples. However, in the GID spectra the number of multiple peaks
decreases more rapidly with decreasing ion energy compared to GISAXS. This means the
crystalline part of ripples at 1200 eV is less ordered compared to the ripple surface. From a
comparison of the background intensities of GISAXS and GID spectra it seems that the
interface is more homogeneous compared to the surface (the GISAXS background
intensity has a more curved form compared to GID, for clarity see the graphs for
6
For the GISAXS and GID spectra with equidistant peaks the separation between peaks is equal with the
position of the first order peak. For this reason both notations will be used in the discussion. The situation
changes when the spectra have different peak separation like in the case of dot spectra.
74
6.4: GISAXS and GID
13
13
10
10
Eion = 1200 eV
9
10
5
5
10
10
GISAXS: intensity [a.u.]
1
1
10
9
10
10
GISAXS qy
Eion = 1500 eV
5
5
10
10
1
1
10
10
Eion = 2000 eV
9
10
5
10
9
10
5
10
1
10
9
10
GID: intensity/10 [a.u.]
9
GID: qang at (-220)
simulations
10
1
-0.12
-0.06
0.00
0.06
0.12
10
-1
q [Å ]
Figure 6.30: Comparison between GISAXS spectra and angular GID spectra of the ( 2 20) Bragg reflection
for different ion energies. Also given are the simulated curves using Eq. (6.4). The dashed lines (at Eion =
2000 eV) indicate the different background intensities of GISAXS and GID.
Eion = 2000 eV in Fig. 6.30).7
A summary of inter peak distances and their width, deduced from the GISAXS and GID
spectra is given in Table 6.1. The results show an increase of the ripple wavelength, for
both GISAXS and GID data, with ion energy similar to AFM results. However the values
deduced with small angle X-ray scattering methods are about 20 % smaller compared to
AFM. Further, the peak width δq decreases with ion energy indicating an improved
ordering of ripples with Eion. For comparison the peak width deduced from the PSD spectra
(using AFM images) are given. Taking in account the low statistics due to the scan
7
The roughness on the surface is higher than in the interface.
75
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
Table 6.1: The inter peak distances, peak width and the corresponding ripple wavelengths for GISAXS and
GID spectra for different samples are given. Also the wavelengths of ripples and the peak width deduced
from PSD spectra of AFM images are given for comparison.8
Eion
(eV)
δqy
(Å-1)
∆qy
(Å-1)
λGISAXS
(nm)
∆qang
(Å-1)
δqang
(Å-1)
λGID
(nm)
λAFM
(nm)
δqPSD
(Å-1)
1200
± 0.0127 0.00186
48
± 0.0133
0.00317
47
56
0.0011
1500
± 0.012
0.00168
52
± 0.012
0.00156
52
63
0.0010
2000
± 0.0113 0.00162
56
± 0.0115
0.00135
55
68
0.00107
size limit in the AFM data, δqPSD given in Table 6.1 are in the range of δq values from the
GISAXS and GID data.
Additional to the experimental data, the corresponding simulated curves are also plotted
in Fig. 6.30. The simulations are performed using the expressions for the correlation
function and the form factor of cone given in Appendix A2 and the assumption that the
peaks have a Gaussian distribution [111,131]. From the simulated curves the model
reproduces quite good the distance between peaks as well as the number of multiple peaks.
However, with this model the width of the first peak is underestimated by at least an order
of magnitude. Probably the short spatial frequency corrugations, the defects of ripples and
their asymmetry should be included in the model. Beside this, by analyzing the fitting
parameters listed in Table 6.2 the model seems to be a good approximation. Thus, the
fitting parameters for the form factor, the radius Rmean (base), the height h and the angle γ
of the cone (Fig. A2.2) correlate very well to the results deduced from AFM and
Table 6.2: The parameters used in the fitting procedure for GISAXS and GID data and the calculated lateral
correlation lengths using Eq. (6.7). For comparison the mean height of ripples hAFM deduced from the AFM
images is given.
Eion
(eV)
S
λGISAX
S
σ SGISAXS
λGID
S
σ SGID
(nm)
ξGISAXS
(nm)
(nm)
(nm)
ξGID
(nm)
(nm)
1200
47
3.1
5600
45
5.1
1700
15
10
2
2.5
1500
52
2.8
8600
53
3.7
5500
15
15
4
3.4
2000
56
2.8
11200
56
2.8
11200
20
15
5.3
4.2
8
Rmean
γ
h
hAFM
(nm) (deg) (nm) (nm)
Usually for data evaluation from the X-ray scattering techniques in the reciprocal space the Å-1 as a unit
is used, which is the case also here. However, for the rest of the data (including AFM data) the unit nm will
be used.
76
6.4: GISAXS and GID
HRTEM images for ripples in Ge. For comparison the mean height deduced from AFM
images is also given in Table 6.2. For the correlation function the wavelength of
nanostructures λ (chosen to coincide with the experimental values) and the size deviation σ
are used as fitting parameters. By changing the value of the size distribution parameter σ
the number of multiple peaks and their intensity vary simultaneously, i. e. they are related
to each other. The size distribution is in the range between 5 % and 7 %. With these values,
applying Eq. (6.5), the lateral correlation length ξ is calculated. ξ increases with Eion and
can reach up to ξ = 11.2 µm for 2000 eV, which seems significantly high. Another
possibility to prove the validity of the SRO model is to plot the peak widths from the
measured spectra, as a function of q. This will be done on the example of the GID data for
Eion = 2000 eV. The data are fitted with multiple Gauss profiles (Fig. 6.31(a)) to determine
the peak width. The results are plotted in Fig. 6.31(b), and show an increase of the peak
width with peak number. With a quadratic fit to the data using Eq. (6.4) a standard
deviation σ = 3.3 nm is maintained. This value is very close to σ = 2.8 nm deduced from
the simulations. In general, from the above discussion the SRO model seems a good
approximation for predicting the main features of the experimental observations although
the correlation length is quite high.
Further, a two-dimensional scan of the crystalline part of ripples in the reciprocal space
using the GID geometry is done. By performing for example ω-scan for different 2θ steps.
The scattering geometry for GID studies is presented in Fig. 6.32 with intensity peaks in
the reciprocal space. From the geometry only scans at certain directions contribute to the
spectra with distinguished intensity peaks. Fig. 6.33 shows a GID map performed for
ripples in Ge along the (220) and ( 2 20) Bragg reflections. The maps are recorded for the
sample sputtered at Eion = 1500 eV. The distance between the intensity peaks in the
reciprocal space corresponds to the ripple wavelength. By comparing the maps with the
0.020
6
a)
GID: Exp. data
fit using Eq. (6.4)
0.015
-1
4
10
b)
σ = 3.3 nm
ξ = 7.6 µm
5
10
δq [Å ]
intensity [a.u.]
10
GID: Exp. data
Gauss fit
0.010
0.005
3
10
0.000
2
10
-0.10
-0.05
0.00
0.05
-0.10
0.10
-0.05
0.00
0.05
0.10
-1
-1
qang [Å ]
qang [Å ]
Figure 6.31: a) Multiple Gaussian fit of the experimental data from Fig. 6.23 for Eion = 2000 eV. b) Peak
widths deduced from the Gaussian profiles as a function of qang and the fitted curve using Eq. (6.4).
77
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
( 2 20)
x
qang
y
qrad
(220)
( 2 2 0)
Figure 6.32: Drawing of the scattering geometry in the reciprocal space. The scattered intensity is collected
using the radial scan for the (220) and the angular scan for the ( 2 20) , respectively (22 0) Bragg reflection.
scattering geometry in Fig. 6.32, the intensity peaks appear almost parallel to the (220),
respectively perpendicular to ( 2 20) . The misalignment of peaks by 5 deg and 85 deg,
corresponds to the misalignment of ripples with respect to the crystallographic planes. This
supports the fact that the ripple formation is dictated by processes that take place in the
surface and near surface region, and not from the crystallographic orientation.
As will be discussed in Chapter 7 dot patterns showing a large area hexagonal lateral
ordering form on Ge surfaces. Fig. 6.34 shows the GISAXS and GID radial spectra for
these dots. The GISAXS data are performed for two different azimuth angles ω = 0° and
-1
°
1
30°. The first peak for all scans appears at q 0y°1 = q 30
y 1 = q rad = ± 0.0148 Å representing the
(-220)
0,04
4E3
4E3
0,02
-1
85 deg
qx [Å ]
-1
qx [Å ]
0,02
0,00
(220)
0,04
1E4
2E4
0,00
-0,02
-0,02
6E4
5E4
-0,04
-0,04
-0,04 -0,02
0,00
0,02
5 deg
-0,04
0,04
-1
qy [Å ]
-0,02
0,00 0,02
-1
qy [Å ]
0,04
Figure 6.33: Reciprocal space maps recorded for ripples in Ge (Eion = 1500 eV) using the GID geometry at
(220) respectively ( 2 20) Bragg reflection. The lines are to visualize the misalignment of ripples with respect
to the crystallographic plane.
78
6.4: GISAXS and GID
mean separation of dots λ = 43 nm which is close to λ = 46 nm obtained from the PSD
spectra using AFM images. The second peak, for ω = 0° scan, appears at
q 0y°2 = ± 0.0297 Å-1 and is a multiple (double) of the first one. For ω = 30° the second peak
-1
°
30°
appears at q 30
which is characteristic for a hexagonal dot
y 2 = 2( 3 / 2 )q y 1 = ± 0.0255 Å
lattice
(see
inset
in
Fig. 6.34).
Furthermore,
the
third
peak
at
-1
°
30°
q 30
is visible. The hexagonal ordering of dots is observed
y 2 = 3( 3 / 2 )q y 1 = ± 0.0386 Å
qy at ω = 0°
simulated data
11
10
9
10
q−0°1 y
7
10
q10°y
q20°y
q−0°2 y
5
q30°y
10
qy at ω = 30°
simulated data
10
intensity [a.u.]
10
ω = 30°
8
10
6
10
q130y°
q1 y
4
10
qrad at (220)
simulated data
7
5
10
10
q330y°
30°
q−302°y q−1 y
10
3
q230y°
−3
rad
q
−2
qrad
−1
qrad
-0.05
1
qrad
0.00
2
qrad
3
qrad
0.05
-1
q [Å ]
Figure 6.34: Experimental and simulated data for dots on Ge (Eion = 2000 eV, αion = 20 deg) using GISAXS
at azimuth angles ω = 0° and 30° and GID at (220) Bragg reflection. Inset: schematic drawing of a
reciprocal lattice point showing the hexagonal ordering.
79
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
Table 6.3: Fitting parameters used for the simulations and the calculated ξS. For comparison the mean
height hAFM deduced from the AFM images is given.
GISAXS and
GID spectra
Eion
(eV)
λS
(nm)
σS
(nm)
ξS
(nm)
Rmean
(nm)
γ
(deg)
h
(nm)
qy at ω = 0°
2000
41
4.2
3829
17
16
4.8
qy at ω = 30°
qrad at (220)
2000
48
2.2
10450
17
20
5.4
hAFM
(nm)
4.5
also for the crystalline part of dots by performing a GID scan at (220) Bragg reflection
(Fig. 6.34).
The experimental curves are simulated using Eq. A2.4 for the correlation function and
the cone form factor (Eq. A2.3). The parameters used for the fitting procedure are given in
Table 6.3. The simulated curve for ω = 0° fits well to the experimental data by predicting
the peak position, width and the number of peaks. For ω =30° and GID scans only the first
and second peak are reproduced but not the third peak. Additionally, the width of the first
peak is underestimated. From the fitting parameters a lateral correlation of dots up to
10 µm is deduced comparable to that of ripples (Table 6.3). A large difference between ξ
values for ω = 0° and ω = 30° is found. Probably a better model should be applied that can
predict the large area hexagonal ordering of dots.
6.4.2 Si
In Fig. 6.35(a,b) a GISAXS map of ripples on Si for two different ion energies is
presented. The samples are sputtered using Ar+ ions at αion = 15 deg and Φ =
20
a)
b)
16
2E5
ω [deg]
ω [deg]
4E3
24
12
8
1E7
2E5
16
2E7
8
4
0
32
4E3
-0,06
-0,03
0,00
0,03
-1
qy [Å ]
0
0,06
-0,06
-0,03
0,00
0,03
0,06
qy [Å]
Figure 6.35: A GISAXS map of ripple samples in logarithmic scale sputtered for different ion energies a)
1200 eV, b) 2000 eV. The samples are scanned for different azimuth angles ω. The peaks have equidistant
spacing with a) ∆q = ± 0.0141 Å and b) ∆q = ± 0.0101 Å. The central peak (white line) in the map is due to
the specular beam.
80
6.4: GISAXS and GID
Table 6.4: The distance between peaks ∆q and the peak width δq deduced from GISAXS and GID scans are
given together with the calculated wavelength of ripples. For comparison also the values deduced from the
PSD spectra of the AFM images are given.
Eion
(eV)
δqGID
(Å-1)
∆qGID
(Å-1)
λGID
(nm)
∆qy
(Å-1)
δqy
(Å-1)
λGISAXS
(nm)
λAFM
(nm)
δqPSD
(Å-1)
1200
± 0.0142 0.00149
44
± 0.0141
0.00186
45
47
0.00161
2000
± 0.0103 0.00146
60
± 0.0101
0.00165
62
64
0.00167
6.7 × 1018 cm-2. The scans are recorded for different azimuth angles ω. The intensity lines
along qy are clearly visible. The number of multiple lines proves the high lateral ordering
(alignment) of ripples. The maps indicate an improved ordering of ripples at Eion =2000 eV
compared to Eion =1200 eV, contrary to the results in Fig. 6.3. Additionally, there is an
asymmetry in the intensity distribution and the number of peaks. This confirms the
asymmetric form of ripples similar to the AFM line profiles and HRTEM images. The
distance between intensity lines is equal to the ripple wavelength. A line profile for a
particular ω value (GISAXS in Fig. 6.36) gives a ripple wavelength of λ = 45 nm and λ =
62 nm, respectively. This corresponds quite good with the wavelength deduced from the
PSD spectra of AFM images (see Table 6.4).
From the map also the angular distribution of ripples can be deduced by taking a line
profile for a given qy value, and determine the FWHM of the peak (Fig. 6.36). This is done
by making a Gaussian fit to the experimental data. The angular distribution of ripples is
1,5
+
Ar : Eion = 1200 eV, FWHM = 13 deg
+
Ar : Eion = 2000 eV, FWHM = 19 deg
normalized intensity
1,2
+
Kr : Eion = 1200 eV, FWHM = 8 deg
Gaussian fit
0,9
0,6
0,3
0,0
0
8
16
24
32
40
ω [deg]
Figure 6.36: Angular distribution of ripples for samples using a) and b) Ar+, and c) Kr+ ions. The data are
taken from GISAXS maps for a given qy for different azimuth angles ω. The solid lines are a Gaussian fit of
the data.
81
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
13 deg for Eion = 1200 eV and 19 deg for Eion = 2000 eV. In Fig. 6.36 the angular
distribution, which is about 8 deg, for the sample presented in Fig. 6.13(c), sputtered for
larger fluences is also presented (Φ = 1.3 × 1019 cm-2, Eion = 1200 eV, αion = 15 deg, with
Kr+ ions). A comparison of results indicates that the angular distribution of ripples
decreases, i. e. their alignment increases, with ion fluence.
Further, the crystalline part of ripples is studied for different crystallographic planes.
Examples of the measured spectra are given in Fig. 6.37 for (220) , ( 2 20) , and (22 0)
Bragg reflections. The scattering geometry for GID studies is presented in Fig. 6.32.
Results in Fig. 6.37 reveal no difference between the GID spectra measured at different
12
10
10
10
8
10
Eion = 1200 eV
exp. data
simulations
GISAXS
6
10
4
10
2
intensity [a.u]
10
qrad at (220)
qang at (-220)
qang at (2-20)
0
10
11
10
Eion = 2000 eV
exp. data
simulations
9
10
GISAXS
7
10
5
10
3
10
1
10
qrad at (220)
qang at (-220)
qang at (2-20)
-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15
q [Å]
Figure 6.37: GID scans performed on different Bragg reflection planes for samples sputtered at different ion
energies. Also shown are GISAXS spectra deduced from Fig. 6.27 for comparison. The simulated curves are
plotted for two characteristic examples.
82
6.4: GISAXS and GID
Table 6.5: Fitting parameters used for simulating the GID data near the ( 2 20) Bragg reflection.
Eion
λS
σS
ξS
Rmean
γ
h
hAFM
(eV)
(nm)
(nm)
(nm)
(nm)
(deg)
(nm)
(nm)
1200
45
2.7
6250
20
13
3.5
2.5
2000
61
3.5
9264
22
15
5.3
5
reflection planes. In all measurements the asymmetry in the intensity distribution is visible.
The mean peak separation ∆q (for Eion = 1200 eV: ∆q = ± 0.0142 Å; Eion = 2000 eV: ∆q =
± 0.0103 Å) and the peak width coincide quite good with the GISAXS and AFM data
(Table 6.4). Using the same explanation like for Ge, the SRO model can be applied to
evaluate the experimental data. The simulations are performed on behalf of two examples
by using the experimental data of the ( 2 20) Bragg reflection for Eion = 1200 eV and
2000 eV, respectively. The fitting parameters used for the simulations are summarized in
Table 6.5. From Eq. (6.5) a lateral correlation of 6250 nm and 9264 nm is deduced,
respectively. Also the form factor fitting parameters have reasonable values compared to
the AFM height data. However the model can not predict the asymmetry of the
experimental data and the width of the first peak is narrower than the experimental one.
A question that usually rises is, if there is strain involved in the process of structure
formation? This can be addressed by analyzing the intensity contribution in GID geometry
by performing a radial scan qrad. The spectra along the (220) Bragg reflection for
9
10
0°
180°
GID radial at (220)
7
Intensity [a.u.]
10
5
10
3
10
1
10
-0.2
-0.1
0.0
0.1
0.2
qrad [Å]
Figure 6.38: Radial GID scan of ripples in Si (Eion = 1200 eV) along the (220) Bragg reflection. The scans
are performed for 0 deg and 180 deg.
83
Chapter 6: Ripple and Dot Patterns on Si and Ge surfaces
ripples on Si (sputtered at Eion = 1200 eV) are plotted in Fig. 6.38. The spectrum at 0 deg
indicates an asymmetry in the intensity peaks. If there would have been strain involved,
than by rotating the sample at 180 deg and performing again the same scan one should
receive the identical spectrum like for 0 deg. Obviously this is not the case. From this short
discussion it seems that there is no strain involved in the process of structure formation
presented in this work.9
In Fig. 6.39 the GISAXS and GID spectra for dots in Si formed at αion = 75 deg are
plotted. The spectra reveal similar results for GISAXS and GID studies, i.e. the same
ordering is observed for the amorphous and the crystalline part of dots. The GISAXS
11
10
9
10
7
intensity [a.u.]
10
Eion = 1000 eV
qy ω = 0 deg
qy ω = 18 deg
simulation
-q1
-q2
qrad at (400)
qang at (400)
+q1
+q2
5
10
12
10
10
10
Eion = 2000 eV
qy ω = 0 deg
qy ω = 50 deg
simulations
8
10
-q2
qrad at (400)
qang at (400)
-q1
+q1
6
+q2
10
4
10
2
10
-0,06 -0,04 -0,02
0,00
0,02
0,04
0,06
-1
q [Å ]
Figure 6.39: GISAXS and GID scans for Si samples sputtered for two different ion energies. GISAXS scans
performed for different azimuth angles ω. GID radial and angular scans are performed along the (400)
Bragg reflection. Also the simulated intensity curves for the GISAXS data are plotted.
9
However, for a detailed investigation more experiments are needed. The best possible way would be to
perform in-situ measurements.
84
6.4: GISAXS and GID
Table 6.6: Parameters about the mean distance between dots deduced from the experimental data using Xray scattering techniques and AFM are given. Also given are the parameters used for simulations.
Eion
(eV)
q1
(Å-1)
λ
(nm)
λAFM
(nm)
λS
(nm)
σS
(nm)
ξS
(nm)
Rmean
(nm)
γ
(deg)
1000
± 0.0178
35
35
34
4.6
928
15
25
7
8
2000
± 0.0155
41
45
40
4.1
1903
17
35
11
10
h
hAFM
(nm) (nm)
scans, performed for different azimuth angles, deliver the same intensity profile indicating
the isotropic distribution of dots on the surface due to sample rotation during sputtering.
Also in the GID studies the intensity peaks are observed for both the radial and the angular
scan performed along the (400) Bragg reflection.10 The first side maximum appears at q1 =
qy = qrad = qang = ± 0.0178 Å for Eion = 1000 eV and at q1 = qy = qrad = qang = ± 0.0155 Å
for Eion = 2000 eV. The second peak position (q2 = ± 0.0288 Å for Eion = 1000 eV and q2 =
± 0.029 Å for Eion = 2000 eV) is between
2q1 representing a square lattice and
3q1 by
assuming a hexagonal lattice. This observations correlate with the AFM images where dots
are arranged in domains showing square respectively hexagonal ordering. The fitting
model used for dots on Si is applied on behalf of two examples on GISAXS data. The
model predicts the first peak position and the peak width. However, there are difficulties in
reproducing the second intensity peak. This is due to the overlap of the intensity
contributions coming from square respectively hexagonal domains of dots. The fitting
parameters together with the experimental values are summarized in Table 6.6. For
comparison also the wavelength and height from the AFM method are given. With this
model a lateral correlation length of dots up to 1.9 µm is obtained.
10
For example, from Fig. 6.32, in the case of ripples a qang scan along the (220) Bragg reflection will not
provide intensity peaks. While for dots the intensity peaks have an isotropic distribution.
85
Chapter 7
Pattern Transitions on Si and Ge Surfaces
Experimental results presented in Chapter 5 and Chapter 6 proved out the influence of
different ion beam parameters on the evolution of the surface topography on Si and Ge.
Particular attention was paid to the conditions were ripple and dot structures evolve. One
of the main demands on these structures is to maintain a large scale lateral ordering. As
will be shown in the next sections there are two parameters of the sputtering process that
contribute significantly to the achievement of such ordering: These are the ion incidence
angle and the secondary ion beam parameters. In Chapter 5, a general discussion of the
role of ion incidence angle on the surface topography is given. However small step
variations of αion show a completely new phenomenon present on the surface. Namely,
transition from ripples to dots. Additionally, the evolving dots have a large scale ordering.
This will be discussed in Section 7.1. Section 7.2 will deal with the role of beam
divergence and the angular distribution of ions within the ion beam on the surface
evolution. This quantity, neglected up to now in the studies for nanostructuring with ion
beam, plays a crucial role in surface evolution processes.
7.1 Role of Ion Incidence Angle
It is well known that the distribution of the deposited energy of ions hitting the surface
depends also on the incidence angle with respect to the surface normal. Consequently, the
amount of the material eroded, i. e. the sputter yield, depends on the ion incidence angle
(see Chapter 2). Therefore, it is expected that αion affects the surface topography. On the
other side, αion is important in order to compare the experimental results with the
continuum theory, describing the process of ripple formation. The results in Section 5.1
revealed that depending on αion, topographies like dots, ripples, and even smooth surfaces
are possible. The amplitude development of ripples and dots on αion was divided mainly in
three regions without discussing in detail the structures forming in these regions.
Indeed, the experimental results that will be presented below give a much more complex
picture of the surface topography with varying αion. Completely new phenomena are
observed, like transitions from ripples to dots. During this transition, there are αion values
where the evolving structures are almost perfectly laterally ordered on a large scale,
covering the entire sample. In the last part of this section the evolution of the wavelength
of ripples and dots with αion for both materials will be discussed. In this context, a
comparison with theoretical values will be performed. There will be a separate presentation
of results for Si and Ge to avoid confusion. The experiments were performed for the case
87
Chapter 7: Pattern Transitions on Si and Ge surfaces
with no sample rotation. As pointed out from the topography diagrams in Section 6.1 a
diversity of structures can evolve for different ion energies. Therefore in this section the
ion energy will be kept constant at Eion = 2000 eV (due to the higher amplitude the
structures posses at 2000 eV).
7.1.1 Influence of Ion Incidence Angle on Pattern Transition on Si
During the experiments, the Si surface is bombarded with Ar+, Kr+, Xe+ ions having an
ion energy Eion = 2000 eV, and ion fluence Φ = 6.7 × 1018 cm-2 (corresponding to 60 min
sputter time by an ion current density of jion ~ 300 µA cm-2). A summary of topographies
evolving for different αion, is given in Fig. 7.1.
2 nm
αion
0 nm
(100) Si
500 nm
a) αion = 0°
10 nm
7 nm
0 nm
0 nm
500 nm
500 nm
c) αion = 23°
b) αion = 5°
2 nm
250 nm
0 nm
0 nm
1500 nm
500 nm
e) αion = 75°
d) αion = 45°
Figure 7.1: Surface topography on Si after Xe+ ion beam erosion without sample rotation, for different ion
incidence angles (Eion = 2000 eV, Φ = 6.7 × 1018 cm-2). The arrows indicate the ion beam direction.
88
7.1: Role of Ion Incidence Angle
At αion = 0 deg the surface is rather smooth with small hillocks.1 By changing the ion
incidence angle to 5 deg with respect to the surface normal the topography changes
completely (Fig. 7.1(b)), i. e. well ordered ripple patterns evolve on the surface. The wave
vector of ripples is parallel to the ion beam projection. By further increasing the ion
incidence angle this topography remains stable up to αion = 23 deg where a mixture of dots
and ripples is observed on the surface (Fig. 7.1(c)). This mixed topography is preserved
until at αion = 45 deg the surface smoothens (Fig. 7.1(d)). At grazing incidence of 75 deg,
columnar structures emerge on the surface. These structures form in the direction of the ion
beam, i. e. 90 deg rotated compared to ripples at near normal ion incidence, and the mean
size increases with ion fluence. However, these structures can not be identified as ripples.2
Below, the surface structures emerging around αion = 23 deg, will be analyzed in detail.
As Fig. 7.1(c) shows, at αion = 23 deg dot and ripple structures form simultaneously on the
surface. Ripples have a slightly curved form (compared to ripples at 5 deg) and are
interrupted by dots. The dots itself form mainly along the ripples, i. e. the alignment of dots
is dictated by the previous alignment of ripples. By increasing the ion incidence angle to
25 deg this coexistence of patterns is retained (Fig. 7.2). However a close look at the AFM
image reveals that ripples with different spatial orientations form on the surface. From the
AFM image three type of ripples can be distinguished. The first type are aligned
perpendicular to the ion beam projection. The other type of ripples make an angle different
8 nm
FFT
0 nm
500 nm
f = - 128 µm-1 … 128 µm-1
αion = 25°, Eion = 2000 eV
Figure 7.2: AFM image of coexisting ripple and dot structures on Si with ripples having different
orientations (Xe+, Φ = 6.7 × 1018 cm-2) (the black arrow indicates the beam projection). Also given is the
corresponding FFT image (white arrows point out the two distinct wave vectors).
1
In the case when Ar+ ions are used to bombard the surface under normal incidence (not shown here), dot
structures appear similar to those reported by Gago et al. [30,31].
2
Detailed experimental studies show that, for low ion fluences ripples with the wave vector oriented
parallel to the ion beam projection evolve on the surface. Additional increase of ion fluence leads to gradual
formation of the columnar structures along the beam. Similar results have already been reported [132].
89
Chapter 7: Pattern Transitions on Si and Ge surfaces
to the ion beam projection. The third type of ripples show a curved form. This is reflected
also on the corresponding FFT image showing peaks with two distinct orientations.
Additionally, the broad angular distribution of the first order spots in the FFT, that posses a
half circle form, is due to the contribution of curved ripples observed by AFM. By a further
increase of αion to 26 deg, the AFM image shows structures aligned mainly in two
directions that cross each other making an angle of 90 deg between them (in the one
direction ripples are still observed) (Fig. 7.3). It is interesting that both dominating
directions are rotated with respect to the ion beam projection. i. e. they are not
perpendicular or parallel to the ion beam projection. Along these directions dots having a
chain like form dominate the surface. The dots show an almost perfect lateral ordering,
with only few defects appearing at the same positions where previously ripple defects
existed (see Section 6.2). This is reflected on the FFT image (Fig. 7.3). The same distance
of the first order spots from the image center implies the same periodicity of dots in the
two directions. Further, the FFT indicates clearly the two wave vectors dominating the
surface with the same size, but oriented perpendicular to each other.
7.1.2 Influence of Ion Incidence Angle on Pattern Transition on Ge
The topography transition from ripples to dots is not only characteristic of Si but is also
observed on Ge. However, for increasing ion incidence angle, in Ge first there is a
transition from dots to ripples and then back from ripples to dots. Figure 7.4 shows the
surface evolution on Ge after Xe+ ion beam sputtering for different ion incidence angles
(Eion = 2000 eV, Φ = 6.7 × 1018 cm-2). The AFM image for normal incidence shows dot
structures evolving on the surface (Fig. 7.4(a)). The ring on the FFT image (in the inset),
FFT
500 nm
f = - 128 µm-1 … 128 µm-1
αion = 26°, Eion = 2000 eV
Figure 7.3: An almost perfect square array of dots dominating the Si surface (Xe+, Φ = 6.7 × 1018 cm-2), the
black arrow indicates the ion beam direction. The corresponding FFT image reveals the square ordering of
dots (white arrows point out the two distinct wave vectors).
90
7.1: Role of Ion Incidence Angle
indicates the presence of dots with uniform size but with a rather poor lateral ordering. At
αion = 5 deg dot nanostructures disappear and well ordered ripples evolve on the surface
(Fig. 7.4(b)). The wave vector of ripples is aligned parallel to the ion beam projection. By
increasing the ion incidence angle to 10 deg ripples are still dominating the surface, but
they start to transform back into dots (Fig. 7.4(c)). Also ripples have a curved form,
reflected also in the FFT image (see inset) where the peaks posses a larger angular width.
Additionally, the FFT image shows a ring representative of dots on the surface. Further
increase of αion toward 20 deg (Fig. 7.4(d)) results in a complete transition of patterns from
ripples into dots. The dots have a hexagonal ordering covering the whole image area (in the
8 nm
12 nm
0 nm
0 nm
300 nm
300 nm
a) αion = 0°
b) αion = 5°
10 nm
7 nm
0 nm
0 nm
300 nm
300 nm
c) αion = 10°
300 nm
d) αion = 20°
2 nm
400 nm
0 nm
0 nm
1000 nm
f) αion = 75°
e) αion = 45°
Figure 7.4: Surface topographies on Ge after Xe+ ion beam erosion for different in incidence angles (Eion =
2000 eV, Φ = 6.7 × 1018 cm-2). The arrows give the ion beam direction. Inset: Corresponding FFT images
calculated from AFM images having 4 µm × 4 µm size.
91
Chapter 7: Pattern Transitions on Si and Ge surfaces
FFT image six equidistant peaks are observed). Furthermore, the FFT image reveals that
ordering is more pronounced in the direction that previously ripples existed as seen from
the second order peaks observed in the direction of the incoming beam. At 45 deg the
surface remains smooth (Fig. 7.4(e)). Until at grazing incidence (αion = 75 deg) the
columnar structures form along the ion beam projection (Fig. 7.4(f)), with the same
characteristics like in Si.
A summary of surface topographies evolving for different ion incidence angles on Si
and Ge surfaces is given in Table 7.1. This Table is valid for the sputtering conditions
given in this section. By varying process parameters also the range of αion for a given
topography will change.
7.1.3 Discussion
In Section 5.1 the amplitude of structures in terms of the rms surface roughness with ion
incidence angle was discussed. Additionally, the wavelength of ripple and dot structures
with ion incidence angle can be analyzed, using the sputtering conditions presented in the
previous two sections. Experimental results concerning the wavelength of ripple and dot
nanostructures with ion incidence angle are summarized in Fig. 7.5 for Si and Ge samples.
From the plots, the wavelength decreases with αion for both Si (Fig. 7.5(a)) and Ge
(Fig. 7.5(b)). Results for different ions (Ar+, Kr+, and Xe+ for Si and Kr+ and Xe+ ions for
Ge) indicate that the wavelength evolution is independent of the ion species used. In
Fig. 7.5 the theoretical values of λ = λ(αion) are also plotted. These values are calculated by
considering ion-induced ESD and IVF as relaxation mechanisms, described in Section 3.1.
The calculated λESD values using Eq. (3.14) are an order of magnitude smaller than the
experimental data. The use of the ESD term implies an increase of wavelength with ion
incidence angle at least for Ar+ and Kr+ ions, while for Xe+ ions a marginal decrease is
observed.3 The λIVF values are calculated using Eq. (3.18) by taking the surface energy
term γ = 1.05 J m-2 from Ref. [133] for Si and γ = 1.9 J m-2 from Ref. [21] for Ge. The
atomic density is equal to N = 5 × 1028 m-3 for both materials. For the thickness of the
Table 7.1: Summary of evolving topographies on Si and Ge surfaces with ion incidence angle.
Region
I
II
III
IV
V
Si (topography)
hillock (αion = 0 deg – 3 deg)
ripples (αion = 4 deg – 22 deg)
ripples + dots (αion = 23 deg – 39 deg)
smooth (αion = 40 deg – 60 deg)
columns (αion = 65 deg – 75 deg)
3
Ge (topography)
dots (αion = 0 deg – 3 deg)
ripples (αion = 4 deg – 12 deg)
dots (αion = 13 deg – 30 deg)
smooth (αion = 31 deg – 60 deg)
columns (αion = 65 deg – 75 deg)
The ESD coefficients in Eq. (3.12) depend on the ratio of the energy distribution parameters a, α and β.
These parameters are deduced using the SRIM code, for Eion = 2000 eV.
92
7.1: Role of Ion Incidence Angle
100
/
/
+
/
/
+
80
+
Ar /Kr /Xe : exp. data
+
+
+
IVF model: Ar /Kr /Xe
40
+
Ar
9
+
Kr
6
+
Xe
3
0
20
40
+
Kr : exp. data
+
Xe : exp. data
+
Xe : IVF model
9
6
+
Xe : ESD model
3
ESD model
0
b)
60
60
wavelength λ [nm]
wavelength λ [nm]
80
a)
0
40
ion incidence angle αion [deg]
0
10
20
30
40
ion incidence angle αion [deg]
Figure 7.5: Experimental and calculated values of λ as function αion using different ion species for Φ =
6.7 × 1018 cm-2, Eion = 2000 eV: a) Si, b) Ge.
amorphous layer a decreasing d with increasing ion incidence angle is expected. A
reasonable description of this decrease is by approximating the layer-thickness parameter
to a cosine function, i. e. d ~ d cos(αion). The λIVF values predict quite good the
experimental data and the decrease of the wavelength with αion. Moreover, there is almost
no difference between the curves plotted for different ion species. For calculations of λIVF,
the ηr is used as a fitting parameter. The λIVF values are calculated by taking the viscous
relaxation term ηr = 4 × 1024 N m-2 cm-2 for Si and ηr = 2 × 1024 N m-2 cm-2 for Ge.4 From
these values, using the expression η = ηr/jion with jion = 300 µA cm-2 a viscosity coefficient
ηSi = 2.13 × 109 N s m-2, respectively ηGe = 1.07 × 109 N s m-2 is deduced. The relaxation
terms ηr have the same order of magnitude as the values presented in Ref. [21]. But, due to
the high fluxes used in this work, the viscosity coefficient η has values that are two orders
of magnitude smaller than those reported in the same reference. The above discussion
suggests that ion induced viscous flow is more reliable to describe the evolution of the
wavelength with ion incidence angle.
In summary, the results in Section 7.1 indicate the importance of ion incidence angle on
the evolution of the surface topography. Depending on ion incidence angle a complexity of
different topographies can evolve on the surface. The most important conclusions of the
above discussion concerning ripples and dots are:
i)
The ripple-dot transition for Si and dot-ripple dot transition for Ge are caused by
small variations of the ion incidence angle.
ii)
The transition is continuous and there are conditions at which a coexistence of both
structures is possible. As shown in Section 6.2 this coexistence remains even for
4
The relaxation term ηr was used as a fitting parameter during the calculations until the λIVF reached
values comparable to the experimental one.
93
Chapter 7: Pattern Transitions on Si and Ge surfaces
large fluences, indicating that these mixed topographies are not meta-stable that
would eventually lead to the domination of one type of structures over the other.
During the transition from ripples to dots, the former serve as a guide for the lateral
ordering of dots.
Although there is a preferred orientation of the ion beam given by the ion incidence
angle structures having isotropic lateral distribution evolve on the surface, at least
on Ge. Additionally, the dominant directions of laterally square ordered dots on Si
are different from that of the ion beam projection.
Ripple structures on both materials evolve at ion incidence angles just a few
degrees off the surface normal. This is in contrast to the discussion in Ref. [134]
that implies a ripple inhibition for incidence angles up to 25 deg. Moreover, in
other studies up to now ripple formation is reported for incidence angles between
35 deg and 70 deg. Our experimental results show that exactly in this region the Si
and Ge surfaces remain smooth.
iii)
iv)
v)
It is obvious that due to different sputtering conditions used to create structures different
results are obtained by various research groups. In this context one very important
parameter not considered up to now, is the characteristics of the ion beam itself. This will
become more clear in the next section.
7.2 Role of Secondary Ion Beam Parameters on the Surface Topography
As already discussed in Section 4.2, there is a number of ion source and beam extraction
parameters that can influence the angular distribution of ions within the ion beam and the
beam divergence [81-84,86,87,92,135,136]. One of the most important parameters that
influences the angular distribution of ions within the ion beam is the acceleration voltage
Uacc applied at the second grid of the broad beam ion source (Fig. 4.2). Throughout this
work, this parameter was kept constant at Uacc = 1000 V.5 As shown in Section 4.1.2 the
increase of Uacc results in an increase of the angular distribution of ions within the beam
and the beam divergence (Fig. 4.4(d-f).6 In this section, the specific role of Uacc on the
surface topography will be discussed in detail. Beyond this, the geometrical parameters of
the ion-optical system will be kept constant to avoid additional influence on the angular
distribution and the beam divergence.7 Eventually, the plasma will be treated as given, i. e.
without discussing plasma properties like plasma density and plasma sheath boundary.
5
This value was taken due to the higher amplitude the structures show at Uacc = 1000 V. Details will be
given below.
6
The impact of Uacc on the beam divergence is valid for the particular ion-optical system.
7
For example, experimental studies show that the transition from ripples to dots, on Si and Ge, for Uacc =
1000 V will happen for larger ion incidence angles by increasing the distance between grids.
94
7.2: Role of Secondary Ion Beam Parameters on the Surface Topography
7.2.1 Secondary Ion Beam Parameters vs. Ion Incidence Angle
Examples for the impact of Uacc on Si and Ge surfaces are given in Fig. 7.6. On Si at Uacc =
200 V dots arranged in chain like form on the surface and by increasing Uacc to 1000 V the
surface is dominated by ripples (Fig. 7.6(a,b)). For Ge, the AFM images show a transition
from smooth to a ripple surface (Fig. 7.6(c,d)). However, a thorough investigation reveals
that the influence of Uacc on the topography is much more complicated, especially if one
studies the behavior of Uacc for different ion incidence angles αion. Such an example is
presented in Fig. 7.7 for Ge. The TD shows the influence of Uacc on the surface topography
for different ion incidence angles. The samples were sputtered with Xe+ ions at Eion =
2000 eV and Φ = 6.7 × 1018 cm-2. The TD presents a complex picture of the surface
topography. In addition to the influence of ion incidence angle also by varying the Uacc a
topography transition can be obtained. This transition is, however, present only at certain
range of ion incidence angles. For example, for αion = 5 deg, topography transitions from
smooth to dots and from dots to ripples are observed with increasing Uacc. Further, for low
Uacc values and independent of αion, the surface remains smooth. Whereas with increasing
αion this parameter region increases to larger Uacc until the surface topography is not
influenced by Uacc.
The boundaries (doted lines) on the TD are guides to the eye used to distinguish
between different topography regions similar to the explanation in Section 6.1.
3 nm
7 nm
0 nm
0 nm
500 nm
500 nm
b) αion = 20°, Uacc = 1000 eV
a) αion = 20°, Uacc = 200 eV
500 nm
2 nm
10 nm
0 nm
0 nm
500 nm
c) αion = 10°, Uacc = 200 eV
d) αion = 10°, Uacc = 1000 eV
Figure 7.6: Surface topography on Si and Ge surfaces for different extraction voltages during Xe+ ion beam
erosion (Eion = 2000 eV, Φ = 6.7 × 1018 cm-2) without sample rotation: Si (a,b); Ge (c,d).
95
Chapter 7: Pattern Transitions on Si and Ge surfaces
acceleration voltage Uacc [V]
1000
s
dot
+
s
ple
rip
800
dots
parallel mode ripples
600
smooth surface
400
200
0
5
10
15
20
25
30
35
ion incidence angle αion [deg]
Figure 7.7: Topography diagram for Ge surfaces for different acceleration voltages Uacc and ion incidence
angles αion. The presented results are for Xe+ ions with Eion = 2000 eV, Φ = 6.7 × 1018 cm-2, without sample
rotation. The symbols represent the experimental data. - smooth surfaces, - parallel mode ripples +
dots, u - parallel mode ripples, and - dots.
Similar investigations were performed on Si surfaces using the same sputtering
conditions like for Ge. Here topographical transitions are also observed, however, the
boundary positions between different parameter regions vary compared to Ge. This can be
attributed to the differences in material properties. The TD in Fig. 7.8 shows that the
1000
600
hillock features
acceleration voltage Uacc [V]
ripples + dots
parallel
mode
ripples
800
dots
400
smooth
200
0
5
10
15
20
25
30
35
40
45
ion incidence angle αion [deg]
Figure 7.8: Topography diagram for Si surfaces for different Uacc and αion. The results are given for Xe+ ions
at Eion = 2000 eV, Φ = 6.7 × 1018 cm-2, without sample rotation. The symbols represent the experimental
data. - smooth surfaces, - parallel mode ripples + dots, u - parallel mode ripples, and ^ - hillock
structures.
96
7.2: Role of Secondary Ion Beam Parameters on the Surface Topography
topography is not influenced too much by the Uacc, at least for small ion incidence angles.
However, at angles between 20 deg and 40 deg a topographical transition is present, from
(ripple + dot) to a dot structure and for lower Uacc the surface smoothens.
It is important to state that this topography transitions are continuous. Thus, between the
ripple and dot parameter regions there is an intermediate region, where ripples and dots
coexist together. By crossing over from a parameter region where structures form a smooth
one the amplitude of structures, i. e. the surface roughness, decreases gradually until the
surface remains smooth.
As next, the rms surface roughness, i. e. amplitude of structures, with Uacc is studied.
The results are plotted in Fig. 7.9 for Si and Ge surfaces, at different ion incidence angles.
In general, one observes an increase of the surface roughness with increasing Uacc for both
materials. At 45 deg the surface topography is independent of the Uacc value. In the case of
Ge, for all incidence angles the surface roughness takes the same value for Uacc = 200 V.
Further, the overall surface roughness decreases with ion incidence angle.
The influence of the acceleration voltage on the dot structures forming on Si surfaces at
grazing incidence (75 deg, SR) is also investigated. An example is given in Fig. 7.10 where
AFM images showing the surface topography after Kr+ ion beam sputtering of Si surfaces
with Uacc = 200 V and 1000 V are depicted. In this case, the Uacc has an impact on the
lateral ordering of dots, i. e. with increasing Uacc the ordering of dots is improved, visible
in the AFM image. Additionally, the long wavelength modulations are suppressed with
increasing Uacc.
7.2.2 Secondary Ion Beam Parameters vs. Ion Energy
The above results are presented for a constant value of Eion. However, if both
parameters Uacc and Eion are varied simultaneously, then the overall picture is even more
complicated. Examples of the surface topography for Ge using Xe+ ions, are given in
2,5
2,5
αion = 5 deg
b) Ge
2,0
αion = 20 deg
1,5
αion = 27 deg
1,0
αion = 34 deg
0,5
0,0
rms roughness w [nm]
rms roughness w [nm]
a) Si
αion = 45 deg
200
400
600
Uacc [V]
800
1,5
αion = 20 deg
1,0
αion = 27 deg
0,5
0,0
1000
αion = 5 deg
2,0
αion = 45 deg
200
400
600
Uacc [V]
800
1000
Figure 7.9: The dependence of surface roughness on Uacc, deduced from the experimental data presented in
the topography diagrams. The results are plotted for different ion incidence angles: a) Si, b) Ge.
97
Chapter 7: Pattern Transitions on Si and Ge surfaces
30 nm
20 nm
0 nm
0 nm
500 nm
500 nm
a) Uacc = 200 eV
b) Uacc = 1000 eV
Figure 7.10: AFM images of Si surfaces after sputtering with Kr+ ions at Eion = 1000 eV, grazing incidence
of αion = 75 deg, Φ = 6.7 × 1018 cm-2 for different acceleration voltages.
Fig. 7.11. The TD are plotted for two ion incidence angles: a) 5 deg and b) 20 deg. From
the TD it can be seen that:
i)
The dependence of the surface topography on Eion is influenced by value of Uacc, or
vice versa.
ii)
There is a parameter region in which a change in orientation of ripples is observed
with increasing Eion (Fig. 7.11(a)).
iii)
In Fig. 7.11(a) with increasing Uacc at αion = 5 deg a transition from dots to ripples
is evident.
iv)
For αion = 20 deg the opposite transition is observed, namely from ripples to dots
with increasing Eion.
b) Ge, Xe+, αion = 20 deg
1000
parallel mode ripples
ripples + dots
accelaration voltage Uacc [V]
acceleration voltage Uacc [V]
a) Ge, Xe+, αion = 5 deg
800
600
400
200
500
perp
endi
dot structures
cula
r mo
de r
ippl
es
smooth surface
1000
1500
ion energy Eion [eV]
2000
1000
800
600
ripples
+ dots
dots
parallel
mode
ripples
smooth
400
200
500
1000
1500
ion energy Eion [eV]
2000
Figure 7.11: Topography diagram for structures on Ge surfaces by varying Uacc for different Eion and for two
αion. The symbols indicate the experimental data. - smooth surfaces, 1 - perpendicular mode ripples, parallel mode ripples + dots, u - parallel mode ripples, and - dots.
98
7.2: Role of Secondary Ion Beam Parameters on the Surface Topography
7.2.3 Summary
The above discussion underlines the importance of the secondary ion beam parameters
on the surface topography. By varying the beam characteristics, different topographies can
form on the surface. One important conclusion from the above results is the use of Uacc as
an additional parameter during the sputtering process for controlling the resulting surface
topography. The influence of Uacc is not only characteristic of Si and Ge surfaces, also on
III/V semiconductors an influence of Uacc was found [137].
The TD plots showed that with increasing beam divergence there is a transition from
dots to ripples. Similar transitions are observed by varying the ion incidence angle. In fact,
the Uacc influences not only the angular distribution of ions but also the effective angle of
ions arriving at the sample surface. In this way the variation of Uacc can be considered as
fine adjustment of ion incidence angle.
Furthermore, the dot-ripple transition with increasing Uacc on Ge, can be compared with
the ripple-dot transition observed with increasing Eion for Ge in Fig. 6.11. This would
correlate with the statement in Section 4.1.2, namely, that the beam divergence decreases
with increasing ion energy, i. e. the opposite of Uacc. Therefore, consideration of such
effects is important in order to explain the different topographical transitions.
The observed dependence of the structure formation on the Uacc is unique to the used
ion source and its plasma and ion beam properties. Other ion sources with other extraction
system geometries probably will produce other structure properties and dependencies.
To come to a comprehensive understanding of the impact of such secondary ion beam
properties on the structure formation more detailed experimental investigations are
necessary. However, in the theoretical models up to now, the inclusion of such parameters
is missing.
At the end, it is worth to mention that the above discussion would explain the different
results obtained from different research groups (and the difficulty to reproduce these
results) for nanostructures on Si and Ge surfaces [69,138-142].
99
Chapter 8
Comparison of Experimental Results with Theory
8.1 Bradley-Harper model and the Nonlinear Extension
A successful theory should be in position to predict the experimental findings and
explain the scaling behavior of sputtering parameters to each other. This task will be
addressed in this Chapter by comparing the experimental results with the continuum theory
presented in Chapter 3. Theoretically, the process of ripple formation is based on the
Sigmund’s linear collision cascade theory of amorphous targets. Bradley and Harper
making use of this theory showed that due to local variations in the surface curvature also
the sputter yield changes locally [61]. This curvature dependent sputtering leads to surface
roughness. On the other side due to the erosion process itself and the mobility of particles
there are different relaxation processes present on the surface.
Concerning roughness, the most important parameters are the coefficients Γx(αion) and
Γy(αion) given in Eq. (3.4) which determine not only the orientation of ripples but also their
wavelength. They depend on the local surface curvature and are completely determined by
the ion incidence angle and the distribution of the deposited energy parameters a, α and β.
Following the discussion in Chapter 3 and the Appendix A1.1 for small ion incidence
angles, with respect to the surface normal, Γx(αion) < Γy(αion). In this case the wave vector
of ripples is along the x-axis i. e. parallel to the ion beam projection. With increasing αion
there is a value for which Γx(αion) > Γy(αion), i. e. the ripples change their orientation and
evolve along the y-axis (perpendicular to the ion beam projection). In Fig. 8.1 the Γx and Γy
coefficients as a function of αion for Si and Ge using Xe+ ions at 2000 eV are plotted. For
both materials a change in orientation of ripples at ~ 45 deg is expected. Monte Carlo
simulations of Koponen et al. [54] confirm this change in orientation that it happens
between 30 deg and 60 deg. At near normal ion incidence theoretical predictions agree
with the experimental results presented in this work, namely, the formation of ripples with
the wave vector parallel to the ion beam projection. However, in the experiments no
change in orientation of ripples with increasing αion is observed. Furthermore, a close
investigation of Si and Ge surfaces at 75 deg shows that for small ion fluences, ripples with
the wave vector parallel to the ion beam projection evolve on the surface.
One prediction of the linear BH model is the exponential increase of the amplitude of
structures with ion fluence. This is contrary to the experimental results presented in
Fig. 6.18 and Fig. 6.19. To account for these observations the nonlinear terms where added
to the BH model (see Section 3.2) and the Kuramoto-Sivashinsky continuum equation was
introduced [59,60]. A detailed description of the continuum equation and nonlinear
101
Chapter 8: Comparison of Experimental Results with Theory
0,3
a = 2.85 nm
Si: α = 1.83 nm
β = 1.13 nm
Γx, Γy [a. u.]
0,0
-0,3
-0,6
a = 2.39 nm
Ge: α = 1.44 nm
β = 0.9 nm
0
20
40
60
ion incidence angle αion [deg]
Γx
Γy
80
Figure 8.1: The evolution of the coefficients Γx and Γy related to curvature dependent sputtering as a
function of αion calculated from Eq. (3.4) for Si and Ge (Xe+ ions at Eion = 2000 eV). Also given are the
energy distribution parameters determined with the SRIM code.
parameters is given by Makeev et al. [66]. It is shown that the tilt dependent sputter yield
parameters, λx and λy, of the nonlinear part of the KS equation, can be expressed by the
energy distribution parameters and αion. While λy is always negative with αion, λx can take
both positive and negative values. Numerical simulations of Park et al. [74] of Eq. (3.20)
showed that for low fluences the linear regime dominates the surface topography. In this
regime ripple structures form on the surface and the surface roughness increases. With
increasing ion fluence the nonlinear terms gain an importance and after a certain fluence
they dominate the sputtering process. This is accompanied by amplitude saturation of
ripples. Additionally, with further increasing fluence ripples disappear and either a new
type of ripples or kinetic roughening on the surface appears. This important conclusion of
the nonlinear theory is in disagreement with the experimental observations. Although the
amplitude saturation agrees with the experiments, the ripple topography remains stable for
fluences many orders larger than the amount needed for the amplitude to saturate (see
Section 6.1.2). The only difference between the evolution of ripples in the linear regime
and of the ripples in the nonlinear regime is the amplitude saturation. Furthermore, by
performing kinetic Monte Carlo simulations of Eq. (3.20) Brown et al. [132] showed that
the wave length of ripples increases with ion fluence. This is also not observed in
experimental studies.
Passing over to the ripple wavelength, given by Eq. (3.9), it is obvious that the value of
λ will depend on the type of the relaxation terms used to describe it. Below a summary of
these terms and their influence on λ will be given.
102
8.2: Surface Relaxation Mechanisms
8.2 Surface Relaxation Mechanisms
1. According to the BH model, by considering thermal diffusion as the main relaxation
mechanism, from Eq. (3.9) it follows
1/ 2
⎛ D th ⎞
⎟⎟ .
λ ~ ⎜⎜
Ja
⎠
⎝
(8.1)
From Eq. (8.1) with the diffusion coefficient Dth being independent of ion flux, the
wavelength λ is a decreasing function of the ion flux i. e. λ ~ 1/J1/2. This is not observed
from the experimental results, which show a rather independent λ of J. Concerning the ion
2m
, i. e. λ is a
energy Eion, from Eq. (8.1) and making use of Eq. (2.8) we have λ ~ 1 / Eion
decreasing function of Eion. This is also contrary to experiments. Furthermore, no timedependent relaxation of ripples is observed after the ion beam is turned off. The above
discussion indicates that thermal diffusion can be ruled out as the main relaxation
mechanism for room temperature experiments presented in this work.
2. Another relaxation mechanism is the ion induced effective surface diffusion ESD
proposed by Makeev et al. [70]. It is temperature independent and can be described
completely from the energy distribution parameters (see Section 3.1 and Appendix A1.2).
By considering the symmetric case α = β for simplicity,1 and assuming that the wave
vector of ripples lies along the x-axis the ripple wavelength is
1/ 2
λESD
⎛ 2D ⎞
= 2π ⎜⎜ xx ⎟⎟
⎝ Sx ⎠
2m
~ a ~ Eion
.
(8.2)
From (8.2) λESD is an increasing function of ion energy that agrees with the experimental
results presented in Section 6.1. The scaling parameter m has a reasonable value, although
a larger energy range is needed to better determine this coefficient. Further, the flux
independent expression of λESD coincides with the experimental results in Fig. 6.21.
However, the value of the ripple wavelength is an order of magnitude smaller than the
experimental one. Also the ESD term, for the given sputtering conditions, predicts an
increasing wavelength with ion incidence angle as shown in Fig. 7.5 (for Ar+ and Kr+
ions). While the experiments show a decrease of λ with αion.
Another point to be discussed is the value of λ for different ion species. The
experimental results give a value of λ independent of used ion species, for the same
sputtering conditions. In Table 8.1 the λESD values calculated for Ar+, Kr+, and Xe+ ions
using Eq. (8.2) are given. The parameters a, α, and β, are calculated for Eion = 1200 eV and
αion = 15 deg [48,123]. From the given values a difference up to 40 % between the
1
In the detailed discussion in Ref. [66] is concluded that there is no difference in the overall qualitative
results between the symmetric and the asymmetric case.
103
Chapter 8: Comparison of Experimental Results with Theory
Table 8.1: Calculated wavelengths λ of ripples for different ion species, and for different relaxation
processes (ESD and IVF). Parameters a, α, and β were calculated using the SRIM code. Coefficients νx and
Dx were calculated using Eq. (A1.3) and (A1.5).
Ion
species
a
(nm)
α
(nm)
β
(nm)
νx
(nm)
Dx
(nm3)
λESD
(nm)
λIVF
(nm)
Ar+
2.5
2
1.6
-0.459
0.332
7.54
44
Kr+
2.2
1.6
1.1
-0.29
0.107
5.4
46
Xe+
2.1
1.4
0.8
-0.18
0.037
4.02
49
wavelengths for Ar+ and Xe+ ions is observed, and indeed it decreases with increasing ion
mass.
The above discussion, assuming ESD as the dominant relaxation process, can be
summarize as follows:
i)
The ESD mechanism agrees with the experiments concerning the ion flux and
the ion energy.
ii)
It disagrees concerning the wavelength value, ion incidence angle and the ion
mass.
3. If ion induced viscous flow IVF is considered as the main relaxation mechanism than the
ripple wavelength can be expressed as (see Section 3.1):
1/ 2
⎛
2N
γ ⎞⎟
.
λIVF = 2πa⎜
(8.3)
⎜ Y0 max Γ x (α ion ), Γ y (α ion ) η r ⎟
⎝
⎠
with the viscosity coefficient given by η = ηr/J. Eq. (8.3) is applicable for the case when
smoothing by viscous flow is restricted to a layer of thickness d comparable with the ion
range in the solid a and the amplitude of structures, but much smaller than the wavelength
[29]. This condition seems more realistic to our experimental results. Further, as shown by
Brongersma et al. [143], the viscous relaxation rate ηr is independent of temperature in the
range from 90 K up to 300 K (the experiments presented in this work were performed at
room temperature), i. e. λIVF is independent in the given temperature range. Additionally,
λIVF is independent of the ion flux. Considering the dependence of λIVF on Eion from Eq.
[
]
3m
, i. e. it is an increasing function of ion energy.
(3.19) it follows that, λIVF ~ Eion
Furthermore, the ripple wavelength is predicted quite good using the IVF mechanism.
Also, as shown in Table 8.1, there are minor changes of λIVF by using different ion species.
Calculations of λIVF for different αion show that it decreases with ion incidence angle similar
to experimental results. Although, the decrease is not so pronounced like in the
104
8.3: Other Models
experiments. However, as discussed in Section 7.1, the λIVF values are deduced by using
the ηr as a fitting parameter.
8.3 Other Models
Another mechanism to be considered is the shadowing effect [144,145]. Mostly
discussed for thin film growth processes, it implies that certain regions on the surface
receive less ions compared to other regions due to the shadowing effects from the
neighboring peaks. While for growth processes this would lead to surface roughening for
etching processes shadowing has the role of a smoothing mechanism [146]. The effect of
shadowing is stronger at grazing incidence angles, where the shadowing of the valleys is
higher.2 This mechanism may be behind the amplitude saturation observed for dots on Si,
at grazing incidence of 75 deg. For the columnar structures (Fig. 7.1(e) and Fig. 7.4(f)))
evolving on the surface in the case without sample rotation other effects (additional to
shadowing) may be identified. First due to the anisotropy given by the direction of the ion
beam an instability is initiated on the surface. As argued by Sigmund [42] once hills form
on the surface sputtering at the top of the hills is lower compared to the hill sides and
valleys. Additionally, due to the grazing incidence angle the ions are hitting the column
walls, the particle reflection will contribute additionally to the erosion of valleys. However,
additional theoretical simulations are necessary to verify the effect of shadowing on the
formation of structures.
Considering the experimental results shown in this work, it is worth to mention that
recently an anisotropic generalized version of a damped non-local Kuramoto-Sivashinsky
equation was proposed to model ion beam erosion under oblique ion incidence [147,148].
The equation has the form
2
2
⎞ ⎛ ∂h ⎞
⎛
⎛ ∂h ⎞
∂h
∂ 2h
∂ 2h
= −⎜⎜ γh + 2 + α 2 + ∇ 4 h ⎟⎟ + ⎜ ⎟ + β ⎜⎜ ⎟⎟ + η ( x, y , t )
∂t
∂x
∂y
⎝ ∂y ⎠
⎠ ⎝ ∂x ⎠
⎝
with
α=
a1 y
a1 x
, β =
a3 y
a3 x
.
(8.4)
(8.5)
A detailed description of Eq. (8.4) is given in Ref. [147]. Here attention will be paid on
parameters α, β and γ (please note that these coefficients are different from the notations
used for the rest of the work). Here, α, β describe the anisotropy due to surface roughening
and tilt dependent sputtering, and γ is a damping parameter [77,80] (coefficients a1x and a1y
are proportional to Γx and Γy, while a3x and a3y to λx and λy). From Eq. (8.5) it results that
2
Due to the nonlocal nature of the shadowing effect only numerical solutions are possible. Numerical
simulations of Drotar et al. [146] showed indeed that shadowing for etching processes can lead to smooth
surfaces. They showed also that the surface roughness saturates with ion fluence similar to our experimental
results.
105
Chapter 8: Comparison of Experimental Results with Theory
40
30
Si
Ge
1.1
1.0
α = Γx/Γy
20
10
0.9
0.8
0
10
20
30
0
-10
0
20
40
60
80
100
ion incidence angle αion [deg]
Figure 8.2: Plotted is the parameter α (taken from the ratio of Γx and Γy coefficients from Fig. 8.1) as a
function of ion incidence angle. The inset is given for better identifying the decrease of the curve.
the parameter α depends on the ratio of the roughening parameters, and as plotted in
Fig. 8.2 (by taking the ratio of curves in Fig. 8.1) it decreases with αion. In their simulations
Vogel and Linz show that by varying the parameter α different topographies can evolve on
the surface. Explicitly, for α =1 at low fluences mounds are formed, while at high fluences
hexagonally ordered dots are observed. With decreasing α below 0.95 (this value depends
on the ion fluence used) this topography transfers into well aligned ripples. An increase of
the ion incidence angle enhances the anisotropy in curvature dependent sputtering and,
therefore, facilitates the formation of hexagonal dot patterns. These transitions are similar
to experimental results by identifying the parameter α with ion incidence angle. Further,
simulations in Ref. [147] predict a saturation of the surface roughness with ion fluence,
similar to experimental results.
By studying the impact of damping parameter it is shown that with increasing γ the
hexagonal ordering of dot structures increases. For even larger γ a previously patterned
surface passes over into a smooth surface. However, the origin of the damping mechanism
is still not known. Potential candidates are re-deposition processes as already suggested
[77,79] or, alternatively, self-sputtering by atoms or particles released by the sputtering
process itself, as discussed shortly in Section 5.2. Nevertheless, the observed transition
from ordered dot pattern to smooth surfaces with increasing ion incidence angle, can be
explained by an enhanced damping which dominates over surface roughening by curvature
dependent sputtering.
Additionally, the behavior of γ can be compared with experimental results, for example,
with the parameter Uacc. As already shown in Section 4.1, an increase of Uacc results in an
increase of the angular distribution of ions and the effective angle of ions arriving on the
106
8.3: Other Models
sample surface. Further, in Section 7.2 it was shown that with increasing angular
distribution, the ordering of structures is improved, and the surface roughness increases.
Therefore, it can be stated that the increase of angular distribution has a similar effect to an
increase of the damping term. However, the results of Vogel and Linz are only preliminary
one. Further investigation are needed, especially to identify the physical origin of the
parameters introduced in this section.
107
Chapter 9
Conclusions
In this work a systematic experimental study of the surface topography evolution on Si
and Ge surfaces during low-energy ion beam erosion is presented. It was demonstrated that
ion beam erosion at low ion energies up to 2000 eV is very well suited for producing
nanostructured surfaces with sizes below 100 nm. Due to self-organization processes, and
for given sputtering conditions, these nanostructures show an almost perfect lateral
ordering covering the whole sample area under treatment. In this context a particular
interest is given to the formation of ripple and dot nanostructures on the surface. For the
sputtering experiments a home built broad beam ion source is used. The major part of the
work deals with the influence of different process parameters on the evolution, amplitude,
lateral size and ordering of these nanostructures. A detailed sample characterization have
been performed by means of atomic force microscopy and small angle X-ray scattering
techniques.
During the experimental investigations it was shown that there are a large number of
process parameters that influence the evolution of the surface topography. Explicitly, a
thorough study of the role of ion incidence angle, ion energy, ion fluence, and ion flux on
the evolution of ripples and dots is performed. These patterns are analyzed in terms of
surface roughness and wavelength (mean size) of nanostructures. Ion incidence angle
investigations, for the case without sample rotation, show ripple patterns with the wave
vector parallel to the ion beam projection evolving on the surface at near normal ion
incidence, comparable to theoretical predictions. However, experimental studies indicate
no change in orientation of ripples with increasing ion incidence angle, as the theory
predicts. Further, a detailed study shows transitions between ripples and dots by varying
the ion incidence angle. This behavior is similar for both materials Si and Ge.
The wavelength (mean size) of nanostructures can be controlled up to a certain range by
varying the ion energy from 500 eV up to 2000 eV, and in fact increase with increasing ion
energy. In this way the wavelength (mean size) of nanostructures can be varied between
30 nm and 70 nm. At the same time, the nanostructures maintain their lateral ordering. This
behavior of the wavelength with ion energy agrees with the theoretical model by
considering ion enhanced surface diffusion or ion induced viscous flow as the main
relaxation mechanisms. However, there are other completely new effects not accounted for
by the continuum theory, namely, the change in orientation of ripples on Si or the
transition from ripples to dots on Ge with increasing ion energy.
Temporal investigations of the evolution of ripples and dots showed that lateral ordering
of nanostructures increases with increasing ion fluence. At the same time the wavelength
of nanostructures remains constant with ion fluence. The independence of the
109
Chapter 9: Conclusions
wavelength with ion fluence is observed at different sputtering conditions. This
observations disagree with the theoretical model as shown from the simulations [74,132].
In the context of this work also the influence of ion mass (Ne+, Ar+, Kr+ and Xe+ ions)
on the surface evolution process is investigated. In general, in order that pattern formation
occurs the incoming ion should have at least the mass of the target material. Thus, no
patterns evolve on Si using Ne+ ions and for Ge using Ne+ and Ar+ ions. However, once
ripples and dots evolve on the surface their dynamics (wavelength, height, lateral ordering)
is not influenced by the ion mass. The independence of the wavelength value from the ion
mass implies ion induced viscous flow as the main relaxation mechanism.
Similar dependencies are observed also for dot structures evolving on Si surfaces at
grazing ion incidence with sample rotation.
In this work, for the first time it was shown that additionally to the ion beam parameters
also secondary ion beam parameters are crucial for the formation of nanostructures at least
on semiconductor materials. Explicitly, the role of the acceleration voltage applied on the
second grid that influences the angular distribution of ions within the beam and the beam
divergence is studied. This parameter is important for: i) the evolution of nanostructures on
the surface, and ii) the lateral ordering of nanostructures. In this way, an additional
parameter for controlling the process of nanostructure formation is introduced.
Another important result is that by combining the secondary ion beam parameter with
the ion incidence angle, at certain sputtering conditions, parameter regions are identified
where transitions between ripples and dots, or vice versa, exist. Especially the transition
from ripples to dots is of particular importance. Due to the previous existence of ripples the
evolving dots have an almost perfect lateral ordering covering the whole sample area.
These investigations show that ion beam sputtering is very well suited as an alternative
method to produce large area nanostructures on the surface. Although the process itself is a
stochastic one the evolving topographies show a remarkably high lateral ordering. The
formation of patterns on Si and Ge surfaces and previous reports on III/V semiconductors
show that this process is a general one. However, the adjustment of the sputtering
conditions to the particular material is very important.
110
Appendices
A1. Details of the Continuum Equation
A1.1 The A, B1, B2, C coefficients presented in Eq. (3.5) can be determined through
2
⎛a⎞
A = ⎜ ⎟ sin α ion
⎝α ⎠
2
2
⎛a⎞
⎛a⎞
B1 = ⎜ ⎟ sin 2 α ion + ⎜⎜ ⎟⎟ cos 2 α ion
⎝α ⎠
⎝β ⎠
(A1.1)
2
⎛a⎞
B2 = ⎜ ⎟ cosα ion
⎝α ⎠
1 ⎡⎛ a ⎞ ⎛ a ⎞
C = ⎢⎜⎜ ⎟⎟ − ⎜ ⎟
2 ⎢⎝ β ⎠ ⎝ α ⎠
⎣
2
2
⎤
⎥.
⎥⎦
A1.2 Makeev et al. using the local parameters of the surface morphology gave a detailed
description of the nonlinear noisy Kuramoto-Sivashinsky continuum equation [66]
∂h
∂ 2h
∂ 2h
∂ 4h
∂ 4h
∂ 4h
= −v0 + S x 2 + S y 2 − Dxx 4 − D yy 4 − D xy 2 2
∂t
∂x
∂y
∂x
∂y
∂x ∂y
+
λx ⎛ ∂h ⎞
2
λ y ⎛ ∂h ⎞
2
⎜ ⎟ + ⎜⎜ ⎟⎟ + η ( x, y , t ).
2 ⎝ ∂x ⎠
2 ⎝ ∂y ⎠
(A1.2)
The general expressions of the coefficients are
⎛a⎞
Fa ⎜ ⎟
⎝α ⎠
Sx =
2α 2 f 3
2
⎧⎪ ⎛ a ⎞ 4 4 ⎛ a ⎞ 4 ⎛ a ⎞ 2 2 2 ⎛ a ⎞ 2 ⎛ a ⎞ 2 2 2 ⎛ a ⎞ 4 ⎫⎪
⎨2⎜ ⎟ s − ⎜ ⎟ ⎜⎜ ⎟⎟ s c + ⎜ ⎟ ⎜⎜ ⎟⎟ c s − ⎜⎜ ⎟⎟c ⎬,
⎝α ⎠ ⎝ β ⎠
⎝α ⎠ ⎝ β ⎠
⎪⎩ ⎝ α ⎠
⎝ β ⎠ ⎪⎭
Fac 2
Sy = −
2f
111
2
⎛a⎞
⎜ ⎟ ,
⎝α ⎠
(A1.3)
(A1.4)
Appendices
DxxESD =
Fa 3 ⎧
a2 2 3
a6 4 2 a2 2 ⎛ 2
a4 2
a8 4 ⎞
⎜
12
s
f
4
s
f
c
f
3
f
6
s
f
s ⎟
−
−
+
+
+
⎨
⎜
α2
α6
α2
α4
α 8 ⎟⎠
24 f 5 ⎩
⎝
⎛ a 2 a 2 ⎞⎛ a 2
a6
a10 ⎞ ⎫
+ 2c ⎜⎜ 2 − 2 ⎟⎟⎜⎜15 2 s 2 f 2 + 10 6 s 4 f + 10 s 6 ⎟⎟ ⎬,
α ⎠⎝ α
α
α
⎠⎭
⎝β
(A1.5)
2
D yyESD =
ESD
xy
D
Fa 3 3β 2c 2
,
24 fα 2
(A1.6)
2
2
2
1 ⎡ ⎛ a ⎞ 2 2 ⎛ a ⎞ 2 ⎛⎜ 2 ⎛ a ⎞ 2 ⎞⎟
− 2⎜ ⎟ s f + ⎜ ⎟ c f + ⎜ ⎟ s f
=
2
3 ⎢
⎟
⎝ α ⎠ ⎜⎝
⎝α ⎠
⎛ a ⎞ f ⎢⎣ ⎝ α ⎠
⎠
4⎜⎜ ⎟⎟
β
⎝ ⎠
Fa 3
(A1.7)
⎛⎛ a ⎞ ⎛ a ⎞ ⎞ ⎛ ⎛ a ⎞
⎛ a ⎞ ⎞⎤
+ 2⎜ ⎜⎜ ⎟⎟ − ⎜ ⎟ ⎟c 2 ⎜ 3⎜ ⎟ s 2 f + ⎜ ⎟ s 4 ⎟⎥,
⎜⎝ β ⎠ ⎝α ⎠ ⎟ ⎜ ⎝α ⎠
⎝ α ⎠ ⎟⎠⎥⎦
⎠ ⎝
⎝
2
λx =
Fc
2f4
2
2
6
⎧ a 10 4
a 10 2 4
a 10 4
2
s
c
s
c
c (1 + 2 s 2 )
(
3
+
2
)
+
4
−
⎨ 8 2
6 4
4 6
α
β
α
β
α
β
⎩
(A1.8)
⎫
⎛ 2a 4
⎞
a4
a 12
− f 2 ⎜⎜ 4 s 2 − 2 2 (1 + 2 s 2 ) ⎟⎟ − 8 4 s 2 c 2 − f 4 ⎬,
α β
⎝α
⎠ α β
⎭
λy =
Fc
2f 2
⎧ a4 2
a4 2
a6 2
2⎫
⎨ 4 s + 2 2 c − 4 2 c − f ⎬,
α β
α β
⎩α
⎭
(A1.9)
with
⎛
JEpa
a 4c2
exp⎜⎜ − 2 2
F≡
αβ 2πf
⎝ 2α β f
⎞
a2
a2
2
⎟⎟ ; s = sin θ ; c ≡ cosθ and f = 2 (cosθ ) + 2 (sin θ ) 2 .
α
β
⎠
112
A2: The Model for GISAXS and GID Simulations
A2. The model for GISAXS and GID Simulations
The simulations performed on the GISAXS and GID data in Chapter 6.2 are based on
some assumptions. The first assumption to be made is the form factor belonging to a single
object. Next is the correlation function that describes the positions of structures and their
correlation to each other. There are sources where a detailed description of the fitting
model is given [111,116,127,128]. According to Lazzari the scattered intensity is given as
a product (in the reciprocal space) of the square of a form factor F (q) of a certain object
and the correlation function C (q) that describes the position of structures (i. e. the
ordering).
I (q) = F (q) 2 C (q)
(A2.1)
For ripples the ordering is described by using the correlation function of the linear
paracrystal model given by
⎛ 1
⎞
1 − exp⎜ − σ 2 q 2 ⎟
⎝ 2
⎠
C ( q) =
⎞
⎛ 1
⎞
⎛ 1
1 + exp⎜ − σ 2 q 2 ⎟ − 2 exp⎜ − σ 2 q 2 ⎟ cos( D0 q )
⎠
⎝ 4
⎠
⎝ 2
(A2.2)
where D0 = d = λ is the mean distance between structures and σ is the mean distance
deviation.
The form factor of a cone in the cartesian frame is given by the expression [127]
H
⎛
z ⎞
F ( q, R, H , α ) = ∫ 2π ⎜⎜ R −
⎟
tan(α ) ⎟⎠
⎝
0
2
⎡ ⎛
z ⎞⎤
J 1 ⎢ q⎜ R −
⎟
tan(α ) ⎠⎥⎦
⎣ ⎝
exp( −iqz )dz.
⎛
z ⎞
q⎜ R −
⎟
tan(α ) ⎠
⎝
(A2.3)
Here R = Rmean = D0/2 is the cone radius at the base, α is the tilt angle (Fig. A2.1). The
height h of the cone is calculated using the relation h = Rmeantan(α). J1 is the first order
Bessel function. The same form factor is used for ripples and for dots. However, by
varying the angle α and the radius R the shape of the cone can be influenced. In the case of
ripples α and R are chosen in such a way that the cone takes a triangle shape (Table 6.2 and
z
γ
h
2R
Figure A2.1: Schematic sketch of a cone with characteristic parameters as used for the simulations. γ, h, and
2R were taken as fitting parameters.
113
Appendices
14
10
2
Eion = 1200 eV
Eion = 2000 eV
10
a)
3
10
b)
12
10
1
12
10
10
C(q)
-3
10
0
10
10
10
-1
10
F(q,R,H,α)
10
Eion = 1200 eV
Eion = 1500 eV
Eion = 2000 eV
10
F(q,R,H,α)
10
C(q)
0
8
8
10
10
-2
10
-6
10
-0.12
-0.06
0.00
0.06
-0.12
0.12
-0.06
0.00
0.06
0.12
-1
-1
q [Å ]
q [Å ]
Figure A2.2: Correlation functions and form factors used for the fitting procedure for ripples: a) Ge, b) Si.
6.4). While for dots the truncated shape of the cone is used by choosing larger angle values
(Table 6.5).
For dots the correlation function of the hexagonal paracrystal is used. This because the
distances between peaks are different. The expression is given by
⎛ 1 + P1 ( q ) ⎞ ⎛ 1 + P2 ( q) ⎞
⎟⎟ Re⎜⎜
⎟⎟
C ( q ) = Re⎜⎜
⎝ 1 − P1 ( q ) ⎠ ⎝ 1 − P2 ( q) ⎠
(A2.4)
with probability distributions:
⎛
3 ⎞
P1 ( q ) = exp (πσ 2 q 2 )exp ⎜⎜ − iD0
q⎟
2 ⎟⎠
⎝
(A2.5)
1 ⎞
⎛
P2 ( q) = exp (πσ 2 q 2 )exp⎜ − iD0 q ⎟
2 ⎠
⎝
(A2.6)
The correlation functions and the form factors used for ripple and dot simulations in Ge
and Si are given in Fig. A2.2 and Fig. A2.3.
11
2
10
2
10
GISAXS: ω = 0°
GISAXS: ω = 30°
GID at (220)
10
a)
1
10
Eion = 1000 eV
Eion = 2000 eV
b)
11
10
10
10
10
-1
10
9
10
-2
10
8
10
8
10
10
-4
-0.06
0.00
-0.06
0.06
-1
0.00
0.06
-1
q [Å ]
q [Å ]
Figure A2.3: Correlation functions and form factors used for the fitting procedure for dots. a) Ge, b) Si.
114
F(q,R,H,α)
9
10
-2
10
10
0
C(q)
C(q)
10
F(q,R,H,α)
10
0
List of Acronyms
AFM
BH
ESD
FFT
FWHM
GID
GISAXS
HRTEM
ISA
ISQ
IVF
KS
LRO
NSR No
PSD
rms
SR
SRIM
SRO
TD
Atomic Force Microscopy
Bradley-Harper
Effective Surface Diffusion
Fast Fourier Transformation
Full Width at Half Maximum
Grazing Incidence Diffraction
Grazing Incidence Small Angle X-ray Scattering
High Resolution Transmission Electron Microscopy
Ionenstrahlätzanlage
Ionenstrahlquelle
Induced Viscous Flow
Kuramoto-Sivashinsky
Long Range Order
Sample Rotation
Power Spectral Density
Root Mean Square
Sample Rotation
Stopping and Range of Ions in Matter
Short Range Order
Topography Diagram
115
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Acknowledgements
There are many persons who helped and supported me during these years, to whom I owe many
thanks. I apology from the beginning in order that I forget somebody. But, I'm in a hurry to catch
the deadline for submitting the work, before summer holidays.
I'm greatly thankful to my thesis advisor, Prof. Dr. B. Rauschenbach, for his readiness to accept
me as a Ph.D. student, for the support, and for the many helpful discussions.
My special thank goes to H. Neumann, Dr. M. Tartz, Dr. B. Faust, and F. Scholze for the very
useful discussions concerning ion sources. Including the technical support for the ion grid system.
Your discussions helped me to make a step further in understanding many tricks about the ion
sources. Additional thank to Michael for performing the very valuable simulations.
I thank Dr. T. Höche for performing the HRTEM measurements and Prof. Dr. U. Gösele for his
approval to perform the measurements at the MPI Halle.
My thank goes also to D. Hirsch and Dr. D. Flamm for their technical support with the AFM
and the ISA equipment, especially at the beginning of my work.
Further, my thank goes to Teresa Lutz, for helping me performing part of the thousands of
thousands of sputtering experiments and AFM measurements.
Many thanks to Dr. T. Metzger and Dr. D. Carbone from the ESRF in Grenoble, for their
support during the measurements and for data evaluation.
I also want to thank Prof. Dr. M. J. Aziz and Prof. Dr. M. P. Brenner from the University of
Harvard, for their useful discussion concerning the process of pattern formation.
I'm grateful to Dr. J. W. Gerlach and Dr. D. Manova for their critical work through parts of the
manuscript.
My thank goes to the colleagues of the ion beam department for their warm welcome and for the
very nice moments during these years. A special thank goes to lunch table companions who helped
me to understand the German way of making jokes.
I don't want to forget Mrs. H. Beck for preparing the filament wires, and Mrs. Herold in the
Chemical lab.
Here I want to thank Dr. E. Schubert, with whom I shared the room for two years, for never
getting bored from my questions. Also, I thank Prof. Dr. M. Schubert who helped me to be here at
IOM, by introducing me to Dr. F. Frost.
I kept my special thanks for the end. I still remember the first day I came to IOM to meet
Frank (Dr. Frost), we were sitting at the seminar room. You were speaking about the project, and I
was thinking to leave the university or not. I'm very thankful to you for accompanying me during
these years. For introducing me to the world of nanotechnology and giving me the possibility to
work in such an interesting and fascinating subject. You were always there for me, even then when
your phone was ringing several times per day. I'm also very thankful to the many discussions,
suggestions and to your keen eyes for details. I'll always be in debt to you.
At the end, a want to thank my family. My two lovely daughters Fatbardha and Ndrina for
making me relax from the long stressful days. To my wife, thank you very much for the patience
during these years, and for sacrificing many holidays.
I dedicate my work to those who are the reason for me to be here were I am, to my parents.
Curriculum vitae
Name:
Bashkim Ziberi
Date of birth:
11. September 1974
Place of birth:
Radiovce (Macedonia)
Nationality:
Albanian
Marital status:
Married since January 1999
Children:
Fatbardha (born July 22, 2004)
Ndrina (born April 23, 2006)
Education and scientific activities:
1981 – 1989
Primary school in Radiovce (Macedonia)
1989 – 1993
Secondary school in Tetovo (Macedonia)
1993 – 1998
Physics studies at the Univeristy of Tirana (Albania)
in the group „Special Physics“ (5 years)
Diploma thesis „Investigation of Microstructures on metalceramic Cu-C-PbO samples with Electron Microscopy“
1998 – 1999
Teaching Assistant at the University of Tetovo (Macedonia)
1999 – 2001
Master of Science studies in Physics at the University of
Leipzig
07/2001
M. Sc. thesis „Ultrasound Monitoring of the Synthesis of
Zeolites in Real Time”
2001 – 2002
Scientific employee at the Faculty of Physics and Earth
Sciences, University of Leipzig
since 05/2002
Scientific employee at the Leibniz-Institut für
Oberflächenmodifizierung e. V. Leipzig and Ph. D. student at
the University of Leipzig, Institute of Experimental Physics II.
Awards:
06/2004
Young Scientist Award of the European Materials Research
Society for the work on ion beam induced self-organization on
semiconductor surfaces
06/2005
348. WE-Heraeus Seminar Poster Prize sponsored by the
“Wilhelm und Else Heraeus – Stiftung” during the Wilhelm und
Else Heraeus-Seminar on “Ions at Surfaces: Patterns and
Processes”
List of Publications
The following articles have been published in the course of this thesis, are submitted, or in
preparation for future publication
1.
Ion beam assisted smoothing of optical surfaces
F. Frost, R. Fechner, D. Flamm, B. Ziberi, W. Frank, A. Schindler
Applied Physics A 78: Materials Science & Processing, 651 (2004)
2.
The shape and ordering of self-organized nanostructures by ion sputtering
F. Frost, B. Ziberi, T. Höche, B. Rauschenbach
Nuclear Instruments & Methods B 216, 9 (2004)
3.
Large area smoothing of optical surfaces by low-energy ion beams
F. Frost, R. Fechner, B. Ziberi, D. Flamm, A. Schindler
Thin Solid Films 459, 100 (2004)
4.
Importance of ion beam parameters on self-organized pattern formation on
semiconductor surfaces by ion beam erosion;
B. Ziberi, F. Frost, H. Neumann, B. Rauschenbach
Thin Solid Films 459, 106 (2004)
5.
Highly ordered self-organized dot patterns on Si surfaces by low-energy ion beam
erosion
B. Ziberi, F. Frost, B. Rauschenbach, T. Höche
Applied Physics Letters 87, 033113 (2005)
6.
Ripple pattern formation on silicon surfaces by low-energy ion-beam erosion:
Experiment and theory
B. Ziberi, F. Frost, T. Höche, B. Rauschenbach
Physical Review B 72, 235310 (2005)
7.
Dot pattern formation on Si surfaces by low-energy ion beam erosion
B. Ziberi, F. Frost, T. Höche, B. Rauschenbach
in Kinetics-Driven Nanopatterning on Surfaces, edited by Eric Chason, George H.
Gilmer, Hanchen Huang, and Enge Wang (Mater. Res. Soc. Symp. Proc. 849,
Warrendale, PA , 2005), KK 6.2.
8.
Pattern transitions on Ge surfaces during low-energy ion beam erosion
B. Ziberi, F. Frost, B. Rauschenbach
Applied Physics Letters 88, 173115 (2006)
9.
Self-organized dot patterns on Si surfaces during noble gas ion beam erosion
B. Ziberi, F. Frost, B. Rauschenbach
Surface Science (im Druck, 2006)
10.
Low-energy ion bombardment induced nanostructures on surfaces
B. Ziberi, F. Frost, B. Rauschenbach
Vacuum (im Druck, 2006)
11.
Formation of large-area nanostructures on Si and Ge surfaces during low-energy ion
beam erosion
B. Ziberi, F. Frost, B. Rauschenbach
Journal of Vacuum Science & Technology A 24, 1344 (2006)
Publikations on other Subjects:
12.
Ion beam sputter deposition of soft x-ray Mo/Si multilayer mirrors;
E. Schubert, F. Frost, B. Ziberi, G. Wagner, H. Neumann, B. Rauschenbach
Journal of Vacuum Science & Technology B 23, 959 (2005)
13.
In situ diagnostics of zeolite crystallization by ultrasonic monitoring
R. Herrmann, W. Schwieger, O. Scharf, C. Stenzel, H. Toufar, M. Schmachtl, B. Ziberi,
W. Grill
Microporous and Mesoporous Materials 80, 1 (2005)
Selbständigkeitserklärung
Hiermit versichere ich, daß die vorliegende Arbeit ohne unzulässige Hilfe und ohne Benutzung
anderer als der angegebenen Hilfsmittel angefertigt und daß die aus fremden Quellen direkt oder
indirekt übernommenen Gedanken in der Arbeit als solche kenntlich gemacht wurden.
Ich versichere, daß alle Personen, von denen ich bei der Auswahl und Auswertung des Materials
sowie bei der Herstellung des Manuskripts Unterstützungsleistungen erhalten habe, in der
Danksagung der vorliegenden Arbeit aufgeführt sind.
Ich versichere, daß außer den in der Danksagung genannten, weitere Personen bei der geistigen
Herstellung der vorliegenden Arbeit nicht beteiligt waren, und insbesondere von mir oder in
meinem Auftrag weder unmittelbar noch mittelbar geldwerte Leistungen für Arbeiten erhalten
haben, die im Zusammenhang mit dem Inhalt der vorliegenden Dissertation stehen.
Ich versichere, daß die vorliegende Arbeit weder im Inland noch im Ausland in gleicher oder in
ähnlicher Form einer anderen Prüfungsbehörde zum Zwecke einer Promotion oder eines anderen
Prüfungsverfahrens vorgelegt und in ihrer Gesamtheit noch nicht veröffentlicht wurde.
Ich versichere, daß keine früheren erfolglosen Promotionsversuche stattgefunden haben.
Leipzig, 30.06.2006
Bashkim Ziberi