Morphology change of the silicon surface induced by Ar+ ion beam sputtering
Two-level modeling for nanoscale pattern formation on silicon target by Ar+ ion sputtering is presented. Phase diagram illustrating possible nanosize surface patterns is discussed. Scaling characteristics for the structure wavelength dependence versus incoming ion energy are defined. Growth and roug...
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Інститут фізики конденсованих систем НАН України
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nasplib_isofts_kiev_ua-123456789-1199812025-02-09T18:03:32Z Morphology change of the silicon surface induced by Ar+ ion beam sputtering Змiна морфологiї поверхнi кремнiю при розпиленнi його iонами аргону Kharchenko, V.O. Kharchenko, D.O. Two-level modeling for nanoscale pattern formation on silicon target by Ar+ ion sputtering is presented. Phase diagram illustrating possible nanosize surface patterns is discussed. Scaling characteristics for the structure wavelength dependence versus incoming ion energy are defined. Growth and roughness exponents in different domains of the phase diagram are obtained. Проводиться теоретичне дослiдження процесiв змiни морфологiї поверхнi кремнiю при розпиленнi його iонами аргону в рамках дворiвневої схеми, що враховує методи Монте-Карло та модифiковану теорiю Бредлi-Харпера. Отримано та проаналiзовано фазову дiаграму у площинi кут падiння налiтаючого iону та енергiя iону, що iлюструє можливi типи поверхневих нано-структур. Отримано узагальнену степеневу залежнiсть довжини хвилi отриманих поверхневих структур вiд енергiї налiтаючих iонiв. Проаналiзовано показник росту i повздовжнiй та поперечний показники шорсткостi отриманих поверхонь. 2011 Article Morphology change of the silicon surface induced by Ar+ ion beam sputtering / V.O. Kharchenko, D.O. Kharchenko // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23602:1-11. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 68.55.-a, 68.35.Ct, 79.20.Rf, 81.16.Rf DOI:10.5488/CMP.14.23602 arXiv:1106.6256 https://nasplib.isofts.kiev.ua/handle/123456789/119981 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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Two-level modeling for nanoscale pattern formation on silicon target by Ar+ ion sputtering is presented. Phase diagram illustrating possible nanosize surface patterns is discussed. Scaling characteristics for the structure wavelength dependence versus incoming ion energy are defined. Growth and roughness exponents in different domains of the phase diagram are obtained. |
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Kharchenko, V.O. Kharchenko, D.O. |
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Kharchenko, V.O. Kharchenko, D.O. Morphology change of the silicon surface induced by Ar+ ion beam sputtering Condensed Matter Physics |
| author_facet |
Kharchenko, V.O. Kharchenko, D.O. |
| author_sort |
Kharchenko, V.O. |
| title |
Morphology change of the silicon surface induced by Ar+ ion beam sputtering |
| title_short |
Morphology change of the silicon surface induced by Ar+ ion beam sputtering |
| title_full |
Morphology change of the silicon surface induced by Ar+ ion beam sputtering |
| title_fullStr |
Morphology change of the silicon surface induced by Ar+ ion beam sputtering |
| title_full_unstemmed |
Morphology change of the silicon surface induced by Ar+ ion beam sputtering |
| title_sort |
morphology change of the silicon surface induced by ar+ ion beam sputtering |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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https://nasplib.isofts.kiev.ua/handle/123456789/119981 |
| citation_txt |
Morphology change of the silicon surface induced by Ar+ ion beam sputtering / V.O. Kharchenko, D.O. Kharchenko // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23602:1-11. — Бібліогр.: 27 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kharchenkovo morphologychangeofthesiliconsurfaceinducedbyarionbeamsputtering AT kharchenkodo morphologychangeofthesiliconsurfaceinducedbyarionbeamsputtering AT kharchenkovo zminamorfologiípoverhnikremniûprirozpilennijogoionamiargonu AT kharchenkodo zminamorfologiípoverhnikremniûprirozpilennijogoionamiargonu |
| first_indexed |
2025-11-29T07:04:35Z |
| last_indexed |
2025-11-29T07:04:35Z |
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| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23602: 1–11
DOI: 10.5488/CMP.14.23602
http://www.icmp.lviv.ua/journal
Morphology change of the silicon surface induced by
Ar+ ion beam sputtering
V.O. Kharchenko, D.O. Kharchenko
Institute of Applied Physics, National Academy of Sciences of Ukraine,
58 Petropavlivska Str., 40030 Sumy, Ukraine
Received April 19, 2011, in final form June 6, 2011
Two-level modeling for nanoscale pattern formation on silicon target by Ar+ ion sputtering is presented. Phase
diagram illustrating possible nanosize surface patterns is discussed. Scaling characteristics for the structure
wavelength dependence versus incoming ion energy are defined. Growth and roughness exponents in differ-
ent domains of the phase diagram are obtained.
Key words: ion-beam sputtering, surface morphology, nanoscale structures
PACS: 68.55.-a, 68.35.Ct, 79.20.Rf, 81.16.Rf
1. Introduction
It is well known that low and medium energy ion sputtering may induce a fabrication of
periodic nanoscale structures on an irradiated surface [1]. Depending on the sputtered substrate
characteristics and sputtering conditions, different types of nanoscale structures such as ripples,
nanoholes and nanodots can grow on a target during ion beam sputtering [2–6]. These patterns have
been found on both amorphous and crystalline materials including insulators, semiconductors and
metals (see reference [7] and citations therein). Main theoretical models describing ripple formation
are based on the results of famous works by Bradley and Harper [4], Kardar et al. [8], Wolf and
Villian [9], and Kuramoto et al. [10]. The main control parameters in these models reduced to
surface tensions, tilt-dependent erosion rates and diffusion constants are determined by sputtered
substrate characteristics and sputtering conditions (see for example [7]).
Among theoretical investigations there are a lot of experimental data manifesting a large class
of patterns formed due to the self-organization process. It was experimentally shown that the main
properties of pattern formation processes depend on ion-beam parameters such as ion flux, energy
of deposition, angle of incidence, and temperature of the substrate (target). Therefore, to study
the ion beam sputtering processes theoretically one needs to determine the mentioned parameters
of the model according to the physical conditions related to concrete materials.
One of the most frequently used materials for ion beam sputtering is silicon because it is the
mainstream material in modern microelectronic industry and it is readily available with high purity
and quality. Nanostructuring of silicon has received much attention due to its potential application
in developing the Si light sources [11]. Various techniques such as acid etching, ion implantation,
reactive evaporation, chemical vapor deposition and molecular beam epitaxy have been used in
developing the Si nanomaterials (porous Si, Si nanocrystal-doped dielectrics and Si quantum dots)
(see reference [11] and citation therein).
In this paper we study the properties of the formation of nanoscale patterns on a silicon target
sputtered by Ar+ ions. To this end, we use a two-level scheme, based on Monte-Carlo simulations
and the modified Bradley-Harper theory. In the first approach we compute the ion energy dependent
penetration depth, widths of the ion energy distribution and sputtering yield. Next, we exploit these
characteristics as input data for the continuum approach describing the evolution of the surface
height field. We define the domains of values for the angle of incidence and ion energy where
different nanoscale structures can be formed. The dynamics of nanoscale pattern formation is
c© V.O. Kharchenko, D.O. Kharchenko, 2011 23602-1
http://dx.doi.org/10.5488/CMP.14.23602
http://www.icmp.lviv.ua/journal
V.O. Kharchenko, D.O. Kharchenko
discussed. We will show that at fixed values for the incidence angle, one has two scaling exponents
for wavelength related to small and large values for ion energy. In addition, we obtain roughness
and growth exponents.
The work is organized in the following manner. In section 2 we present the theoretical model in
the framework of the modified Bradley-Harper approach. In section 3 using Monte-Carlo modeling
we compute the main characteristics incorporated into the continuum theory. The related phase
diagram, the dynamics of the formation of nanoscale structures, the ion energy dependent wave-
length and scaling properties of the surface morphology are discussed in section 4. We conclude in
the last section.
2. Theoretical model
Let us consider a d-dimensional substrate and denote with r the d-dimensional vector locating
a point on it. The surface is described at each time t by the height z = h(r, t). If we assume that
the surface morphology changes during ion sputtering, then we can use the model for the surface
growth proposed by Bradley and Harper [4] and further developed by Cuerno and Barabasi [2].
We consider the system where the direction of the ion beam lies in x− z plane at an angle θ to the
normal of the uneroded surface. Following the standard approach one assumes that an averaged
energy deposited at the surface (let say at point O), due to the ion arriving at the point P in the
solid, follows the Gaussian distribution [4] E(r) =
[
ε/(2π)3/2σµ2
]
exp
[
−z2/2σ2 − (x2 + y2)/2µ2
]
;
ε denotes the kinetic energy of the arriving ion, σ and µ are the widths of distribution in directions
parallel and perpendicular to the incoming beam. Parameters σ and µ depend on the target material
and can vary with physical properties of the target and incident energy. The erosion velocity at
the surface point O is described by the formula v = p
∫
R
drΦ(r)E(r), where integration is provided
over the range of the energy distribution of all ions; here Φ(r) are the corrections for the local slope
dependence of the uniform flux J . The material constant p is defined as: p = 3/(4π2)(NU0C0)
−1,
where U0 and C0 are the surface binding energy and the constant proportional to the square of the
effective radius of the interatomic interaction potential, respectively [12]. The general expression
for the local flux for surfaces with non-zero local curvature is [13]:
Φ(x, y, h) = J cos
[
arctan
(
√
(∇xh)2 + (∇yh)2
)]
.
Hence, the dynamics of the surface height is defined by the relation ∂th ≃ −v
(
θ −∇xh,∇2
xh,∇2
yh
)
and is given by the equation ∂th ≃ −v(θ)
√
1 + (∇h)2, where 0 < θ < π/2 [2–4, 8, 14]. The linear
term expansion yields ∂th = −v0 + γ∇xh + να∇2
ααh; where ∇ = ∂/∂r, ∇α = ∂/∂α, α = {x, y}.
Here v0 is the surface erosion velocity; γ = γ(θ) is a constant that describes the slope depending
erosion; να = να(θ) is the effective surface tension generated by erosion process in α direction.
If one assumes that the surface current is driven by differences in chemical potential µ, then
the evolution equation for the field h should take into account the term −∇ · js in the right hand
side, where js = K∇(∇2h) is the surface current; K > 0 is the temperature dependent surface
diffusion constant. If the surface diffusion is thermally activated, then we have K = Dsκρ/n
2T ,
where Ds = D0 exp (−Ea/T ) is the surface self-diffusivity (Ea is the activation energy for surface
diffusion), κ is the surface free energy, ρ is the areal density of diffusing atoms, n is the number
of atoms per unit volume in the amorphous solid. This term in the dynamical equation for h is
relevant in high temperature limit which will be studied below.
Assuming that the surface varies smoothly, we neglect spatial derivatives of the height h of
third and higher orders in the slope expansion. Taking into account nonlinear terms in the slope
expansion of the surface height dynamics, we arrive at the equation for the quantity h′ = h+ v0t
of the form [2, 4]
∂th = γ∇xh+ να∇2
ααh+
1
2
λα(∇αh)
2 −K∇2(∇2h) + ξ(x, y, t), α = {x, y}, (1)
where we drop the prime for convenience. Here we introduce the uncorrelated white Gaussian noise
ξ with zero mean mimicking the randomness resulting from the stochastic nature of the ion arrival
23602-2
Morphology change of the silicon surface
to the surface. In equation (1) the effective surface tensions νx and νy generated by the IBS, the
tilt-dependent erosion rates λx and λy are defined through the incident angle θ, penetration depth
of incident ion a, distribution widths σ, µ and the sputtering yield Y0 as follows [2, 4]:
γ = F0
s
f2
[
a2σa
2
µc
2
(
a2σ − 1
)
− a4σs
2
]
,
νx = F0a
a2σ
2f3
(
2a4σs
4 − a4σa
2
µs
2c2 + a2σa
2
µs
2c2 − a4µc
4
)
,
νy = −F0a
c2a2σ
2f
,
λx = F0
c
2f4
{
a8σa
2
µs
4
(
3 + 2c2
)
+ 4a6σa
4
µc
4s2 − a4σa
6
µc
4
(
1 + 2s2
)
− f2
[
2a4σs
2 − a2σa
2
µ
(
1 + 2s2
)]
− a8σa
4
µc
2s2 − f4
}
,
λy = F0
c
2f2
(
a4σs
2 + a2σa
2
µc
2 − a4σa
2
µc
2 − f2
)
. (2)
Here we have used the following notations:
F0 ≡ JεY0pa
σµ
√
2πf
exp
(
−a2σa2µc2
2f
)
, (3)
aσ ≡ a
σ
, aµ ≡ a
µ
, s ≡ sin(θ), c ≡ cos(θ), f ≡ a2σs
2 + a2µc
2. (4)
Let us perform the stability analysis for a system with additive fluctuations. To this end, we
average the Langevin equation (1) over noise and obtain
∂t 〈h〉 = γ∇x 〈h〉+ νx∇2
xx 〈h〉+ νy∇2
yy 〈h〉+
λx
2
〈
(∇xh)
2
〉
+
λy
2
〈
(∇yh)
2
〉
−K∇4 〈h〉 . (5)
Considering the stability of the smooth surface characterized by 〈h〉 = 0, we can rewrite the
linearized evolution equation in the standard form:
∂t 〈h〉 =
(
ν̂ef + K̂ef
)
〈h〉 , (6)
with notations
ν̂ef = γ∇x + νx∇2
xx + νy∇2
yy , K̂ef = −K∇4. (7)
It is easy to see that equation (7) admits a solution of the form 〈h〉 = A exp[i(kxx+kyy−ωt)+χt].
Indeed, substituting it into equation (7) and separating real and imaginary parts we get
ω = −γ (θ) kx , (8)
χ = −νx (θ) k2x − νy (θ) k
2
y −K
(
k2x + k2y
)2
. (9)
As far as F0 , f , a are positive values, hence one has νy < 0, whereas νx can change its sign.
Therefore, the Bradley-Harper model does not provide for stable smooth surface. Hence, we can
conclude that if νx > 0, then ripples (wave patterns) appear in x-direction. On the contrary, when
νx < 0, equiaxed structures (nanodots/nanoholes) can be formed on an eroded surface. In addition,
the sign of the product λx · λy can play a crucial role in ripple formation processes [15].
For the noiseless nonlinear model (1) it was shown that as the sets να and λα are the functions
of the angle of incidence θ ∈ [0, π/2] there are three domains in the phase diagram (aσ , θ) where
νx and λx change their signs, separately [2]. This results in the formation of ripples in different
directions x or y varying aσ or θ.
One needs to note that in the Bradley-Harper approach describing the processes of ripple
formation on amorphous substrates the penetration depth can be approximated as a(ε) ∼ ε leading
23602-3
V.O. Kharchenko, D.O. Kharchenko
to the power-law asymptotics for the wavelength of the ripples Λ versus ion energy as follows:
Λ ∼ ε−1/2 [4]. In the next section, performing calculations for the sputtering of the silicon target
by Ar+ ions we shall show that there are deviations from these asymptotics due to power-law
dependence of a, σ, µ versus ion energy ε. Moreover, it will be shown that the sputtering yield
depends on both incident angle θ end ion energy ε in a power-law form. To define a, σ, µ and the
sputtering yield, we shall use Monte-Carlo approach.
3. Monte-Carlo modelling
To study the evolution of the silicon surface morphology during Ar+ ion beam sputtering one
needs to know the energy of ions and target characteristics such as: penetration depth of the Ar+
ions into the silicon target a; widths of the distribution in parallel and perpendicular directions of
the incoming beam (σ and µ) and sputtering yield Y0 . Moreover, to simulate the target morphology
evolution one should define temperature T , uniform flux J , atomic density of the target N , surface
binding energy U0 and the effective radius of interatomic interaction potential. Using data from
reference [5] at T = 550 C one has K = C′8.49 × 103 nm4/s, where addimer concentration
is C′ = 0.04 atoms/site (≈ 4% coverage), or, C′ = 0.07 atoms/nm2. To define the material
constant p we shall use N ≃ 50 atoms/nm3, U0 = 4.73 eV. For the effective radius of interatomic
interaction potential, we put the length of the main diagonal of silicon primitive cell with lattice
parameter 0.5437 nm. To compute the time evolution of the silicon surface morphology, we should
calculate the effective surface tensions νx and νy generated by the IBS, and the rates λx and λy
[see equation (2)]. Following relations (3) and (4) dependent on ion energy, parameters a, σ and µ,
as far as Y0 = Y0(θ, ε), can be computed. From experimental point of view, the control parameters
at IBS are the energy of ion beam, off-normal incidence angle and ion flux. In all our calculations
we put J = 20 ions/(nm2 s). Thereafter we vary the ion energy in the interval 100 eV÷10 keV and
use intermediate off-normal incidence angles θ ∈ [40◦, 65◦].
In further study, we use the well-known program codes (TRIM and SRIM) to calculate the
stopping range of ions in matter and transport range of ions in matter. Description of algorithms
and the basic principles for Monte-Carlo calculation of both the transport range of ions in matter
and the stopping range of ions in matter can be found in [16]; TRIM and SRIM codes can be found
on the web-site www.srim.org.
Values for parameters a, σ and µ for silicon target sputtered by Ar+ ions were obtained with
the help of SRIM code (a program for calculating the stopping range of ions in matter). Results
for relative penetration depths aσ ≡ a/σ and aµ ≡ a/µ versus ion energy are shown in figure 1
(dependencies a(ε), σ(ε) and µ(ε) are shown in the insert). In figure 1 it is seen that longitudinal
0.1 1 10
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
0.1 1 10
3
6
9
12
a
a
(
n
m
)
(keV)
a
a
a
,
a
(keV)
Figure 1. Relative penetration depths of the Ar+ ions into the silicon target versus ion energy
(penetration depth and distribution widths versus ion energy are shown in insertion).
23602-4
www.srim.org
Morphology change of the silicon surface
and transverse widths σ and µ, respectively, as far as aσ and aµ satisfy the following relations, are
as follows: σ < µ and aσ < aµ . In reference [17] it was shown that the rotated ripple structures
formed when λxλy < 0 with rotation angle ϕ = tan−1
√
−λx/λy can be observed at small incidence
angles θ when aµ < aσ (aσ = 1) and at intermediate and large θ when aµ > aσ . Hence, one can
expect the appearance of rotated ripple structures in our system.
It is principally important that dependencies of penetration depth a and longitudinal and
transverse widths σ and µ, respectively, versus ion energy ε deviate from the linear law predicted
by Bradley and Harper [4]. For a silicon target sputtered by Ar+ ions we have obtained a power-law
approximation of the form: φ(ε) = A+BεC , where φ = {a, σ, µ}, constants A, B and C are fitting
parameters. So, we can expect that the wavelength dependence Λ(ε) ∼ ε−δ can be characterized
by the exponent δ 6= 1/2. In our further continuum approach we shall use the obtained power-law
asymptotics for a, aσ and aµ from Monte-Carlo simulations.
To compute the dependence of the sputtering yield versus ion energy and angle of incidence we
use Monte-Carlo approach realized in TRIM code (program for the calculation of transport range
of ions in matter). The results of calculations for sputtering yield versus incident angle at fixed
ion energy and sputtering yield versus ion energy at fixed incident angle are shown in figures 2 (a)
and 2 (b), respectively. In figure 2 it is seen that sputtering yield depends on both ion energy and
incidence angle in accordance with a power law as follows Y0(ψ) = A′ + B′ψC′
, where ψ = {θ, ε},
constants A′, B′ and C′ are fitting parameters. Therefore, all parameters (νx , νy , λx , λy) required
to monitor the time evolution of silicon surface morphology during IBS are well defined.
40 45 50 55 60 65
0
1
2
3
4
5
6
= 0.2 keV
= 0.5 keV
= 2 keV
= 10 keV
Y
0
(
a
to
m
s
/i
o
n
)
(deg)
2 4 6 8 10
0
1
2
3
4
5
6
= 40
0
= 48
0
= 56
0
= 64
0
Y
0
(
a
to
m
s
/i
o
n
)
(keV)
(a) (b)
Figure 2. Sputtering yield for Ar+ in Si at (a) fixed ion energy, (b) fixed incidence angle.
4. Surface morphology change during sputtering
4.1. Phase diagram and typical patterns
Firstly, let us compute a phase diagram ε(θ) defining domains for different surface patterns of
silicon sputtered by Ar+ ions. To this end, we shall monitor a sign change of surface tension νx and
tilt-dependent erosion rates λx and λy (as it was mentioned above νy is always less than 0). The
corresponding phase diagram indicating possible patterns is shown in figure 3. We need to stress
that in the related interval for both the ion energy and the incidence angle except νy < 0, one has
λy < 0. From figure 3 it follows that plane (θ, ε) is divided by three curves into five domains A,
B, C, D and E. If one crosses the dash-dot curve, then quantity νx changes it sign. Therefore, in
the linear regime at small incidence angles θ (domain A), instability of the silicon surface occurs
in both x and y directions due to νy < 0 and νx < 0. In the domain E (at large θ) in the linear
regime, patterns are stable in x-direction due to νx > 0. At large times (nonlinear regime) the
23602-5
V.O. Kharchenko, D.O. Kharchenko
surface morphology is governed by nonlinear parameters λx and λy . Solid curve in figure 3 divides
domains characterized by λx < 0 and λx > 0. Therefore, between solid and dash-dot lines only
λx is positive (domains C and D), whereas in the domain E both νx and λx are positive. Dash
curve corresponds to the condition νx = νy . Hence, before the dash curve (domains A and C)
when νx < νy, vertical elongated surface structures should be formed, whereas after the dash curve
(domains B, D and E), the corresponding structures should be of a horizontal elongated type.
Figure 3. Phase diagram and typical surface patterns.
To illustrate typical structures in each domain in figure 3 we numerically solve equation (1)
on quadratic lattice L × L of the linear size L = 256 with periodic boundary conditions. Spa-
tial derivatives of the second and fourth orders were computed according to the standard finite-
difference scheme; the nonlinear term (∇h)2 was computed according to the scheme proposed
in references [18, 19]. We have used Gaussian initial conditions taking 〈h(r, t = 0)〉 = 0 and
〈(δh)2〉 = 0.1; the integration time step is ∆t = 0.005 and the space step is ℓ = 1.
Typical surface patterns in domains (A–E) are shown in figure 3. It is seen that on the left hand
side of the solid curve when νy < 0 and νx < 0, pattern type of holes is realized (see snapshots
A and B). It follows that patterns realized at high energy ions are characterized by small size
(see snapshot A), whereas at small ε one has large-scale patterns (see snapshot B)1. Moreover,
orientation of holes in points A and B is different. It is defined by a minimal value of both νx
and νy . Structures, shown by snapshots C and D (ripples) are characterized by positive value of
parameter λx , which defines nonlinear effects in x-direction. An orientation of the corresponding
ripples is defined by a minimal value of both νx and νy as in the previous case. Hence, as far as
νx < νy from the left of the dashed curve in figure 3, the related patterns in domains A and C are
elongated in y direction. On the contrary, in snapshots B, D, and E there are horizontal elongated
structures. Structures in snapshots A, B, C and D are characterized by instabilities in both x and y
directions due to νx < 0 and νy < 0. In the domain, indicated by point E due to νx > 0, structures
are stable in x direction.
The obtained phase diagram is in good correspondence with the results of experimental studies
of the dynamics of the surface Si(001) sputtered by Ar+ ions [20], where according to the exper-
imentally obtained a phase diagram in the plane “ion energy – angle of incidence” it was shown
that if the angle of incidence or ion energy varies, then orientation of ripples can be changed.
1Dependence of wavelength versus ion energy at fixed values for incidence angle will be discussed later.
23602-6
Morphology change of the silicon surface
It is important that in the considered interval of incidence angle, the obtained phase diagram in
figure 3 is topologically similar to the experimental one. However, nonlinear KS equation (1) with
parameters defined by equations (2)–(4) does not presume a stable smooth surface because νy is a
negative quantity.
Figure 4. Relative number of islands at ε = 2 keV and characteristic snapshots at θ = 50
◦ (top)
and θ = 63
◦ (bottom).
To prove that holes and ripples are stable in time, let us consider the dynamics of the surface
morphology change. We analyze two representative kinds of patterns shown in figure 3 as snapshots
A and E and compute the number of islands for each pattern in time. To this end, we have cut
the surface h(x, y) at an average height level 〈h〉 and calculated the relative number of islands
N/Nmax at fixed times, where Nmax is a maximal value of islands. In our computation scheme we
used the following definition for the island: all points on the surface with h < 〈h〉 belonging to
one manifold having a closed boundary, form an island. The corresponding boundary of the island
was obtained according to the percolation model formalism. Results for relative number of islands
were averaged over 20 independent runs. Typical evolution of the number of islands is shown in
figure 4 at ε = 2 keV for θ = 50◦ and θ = 63◦. It is seen that the relative number of islands grows
at small time interval that corresponds to processes of the formation of islands. At intermediate
times, the relative number of islands decreases which means a realization of coalescence processes.
It is important that in the process of ripple formation the coalescence regime is well pronounced
(see empty circles). On the contrary, for the process of nanohole formation (filled circles), this such
regime is only weakly observed. At large times one has a stationary behavior of the relative number
of islands. Hence, processes of ripple and nanohole formation are stationary ones: at large time
intervals the averaged number of islands does not change in time. Snapshots of the silicon surface
morphology for θ = 50◦ and θ = 63◦ at t = 0, 40, 100 and 400 seconds are shown in figure 4 in the
top and in the bottom of the figure, respectively.
23602-7
V.O. Kharchenko, D.O. Kharchenko
4.2. Wavelength dependence on the ion energy
Next, let us study the wavelength dependence on the incident ion energy and on the angle of
incidence. As it was shown earlier in the Bradley-Harper theory, a relation between parameters
νx and νy determines the orientation of surface patterns. The wavelength of selected patterns
in the corresponding direction is defined as follows: Λx,y = 2π
√
2K/|νminx,y|, where νmin x,y =
min(νx , νy). One needs to note that following the phase diagram shown in figure 3, a variation
in the ion energy at fixed angles of incidence causes a change in the orientation of structures: at
small ε one has structures elongated in y direction, whereas at large ε, structures are horizontally
elongated. Corresponding dependencies of the wavelength versus ion energy at fixed values for
incidence angle are shown in figure 5. It is seen that the wavelength decreases with the ion energy
Figure 5. Dependence of the wavelength on the ion energy (the scaling exponent δ dependence
on the incident angle is shown in insertion).
growth according to a power law and varies in the interval from 100 nm to 1 µm. This result is in
good correspondence with experimental data for sputtering of the silicon target by Ar+ ions [5].
It is principally important that as far as the penetration depth depends on the ion energy in a
nonlinear manner (see the insert in figure 1) one can expect a deviation from the Bradley-Harper
wavelength asymptote Λ ∼ ε−1/2. In figure 5 it is seen that at small and large incidence angles
(see dot and dash lines, respectively) one has linear dependencies in log-log plot characterized by
the corresponding unique slope. However, at intermediate values for θ related to the dash curve in
figure 3, the dependence Λ(ε) has a kink. This kink means a change in the orientation of patterns.
In such a case one has two slopes at small energies, i.e., before kink, one has selected the patterns
characterized by Λx , whereas at large energies the patterns are defined by Λy (see solid line and
asymptotics in figure 5). Therefore, at small ion energies the patterns are oriented in y direction,
whereas at large ion energies they are oriented in x direction. Hence, for the wavelength dependence
on the ion energy one can write Λ ∼ ε−δ where the scaling exponent δ is defined as a slope of the
dependence Λ(ε) in double logarithmic plot before and after the kink. The dependence of the scaling
exponent versus incidence angle is shown as an insert in figure 5. One can see that for the described
interval for the angle of incidence δ > 1/2. Moreover, at small and large θ the exponent δ does not
essentially change it values, whereas in the interval for θ when Λ(ε) has a kink, the exponent δ varies
from 0.65 toward 1.05. We should note that the obtained picture is realized when the incoming ion
flux J and temperature T are constants. In the opposite case, variation in J and T leads to the
known asymptotes: Λ ∝ J−1/2, Λ ∝ T−1/2 exp(−Ea/2T ), where Ea is an activation energy.
23602-8
Morphology change of the silicon surface
4.3. Scaling properties of patterns
Finally, let us study the scaling properties of the surface patterns, computing growth and
roughness exponents. To this end, we analyze a height-height correlation function Ch(r, t) = 〈[h(r+
r′, t)−h(r′, t)]2〉. In the framework of dynamic scaling hypothesis following references [21, 22], one
arrives at scaling relations Ch(t) ∝ t2β , Ch(r) ∝ r2α, allowing one to define the growth exponent
β and the roughness exponent α.
In reference [23] it was shown that there is a set of exponents {β} describing the universal
behavior of the correlation function at early stages of the system evolution. At late times where
a true scaling regime is observed there is a unique value for β. The roughness exponent α takes
similar values at different time windows and can be considered as a constant depending on the
system parameters only. From practical viewpoint, the analysis of the surface growth is urgent at
large time intervals where the true scaling regime is observed and there is no essential difference
in values β at different time windows. It is known that anisotropic surfaces studied in this paper
may exhibit a more complex dynamic scaling behaviour than isotropic ones because anisotropy
of the surface is reflected in lateral correlations of the surface roughness [24, 25]. In reference [6]
it was proposed to use local roughness scales αx , αy in the directions normal and parallel to the
projection direction of the ion beam. Values for growth and roughness exponents together with
surface tensions νx and νy at ε = 2 keV and fixed values for incidence angle are presented in table 1.
It is seen that when νx < νy, a relation αx > αy is realized due to orientation of the structures in
y direction. On the contrary, if νx > νy holds, then one has αx < αy . Hence, making an analysis
of the obtained scaling exponents, one can conclude that if structures are oriented in y direction,
then roughness is larger in x-direction and vice versa (compare patterns in snapshots A, C, D and
E in figure 3 with exponents in table 1). The obtained results for growth and roughness exponents
are in good correspondence with experimental studies of the silicon target sputtered by Ar+ ions
(see [6, 26]).
Table 1. Growth and roughness scaling exponents at ε = 2 keV.
θ αx αy β νx νy
50◦ 0.90 0.82 0.23 –0.222 –0.151
55◦ 0.94 0.90 0.22 –0.137 –0.127
58◦ 0.90 0.95 0.21 –0.067 –0.112
63◦ 0.89 0.99 0.17 0.086 –0.087
5. Conclusions
Two-level modeling for nanoscale pattern formation on silicon target induced by Ar+ ion sput-
tering has been reported. We have used Monte-Carlo simulations and a continuum approach based
on the Bradley-Harper theory. It was shown that for the described system, the dependencies of
the averaged penetration depth of the incident ion and the corresponding distribution widths of
the deposited energy in directions parallel and perpendicular to the incoming beam versus ion
energy are of the power-law form. Varying the incoming ion energy and ion incidence angle, we
have defined the sputtering yield with the help of Monte-Carlo simulations. The obtained results
have been used in the modified Bradley-Harper theory within the framework of two-scale modeling
scheme.
We have computed a phase diagram for control parameters: i.e., incidence angle and ion energy
that defines possible patterns on silicon target sputtered by Ar+ ions. It was shown that at small
incidence angles, nanohole patterns are realized, whereas at large incidence angles, pattern type of
ripples is observed. Analyzing the morphology change of silicon surface we have shown that during
the system evolution, the number of nanoholes/ripples becomes constant, indicating stability of
the obtained structures in time.
23602-9
V.O. Kharchenko, D.O. Kharchenko
We have found that there are deviations from the Bradley-Harper asymptotics for the wave-
length dependence on the ion energy. Moreover, when the orientation of patterns changes, a kink
is realized in such asymptotics. The exponent of such power-law asymptotics depends on the angle
of incidence. At fixed values for incidence angle one has two scaling exponents related to small
and large values for the ion energy according to a change in the orientation of structures. While
studying the scaling characteristics of the height-height correlation function, the growth exponent
together with longitudinal and transverse roughness exponents are obtained for different values of
incidence angle at a fixed ion energy. It was shown that relations between roughness exponents are
defined through relations between corresponding effective surface tensions.
The results obtained in a two-scale modeling scheme are in good correspondence with the known
theoretical and experimental data for sputtering of silicon target by Ar+ ions in the considered
interval of values for incidence angle of ions, the incoming ion energy, temperature and ion flux [5,
6, 20, 26, 27].
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Morphology change of the silicon surface
Змiна морфологiї поверхнi кремнiю при розпиленнi його
iонами аргону
В.О. Харченко, Д.О. Харченко
Iнститут прикладної фiзики НАН України, вул. Петропавлiвська 58, 40030 Суми, Україна
Проводиться теоретичне дослiдження процесiв змiни морфологiї поверхнi кремнiю при розпиленнi
його iонами аргону в рамках дворiвневої схеми, що враховує методи Монте-Карло та модифiко-
вану теорiю Бредлi-Харпера. Отримано та проаналiзовано фазову дiаграму у площинi кут падiння
налiтаючого iону та енергiя iону, що iлюструє можливi типи поверхневих нано-структур. Отримано
узагальнену степеневу залежнiсть довжини хвилi отриманих поверхневих структур вiд енергiї на-
лiтаючих iонiв. Проаналiзовано показник росту i повздовжнiй та поперечний показники шорсткостi
отриманих поверхонь.
Ключовi слова: iонне розпилення, морфологiя поверхнi, нано-структури
23602-11
Introduction
Theoretical model
Monte-Carlo modelling
Surface morphology change during sputtering
Phase diagram and typical patterns
Wavelength dependence on the ion energy
Scaling properties of patterns
Conclusions
|