UNIVERSITY OF MINNESOTA
This is to certify that I have examined this copy of a master’s thesis by
Mayur Suri
and have found that it is complete and satisfactory in all respects,
and that any and all revisions required by the final
examining committee have been made.
Traian Dumitrică
Name of Faculty Adviser(s)
Signature of Faculty Adviser(s)
Date
GRADUATE SCHOOL
Phase-Transition Assisted Deposition of
Passivated Nanospheres
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Mayur Suri
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
Traian Dumitrică, Adviser
c Mayur Suri 2008
i
Abstract
Large-scale atomistic simulations considering a 5 nm in radius H-passivated
Si nanosphere that impacts with relatively low energies onto a H-passivated
Si substrate reveal a transition between two fundamental collision modes. At
impacting speeds of less than ∼ 1000 m/s, particle-reflection dominates. At
this lower-impact speed, no significant phase transition within the particle is
observed. Adhesion is primarily motivated by bond formation between particle
and substrate. Therefore, passivated nanoparticles could not be captured. At
increased speeds the partial onset in the nanosphere of a β-tin phase on the
approach followed by amorphous-Si phase on the recoil is an efficient dissipative route that promotes particle-capture. In spite of significant deformation,
the integrity of the deposited nanosphere is retained. This project explains
the efficient fabrication of nanoparticulate films by deposition under hypersonic speeds. Preliminary impact simulations are also performed for C and SiC
nanoparticles.
CONTENTS
ii
Contents
List of Figures
1 Introduction to Molecular Dynamics
v
1
1.1
The Molecular Dynamics Method . . . . . . . . . . . . . . . . . . . .
2
1.2
The velocity-Verlet Time Evolution Algorithm . . . . . . . . . . . . .
3
1.3
Periodic Boundary Conditions and Linked-List Method . . . . . . . .
6
1.4
Temperature Control Algorithm . . . . . . . . . . . . . . . . . . . . .
8
1.5
The Interatomic Potential . . . . . . . . . . . . . . . . . . . . . . . .
11
2 The Tersoff Potential
2.1
The Functional Form of the Tersoff Potential . . . . . . . . . . . . . .
13
14
3 Experimental Deposition of Nanoparticles
21
4 Modeling the Nanoparticle – Substrate System
27
4.1
The Nanocluster and Substrate . . . . . . . . . . . . . . . . . . . . .
27
4.2
Surface Passivation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.3
Simulation Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
5 Molecular Dynamics Results - Impact of Hydrogenated Si Nanoparticles on Bare and Hydrogenated Si Substrate
41
6 More Molecular Dynamics Results - Impact of Si31,075 N413 , C90,326 and
Si12,080 C12,080 Nanoparticles on Bare Si Substrate
54
6.1
Si31,075 N413 Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . .
54
6.2
C90,326 and Si12,080 C12,080 Nanoparticles . . . . . . . . . . . . . . . . .
58
CONTENTS
iii
7 Future Work
62
References
64
LIST OF FIGURES
iv
List of Figures
1
Flow chart for the molecular dynamics method . . . . . . . . . . . . .
4
2
2-D representation of Periodic Boundary Conditions . . . . . . . . . .
7
3
Dynamics of linked-list assignment . . . . . . . . . . . . . . . . . . .
9
4
Comparison of cohesive energies of various bulk-Si structures . . . . .
14
5
Structure of the highly compressed BC-8 state of Si . . . . . . . . . .
17
6
Bonding in the Murty-Atwater Classical Potential . . . . . . . . . . .
18
7
Tersoff Paramteres for Si, H, and N used in this work . . . . . . . . .
19
8
Test Molecules for Tersoff Potential . . . . . . . . . . . . . . . . . . .
20
9
Schematics for the hypersonic plasma particle deposition process . . .
22
10
Schematics for the ocused deposition process . . . . . . . . . . . . . .
23
11
Gear filled with SiC nanoparticles . . . . . . . . . . . . . . . . . . . .
23
12
Ball-and stick representation of a Si nanoparticle of 1 nm in radius . .
28
13
Perspective on the simulated nanoparticle − surface system . . . . . .
29
14
Center of mass (z) with respect to height of the substrate (d) . . . . .
33
15
Example of a Phase Space Plot . . . . . . . . . . . . . . . . . . . . .
33
16
Plot of instantaneous particle temperature at 2000 m/s impact . . . .
34
17
Shape of the plane of contact . . . . . . . . . . . . . . . . . . . . . .
35
18
Evolution of contact radius for 900 m/s impact . . . . . . . . . . . .
36
19
Region of contact stress in the particle (shown in red) . . . . . . . . .
37
20
Evolution of contact stress for 2000 m/s impact . . . . . . . . . . . .
37
21
Atoms in strained nanocluster as a function of cohesive energy − grey
atoms correspond to CE of 4.6 eV while light blue atoms have a CE of
∼ 4.3 eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
LIST OF FIGURES
v
22
Atoms with coordination number 6 versus time . . . . . . . . . . . .
39
23
Radial distribution function . . . . . . . . . . . . . . . . . . . . . . .
40
24
Phase space trajectories for three impacting speeds: (a) 900 m/s, (b)
1, 300 m/s, and 2, 000 m/s. The nanosphere approaches the substrate
with positive V , touches down at δ = 0, reaches V = 0 at maximum
penetration, and recoils with negative V
25
. . . . . . . . . . . . . . . .
43
MD simulations of H-passivated Si nanosphere impacting onto a Hpassivated Si substrate show two collision modes: (a) reflection and
(b) capture. The middle frames show the maximum penetration instant. Only a cross-section is shown and H atoms are not represented.
The color code carries the local PE, with blue (gray) and pink (light
gray) representing atoms with PE larger and smaller than 4.4 eV/atom,
respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
45
Time evolution of (a) contact radius and (b) contact-stress. Variation
of nanosphere’s (c) temperature, and (d) potential energy under two
impacting speeds, V0 = 900 m/s (lower two curves) and V0 = 2, 000 m/s
(upper two curves), and two surface types, H-passivated nanosphere
and substrate (black) and H-passivated nanosphere and bare substrate
(gray). Down arrows mark the particle release instants . . . . . . . .
47
LIST OF FIGURES
27
vi
Pair-correlation function of the H-passivated Si nanosphere at maximum penetration for (a) V0 = 900 m/s and (b) V0 = 2, 000 m/s. Vertical bars mark the neighbor position in cubic Si. (c) Pair-correlation
and (d) bond-angle distributions in the most deformed conical region.
Vertical bars mark neighbor positions and bond angles in bulk β-tin
Si. (e) Unit cells for cubic-diamond (left) and β-tin Si (right) . . . . .
28
RADF at the end of 30 ps for impact at 2000 m/s with hydrogen on
both surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
49
51
BADF at the end of 30 ps for impact at 2000 m/s with hydrogen on
both surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
30
Time evolution of contact energy . . . . . . . . . . . . . . . . . . . .
55
31
Time evolution of contact area for different surface chemistries . . . .
56
32
Time evolution of coordination number plot for SiN . . . . . . . . . .
57
33
Phase space trajectories at 2,000 m/s . . . . . . . . . . . . . . . . . .
57
34
Phase space trajectories for C90,326 nanoparticle . . . . . . . . . . . .
59
35
Restitution for impact of Si nanoparticle with hydrogen on both surfaces 60
36
Phase space trajectories for Si12,080 C12,080 nanoparticle . . . . . . . . .
61
1
Introduction to Molecular Dynamics
For nearly seven decades now, computers have been used as a mathematical laboratory for simulating real-world experiments. A plethora of microscopic simulation
techniques have evolved over time, examples of which are [39]:
− Monte Carlo (MC) simulations
− Classical molecular dynamics (MD)
− Molecular dynamics combined with density function theory (DFT)
− Discrete methods such as Cellular automata, Lattice-Boltzmann method.
Broadly speaking these methods can be largely classified into two categories
Stochastic and Deterministic. Monte Carlo is a prime example of a stochastic method.
MC simulations probe the system by small trial changes to the particle positions. The
energy difference between one configuration and the next is measured, and it is this
difference in energy between successive configurations that is used as a benchmark
for accepting or rejecting the trial move [43]. This algorithm is called the Metropolis
algorithm. MD on the other hand is a deterministic method. It is a technique for
computing the equilibrium and transport properties for many body systems [2]. In
classical MD, the motion of individual atoms within the system is determined by
solving classical equations of motion. The idea that Newtonian mechanics − given
a known potential constraint and initial state of a system − can effectively predict
molecular motion is, in fact, an eighteenth century concept. Laplace [39] foresaw this
when he wrote:
“Given for one instance an intelligence which could comprehend all
the forces by which nature is animated and respective situations of the
1
1.1 The Molecular Dynamics Method
beings who compose it an intelligence sufficiently vast to submit these
data for analysis it would embrace in the same formula the movements
of the greatest bodies of the universe and those of the lightest atoms”
1.1
The Molecular Dynamics Method
Solving the classical equations of motion, step-by-step, is the basis of the Molecular
dynamics method. The equations are:
mi
∂ 2 ri
= fi
∂2t
fi = −
∂U (RN )
∂ri
(1.1)
(1.2)
where mi stands for the mass of the atom, ri is the coordinate of the atom i. The
force acting on the atom is the negative derivative of the potential energy function
U (RN ). RN = (r1 , r2 ....rN ) encompasses the whole set of N atoms.
Using a Molecular Dynamics simulation entails making some choices. The most
important choices are listed below.
1. Choice of Interatomic potential − U (RN ).
2. Choice of time integrator to solve equation (1.1).
3. Choice of Periodic Boundary Conditions (PBC).
4. Statistical constraints, like constant energy or temperature, imposed on the
system.
Each of theses choices will be explained in some detail in later sections. In particular, parameters that were used in this project will be emphasized.
2
1.2 The velocity-Verlet Time Evolution Algorithm
A flow chart to sequentially map out steps in a MD simulation is good way of
understanding the process.
Figure 1: Flow chart for the molecular dynamics method
1.2
The velocity-Verlet Time Evolution Algorithm
Since the central equation relating force on atoms to acceleration is a second order differential equation, namely (1.1), a suitable time integrator is crucial to the
3
1.2 The velocity-Verlet Time Evolution Algorithm
simulation’s success. Choice of the integrator dictates, among other things, energy
conservation (short term or long term), accuracy in trajectory, length of permitted
step size, storage requirement etc. Ever since it was it was first introduced in 1960,
the Verlet algorithm has been a popular choice. For simplicity the algorithm is discussed for a single particle of coordinate x̃ and velocity ṽ. The following description
of the algorithm closely follows references [15] and [20].
The Verlet equations are:
x̃(t + 4t) = x̃(t) + ṽ(t)4t +
ã(t)4t2 b̃(t)4t3
+
+ o(4t4 )
2
6
(1.3)
x̃(t − 4t) = x̃(t) − ṽ(t)4t +
ã(t)4t2 b̃(t)4t3
−
+ o(4t4 )
2
6
(1.4)
Adding equations (1.3) and (1.4) gives:
x̃(t + 4t) = 2x̃(t) − x̃(t − 4t) + ã(t)4t2 + o(4t4 )
(1.5)
To calculate the velocity from the equations for position, the following equation is
used:
ṽ(t) =
(x̃(t + 4t) + x̃(t − 4t)) + o(4t3 )
24t
(1.6)
Note that the above equation does not estimate the instantaneous velocity for a
given position and also requires the storage of two timesteps to estimate the velocity.
Further, it is one order less accurate than the equations for position.
The TROCADERO program [40], used for all simulations here, uses a slightly
modified version called the velocity-Verlet algorithm. It is a related scheme for time
progression which stores information from only one time step to estimate the velocity.
4
1.3 Periodic Boundary Conditions and Linked-List Method
Reference [15], details the derivation of the velocity-Verlet equations. For brevity, only
the equations are listed below :
x̃(t + 4t) = x̃(t) + ṽ(t)4t +
ṽ(t + 4t) = ṽ(t) +
1.3
ã(t)4t2
2
ã(t) + ã(t + 4t)
4t
2
(1.7)
(1.8)
Periodic Boundary Conditions and Linked-List Method
Periodic Boundary Conditions are a crucial concept necessary to maintain the relevance of simulation results to empirical observations. The following summary is based
largely on reference [2].
Consider a simulation of about 1000 atoms, arranged in a 10×10×10 cubic matrix.
As can be readily noted this arrangement leaves nearly half of all atoms in the system
on the surface of the cube. Surface atoms carry the baggage of more reactivity, due to
the presence of unsaturated bonds. Such fundamental physical differences will alter
the properties of the system, compared to a experimentally observed nanocrystal of
1000 atoms, which is likely found within a larger arrangement of atoms.
To tackle this problem, the cube is embedded in a matrix of its own replicas
and each atom sees its closest neighbors. If an atom leaves the basic simulation box,
another incoming image atom substitutes for it. This artificially imposed periodicity is
called periodic boundary conditions (PBC). The figure below gives a two dimensional
representation of PBC. The central cell is the actual simulation box, enmeshed by its
replicas. Note the circle in the final frame that indicates the newly - paired atoms.
Most classical potentials have a finite spatial range, beyond which they do not
5
1.3 Periodic Boundary Conditions and Linked-List Method
Figure 2: 2-D representation of Periodic Boundary Conditions
affect the interatomic forces. Mathematically this interatomic potential, V (rij ), can
be written as:
V (rij ) = 0 for rij > rcutof f
rcutof f corresponds to the maximum distance for which the potential function is
effective.
During the course of the simulation, a large number of pair-wise distance calculations would be necessary, just to determine whether a pair of atoms do or do not
interact. Just to give an idea about the scale of these iterations, if the system has N
elements, 0.5N (N − 1) distinct pairs of atoms arise. This presents a huge computational stumbling block to faster simulations. In reference [50], Verlet first suggested
the Linked-List algorithm designed to deal with this issue. The main steps are:
1. Creating a skin surrounding each atom, such that the radius of the cut-off sphere
rlist is a slightly greater than rcutof f .
2. During subsequent MD steps, only rij < rlist are considered.
6
1.4 Temperature Control Algorithm
3. These atoms are linked together in a list. A head atom is associated with each
cell.
4. Subsequent iterations consider only the atoms within a linked list.
5. The cells are redefined every few time steps keeping in mind the motion of the
atoms within the system.
Figure 3 represents this process. The red circles represent atoms outside of the
link-list zone. The outer circle indicates the extent of spatial range of the linked-list.
The inner circle indicates the potential’s cut off limit. All atoms within the outer
circle are part of the linked list, but only the blue atoms are allowed to interact with
the central atom. It is important to note that linked lists are a valid concept only as
long as the red atoms do not enter the inner circle. When this happens, the list has
to be remade to account for new entrants.
Once atoms move out to encroach into the rcutof f zone, as the second frame
shows, the list has to be redefined to link only relevant atoms. This is schematically
represented in the third frame. The shape of the cell is shown as a circle only for
convenience of representation. This shape can vary; in fact, TROCADERO uses
cubical cells.
1.4
Temperature Control Algorithm
Two kinds of statistical ensembles used in our simulations are NVE and NVT. NVE
stands for constant number of atoms, constant volume, and constant energy. NVT
stands for constant number of atoms, constant volume, and constant temperature.
Reference [43] lists multiple methods of temperature control such as:
1. Differential Thermostat
7
1.4 Temperature Control Algorithm
(a) Initial cell assignment.
(b) Migration of atoms into cell.
(c) Redefining original cell to form new
linked list.
Figure 3: Dynamics of linked-list assignment
8
1.4 Temperature Control Algorithm
2. Proportional Thermostat
3. Stochastic Thermostat
4. Integral Thermostat.
The Langevin thermostat used in this project is an example of stochastic thermostat. In this method, select degrees of freedom in the system are subject to collisions
with virtual particles which act as “dampers”. This is theoretically similar to simulating Brownian motion in a field of viscous force, and has parallels in the field
of aerosol particulate simulation [8]. The reason for using Langevin Thermostat is
because it has been determined to be well suited for both low speed and high speed
collision simulations [24]. The following description of the Langevin thermostat is
taken mostly from references [6] and [30].
The Langevin stochastic equation is:
∂U
∂vi
=−
− γvi + G
∂t
∂ri
(1.9)
Here, as before, vi stands for a component of velocity, ri is the coordinate of the
atom i and U is the potential energy. γ is the friction coeffecient (similar to viscosity), and G (random Gaussian noise) is determined by second fluctuation-dissipation
theorm, as:
hGi (ti )Gj (tj )i ≥ 2γKb T δij δ(ti − tj )
(1.10)
Here Kb is Boltzmann constant, T represents the instant temperature. As γ
increases, so do thermal fluctuations, and loss of particle memory. Reference [43]
elucidates how this method is applied to MD :
9
1.5 The Interatomic Potential
1. Particles collide against imaginary damping particles. The velocities of these
imaginary particles are determined by a Maxwell-Boltzmann velocity distribution at
To , which is the reference temperature for the system.
2. The memory of trajectory they have is lost (motion is gradually randomized).
3. If collision frequency is too high a stronger loss of particle memory results, leading
to a decay of dynamic system functions.
1.5
The Interatomic Potential
Potential energy functions define the interaction between atoms in a system. If a
potential function is designed to estimate the interaction of N atoms, it is called
a N-body potential function. The simplest form it can assume is, of course, a 2body potential. The very first MD simulations for atoms by A. Rahman in 1964
[33] used the Lennard-Jones potential to describe the interaction of atoms in Noble
gases. The classic Lennard-Jones potential defines interaction only as a function of
distance. Atoms very close to each other have strong repulsion and when farther apart
dipole-dipole interaction causes attraction between atoms. Mathematically this can
be represented as:
h σ
σ i
V (r) = 4 ( )12 − ( )6
r
r
(1.11)
Here σ is the interatomic distance and is the depth of the potential well.
While this form of potential can be used effectively for ionic bonds or metallic bonds, which typically have stable FCC structures, more directional bonds (like
covalent bonds) cannot be suitably described. In this case, the potential function
10
1.5 The Interatomic Potential
will have to include an angle dependent component, and account for bond order.
Examples of potentials suitable for covalent systems are the Tersoff Potential [44],
Environment Dependent Interatomic Potential [4] and Murty-Atwater Potential [36].
Note that the Tersoff potential chosen for our impact simulations of Si, can not only
describe Si in the cubic-diamond form, but also provides a robust description of Si’s
different pressure induced phases and melting. The choice of potential and its form
are described in the following chapter.
11
2
The Tersoff Potential
As mentioned in the previous chapter, the choice of a suitable interatomic potential
is crucial to the credibility of the molecular dynamics simulation. S.J.Cook et al. [10]
and Balamane et al. [3], did a thorough comparison of the commonly used potentials
for Si. The potentials compared were:
(a) 2 and 3 body Tersoff potential (T2, T3) [44]
(b) Biswas and Hamann (BH) [7]
(c) Stillinger and Webber (SW) [42]
(d) Dodson (DOD) [13]
(e) Pearson, Takai, Halicioglu and Tiller (HTPT) [32]
All potentials reasonably described the cohesive energies and elastic mechanical
properties of diamond cubic Si. However when pressure induced changes in structure
were simulated, stark differences were brought out between the potentials. The first
change of structure when Si is subjected to high pressure ( ≥15 GPa), is to β-tin
phase, where each Si has a coordination number 6. The Tersoff potential correctly
predicted the change of phase and was the only potential to correctly estimate the
bulk modulus of the β-tin phase [3]. The figure below indicates the cohesive energies
of consecutive Si structures formed by pressure induced phase transformation. The
vertical axis corresponds to energies derived from using each of the 5 five potentials
listed above, whereas the horizontal axis corresponds to energies derived from using
density function theory. It can be seen that Tersoff potential correctly estimates the
energies of β-tin structure and the BC-8 structure that follows - at 4.3 eV and 4.4 eV
respectively.
Since this project deals with particle-substrate collisions, a capability to describe
12
2.1 The Functional Form of the Tersoff Potential
Figure 4: Comparison of cohesive energies of various bulk-Si structures
also the amorphization of Si is important. References [3] and [27] both use the Tersoff
potential to describe amorphous Si at 300K. It can be found that amorphous Si
density (2.2901 g/cm3 ) and average amorphous coordination (4.16) are consistent
with experimental results [3].
2.1
The Functional Form of the Tersoff Potential
Introduced in 1988 by Tersoff [44], the Tersoff potential is one of the most widely used
potentials to describe Si structures. The functional form of the structure is described
below. Details of parameters used in our simulations and our trial runs with simple
molecules to test them follow. The concept behind this potential is based on LinusPaulings bond order theory. As explained in reference [44], the strength of a bond
depends directly on the local environment, with more bonds inversely affecting the
13
2.1 The Functional Form of the Tersoff Potential
strength.
The basic form of the potential is:
U=
1X
Vij
2 i6=j
(2.12)
Here Vij represents the interaction energy of the bond between atom i and another
atom j located within its neighbor shell.
Vij = fc (rij ) [fR (rij ) + bij fA (rij )]
(2.13)
Here fc , fR and fA are smoothing function, repulsive function, and attractive
function respectively.
fR (r) = A × e−λ1 r fA (r) = −B −λ2 r
Given the condition, (RD ) ≤ r ≤ (R + D),
π r−R
1 1
fC (r) = − × sin
×
2 2
2
D
(2.14)
fC (r) becomes unity or zero on either side of this limit. R and D are chosen to include
1st neighbor shells only for Si.
bij =
1
(1 +
1
β n nij )( 2n
)
(2.14)
bij is the special term here, standing for the attractive component of the potential
function. The term ij defines the coordination number of atom i.
14
2.1 The Functional Form of the Tersoff Potential
ij =
X
3
fC (rij )g(θijk )eλ3 (rij −rik )
3
(2.14)
k6=ij
In the above function θijk represents the angle formed between the atoms i, j and k,
where k is a 3rd atom found within the neighbor shell of i. In other words the energy
of the bond between i and j is altered by the presence of a 3rd atom k at a distance
of rik from i. The function g(θijk ) determines how sharply θijk influences the strength
of Vij .
Figure 5: Structure of the highly compressed BC-8 state of Si
References [27] and [41] both use the Tersoff potential to simulate phase transformations caused by the nano-indentation of Si substrates. Their results, which
closely follows experimentally observed phase transformations with indented Si sub-
15
2.1 The Functional Form of the Tersoff Potential
strates confirm that the Tersoff potential provides a robust description for pressure
induced phase transformations in Si structure. Simulations in this project will not
be restricted to pure Si. To study the effect of surface reactivity on the adhesion of
impacting nanoparticles with the substrate, Si atoms on the particle surface are saturated with hydrogen atoms. To get a better insight into the role of surface chemistry,
impact simulations are also done with nitrogen-saturated Si Nanoparticles. Both of
the above simulations would, consequently, require potentials which are capable of
describing Si-H and Si-N bonds.
The Murty-Atwater 3-body potential [36] which is the most commonly used for SiH interaction fundamentally modifies the form of the Tersoff potential. Also separate
parameters are needed for each of the 9 kinds of 3-body interactions that can occur
between Si and H atoms, shown in the figure below. It is computationally demanding
to analyze each 3-body interaction between Si and H atoms to determine which of
the nine categories they fall into (with each category having its own set of interaction
parameters). Also, this potential is not compatible with Si-N bond descriptions.
Figure 6: Bonding in the Murty-Atwater Classical Potential
The potential described by De Brito Mota and co-workers [11] overcomes these
16
2.1 The Functional Form of the Tersoff Potential
shortcomings by retaining the form of the Tersoff potential and modifying its parameters to describe different bonds. For example, consider the attraction component
of the Tersoff potential, bij . In the de Brito Mota scheme, bij varies according to
the interacting species. When Si interacts with H, bij is reduced by a factor of 0.78,
whereas when Ai interacts with N, bij is reduced by a factor of 0.65. When like
species interact bij is left unmodified. Also, to our knowledge it is the only potential
to describe Si-H-N interactions successfully within the same functional form. The
parameters used for each element are given below. Small molecules of Si with H and
N were simulated with de Brito Motas parameters, and it was found that their bond
lengths and bond angles compare favorably with experimentally found values. For
example, Si-N bond length was found to be 1.1 Å compared to the theoretical value
of 0.98 Å and the angle for both simulated and theoretical values centered around
120o . A sampling of these results are also given below.
Figure 7: Tersoff Paramteres for Si, H, and N used in this work
17
2.1 The Functional Form of the Tersoff Potential
(a) Si2 H6
(b) SiH4
Figure 8: Test Molecules for Tersoff Potential
18
3
Experimental Deposition of Nanoparticles
Nanoparticle deposition by Hypersonic Plasma Particle Deposition (HPPD) is a
method for deposition of Si particles that was developed in the late 1990’s [37]. A
schematic representation of the deposition system is shown in Figure 9.
A brief explanation of the process is as follows: The thermal plasma is injected
with reactants. The plasmas elemental constituents are forced through the high temperature nozzle where they nucleate into nanoparticles - onto a region of considerably
lower pressure. (Typically the nozzle sees a pressure drop from ∼ 50 KPa to 300 Pa
and a 2,000 K temperature difference.) Such huge pressure and temperature gradients
create a hypersonic nanoparticle flow. Indeed, particle velocities range between 1,000
to 2,000 m/s, before they impact with the substrate. HPPD aims to capitalize on
these high impact velocities to make films with high deposition efficiency and density.
Materials successfully depositied by HPPD include combinations of Si, Ti, C and N
[17].
Methods to implement a more focused deposition of nanoparticles onto selected regions of substrates have been developed by McMurry and coworkers [28], [29]. Aerosol
flows are directed through a series of small orifices in thin film plates. Particles with
high Stokes numbers are filtered out because they are unable to follow the rapid
contraction/expansion needed to pass through a series of such aerodynamic lenses.
Ultimately a highly collimated, focused beam exits the final orifice. Aerodynamic
lenses were used to fabricate microstructures by focused nanoparticle-beam deposition. This process is represented schematically in Figure 10 [12].
An example of a micromachined Si-gear filled with SiC nanoparticles deposited
by particle-beam [19] is show in Figure 11.
19
Figure 9: Schematics for the hypersonic plasma particle deposition process
Figure 10: Schematics for the ocused deposition process
20
Figure 11: Gear filled with SiC nanoparticles
The fundamental understanding of nanoparticle-surface collision modes in the
low-energy regime (of less than ∼ 1 eV/atom) is of considerable importance because
achieving efficient sticking to surfaces with preservation of a grainy structure are key
issues for creating new materials. Chemical passivation in the gas-phase is essential
for preventing coalescence and large particle growth but makes the deposition step
challenging. Deposition strategies have been developed, some necessitating the creation of defects in the substrate in order to pin the impinging nanoparticles [31], which
otherwise would be reflected. Remarkably, impact in the newly available regime of
hypersonic speeds leads to efficient capturing of nanoparticles as small as 2 nm in
radii [17].
The achieved deposition of the well-studied surface-coated Si nanospheres [16],
as demonstrated by the produced dense particulate films [17], has been especially
puzzling for a number of reasons: (i) In macrosphere-surface impact, inelastic behavior
is usually due to the nucleation of dislocations [26], which serves as a contact-stress
release mechanism. At the nano-scale this mechanism is unlikely to operate due to
both the small time and size scales. Indeed, the impact duration (τ ) can be estimated
with the classical Hertz theory [26] applied to a nanosphere-plane central collision,
as:
21
τ =(
5π 2/5 ρ 2/5
) ( ) RV −1/5
4
E
In this equation, substituting the density of Si of 2, 330 kg/m3 , a Young’s modulus
value of 127 GPa, R of 1 nm , and V of 1 m/s, we obtain a contact time duration of
∼ 3 ps (1 ps=10−12 s).
The obtain ps durations for τ means that the nanosphere experiences very high
strain-rates. As indicated by MD investigations, under such extreme conditions nanomaterials are exhibiting new behaviors. For example, a high-rate compression of
metallic nanowires creates amorphization [25] rather than dislocations. Furthermore,
it is also accepted that even under slower applied strain-rates, dislocations cannot
be accommodated in nanoparticles with dimensions below a critical size. Supporting
this point, recent MD [49] obtained that bare Si nanospheres experience a first-order
phase-transition under severe compression. (ii) The Si nanospheres have been mechanically compressed with a nanoindentor tip and were found superhard [16], i.e.,
very large pressures, up to four times larger than in bulk were needed to generate
yield. Thus, any plastic mechanism requires high contact pressures and it is not
known whether such pressures are generated during the still low-energetic hypersonic
impact. Finally (iii), the surface chemistry plays an important role as strong adhesion
promotes sticking. In the ideal case, previous MD obtained that bare Si nanospheres
are always captured by the bare Si substrate due to the strong covalent bonding [48].
However, the formation of new Si-Si bonds is hindered when chemically passivated
contacts are involved. Supporting this point, MD indicated that the coalescence of
Si nanoparticles, as seen in references [21] and [22], was significantly slowed down
when surfaces were H-passivated. Thus, a short τ inhibits adhesion and promotes
22
particle reflection.
Hence, pertinent questions to raise here are: How does high velocity impact affect
the structure of the nanoparticle? How does structural transition within the nanoparticle affect adhesion with the substrate? How does particle surface passivity influence
adhesion during hypersonic impact with an equally passive surface? These are the
fundamental questions that this project tries to answer through MD simulations.
23
4
Modeling the Nanoparticle – Substrate System
4.1
The Nanocluster and Substrate
The standard Si nanoparticle used in all simulations has a radius of 5 nm. Si nanoclusters, and in general most nanoclusters formed by covalent atomic bonds like C
and Ge have complex structures. The shapes of these nanoclusters are determined
by the number of atoms that form them. Small clusters have high surface to volume
ratio and a number of free bonds for atoms on the surface. These dangling bonds
can cause radical surface re-alignment. To illustrate, below is a figure of an energetically favorable small Si nanocluster, 1 nm in diameter and comprised of 147 atoms equilibrated at 300 K. The shape of the nanocluster is clearly not spherical
Figure 12: Ball-and stick representation of a Si nanoparticle of 1 nm in radius
24
4.2 Surface Passivation
However at sizes of 5 nm and above, experimental and theoretical studies have
shown that the shape of the Si nanoparticle is spherical, with a diamond cubic structure [16]. To make spherical Si nanospheres, first a cubic block of Si atoms in diamond
structure was created. Spherical particles of the required size were cut off from this
cubic block. These particles were then relaxed, using a conjugate gradient relaxation
scheme, followed by a high temperature (800 K) equilibration over 2000 0.8 fs time
steps, using the Tersoff potential and a velocity verlet time progression scheme. The
cohesive energy of the particle thus obtained was ∼ 4.5 eV/atom.
A (001) type substrate is created with Si atoms arranged in the cubic diamond
configuration. The Tersoff potential has been found to give a good description of this
particular surface orientation [3], with the surface reconstructed with appropriate
dimerization. Substrate size was at 16 × 16 × 5.4 nm. This structure encompassed
71,800 atoms of Si. The bottom 2 layers of the substrate were kept static (i.e. the
3,600 atoms in these layers had their individual velocities nullified). This was done
to withstand the force of particle impact. The layer immediately above, comprising
18,000 atoms had their velocities follow the Langevin dynamics thus enabling them
to act as a thermostat and an energy drain for the system.
4.2
Surface Passivation
One of the central aims of this project is to study the effect that surface passivation
has on the adhesion of nanoparticles to the substrate. This is done by saturating
all the dangling bonds of Si atoms on surface of the particle and substrate with
H atoms. To do this, unsaturated Si bonds on the surface were identified - the
overwhelming majority of surface atoms had just one unsaturated bond, but there
25
4.2 Surface Passivation
Figure 13: Perspective on the simulated nanoparticle − surface system
were a minority of atoms (∼ 90) which had 2 unsaturated bonds. Then, H atoms
were placed at a proximal location of 1.4 Å from the Si atoms. This structure
was put through successive cycles of conjugate gradient optimization, before a 500 K
molecular dynamics run, with a 0.8 fs time step. This time step was found to yield
an energetically stable structure, with the H atoms firmly adhered to the surface Si
atoms. A similar procedure was followed for saturating the substrate. This structure
was then optimized for a stable hydrogen terminated (001) surface. The overall
cohesive energy of this substrate was ∼ 12, 192.4 eV.
Now we can discuss the relevance of choosing 30 ps as the overall MD simulation
time. Upon impact, a fraction of the kinetic energy of the particle is converted to
thermal energy in the substrate. The characteristic time for draining this thermal
energy is a good indicator to determine the suitability of the MD simulation time.
26
4.2 Surface Passivation
Reference [47] carries a more detailed discussion of this process, but in brief - consider
the 1-D heat conduction equation:
∂T
∂2T
=α 2
∂t
∂z
(4.14)
T stands for the substrate temperature. If we insert standard values of Si conductivity (α) at 1000 K, which is 0.24
cm2
s
and a substrate depth of 5.5 nm, the
characteristic heat conduction time would be about 1.5 ps. Thus we choose a net
simulation time which is 20 times this value.
The choice of a 0.8 fs time scale has more to do with the mechanics of the MD
simulation used. Considering that the mass of Si is ∼27.8 times that of a H atom,
using a large time scale at high temperature simulations may mean that the Si atom
has traversed a much smaller distance than the H atom. Energetically unstable
conditions may arise, especially if the distance between the atoms narrows down
enough to trigger very high repulsion. A smaller time scale was found to be more
feasible under such conditions. For simulations involving pure Si, a time steps of 2 fs
could be used without any instability. Reference [21] also gives a similar explanation
for the choice of time step in simulations involving H atoms.
Before each simulation, both the particle and the whole system are equilibrated at
500 K. This temperature is chosen to reflect actual experimental conditions of HPPD
[19]. Finally, the actual impact is initiated by imparting a translational velocity in
the z-direction to the nanoparticle. As a result the particle then drifts down towards
collision with the substrate. Impact happens after 2-3 ps depending on the initial
velocity. For the whole 30 ps a number of indicators are continuously monitored,
27
4.3 Simulation Indicators
to help understand the process of adhesion or reflection. The next section describes
these indicators.
4.3
Simulation Indicators
The quantities which are monitored during the course of the simulation are:
− Phase-Space plots
− Particle temperature
− Particle-substrate contact radius
− Mean distributed contact stress
− Cohesive energy within the particle
− Particle-Surface adhesion energy
− Coordination number within the particle
− Radial distribution function within the particle
− Bond angle distribution function
Phase space plots are an easy graphical way to illustrate the dynamics of particle
impact. Figure 15 is a typical phase space plot for the impact of hydrogentated Si
with a bare substrate at 900 m/s. On the vertical axis is the instantaneous particle
velocity in the z-direction. The horizontal represents the distance between the centre
of the particle and the top layer of the substrate. This quantity is indicated in the
Figure 14. As can be seen in Figure 15, at the beginning of the simulation the particle
hurtles down at velocity of 900 m/s, at a distance of 10 nm above the substrate. When
the radius of the particle is subtracted from the center of mass (CM) values, the plots
give a clear idea of the instantaneous position of the particle with respect to the top
layer of the substrate. In this phase-space plot, the particle ends up bouncing from
28
4.3 Simulation Indicators
the substrate, at a reflected speed of 300 m/s.
Figure 14: Center of mass (z) with respect to height of the substrate (d)
The instantaneous particle temperature:
Instantaneous particle temperature is crucial to understanding the transformation
of translational energy upon impact. This is measured by the following equation:
Np
m X 2
Tp (t) =
v (t)
3Np kB i=1 i
(4.14)
Here, Vi is the instantaneous velocity of the it h atom in the particle. To do
away with the effect of translational velocity, Vi is measured in a coordinate system
that originates at the particle. Figure 16 shows temperature fluctuations within the
particle for 2000 m/s impact.
The particle-substrate contact radius:
When the particle impacts the surface, the contact evolves from a single point
of contact to a circular disk of contact between the particle and the surface. The
adhesion beween the particle and the substrate is largely limited to this contact area,
29
4.3 Simulation Indicators
Figure 15: Example of a Phase Space Plot
Figure 16: Plot of instantaneous particle temperature at 2000 m/s impact
30
4.3 Simulation Indicators
and thus quantifying this area of contact is important. Figure 17 is a picture of the
contact area for 2000 m/s collision between a bare Si particle and a bare substrate.
The figure on the left, is at maximum penetration and the one on the right is at the
final frame of the simulation after 40 ps. Close inspection reveals the contact area
has evolved to a smaller value.
Figure 17: Shape of the plane of contact
Since the contact area is roughly spherical, we can define the contact radius as
the mean distance of the positions of atoms in the plane of contact with respect to
their centre of mass. The equation for this is given by :
Na
2 X
(xi − Xcm )2 + (yi − Ycm )2
a =
N i=1
2
(4.14)
Figure 18 shows contact area radius’ fluctuations within the particle for 900m/s impact.
The mean distributed contact stress:
Quantifying stress due to impact is necessary because, high stress levels are needed
to induce a phase transition within the particle and phase transition is a crucial
channel for the flow of impact energy. Once the contact radius is determined, stress
is simply the force acting downward on the substrate divided by the area of contact.
31
4.3 Simulation Indicators
Figure 18: Evolution of contact radius for 900 m/s impact
In the figure below, the area used to find the contact stress is high lighted in red.
Figure 20 shows contact stress’ fluctuations within the particle for 900 m/s impact.
The particle cohesive energy (CE):
It can be defined as the net binding energy of an atom with its neighbours. An
atom in crystalline Si has a typical cohesive energy of
4.5 eV, whereas a high
pressure form the β-tin form has a lower cohesive energy of ∼ 4.3 eV [49]. Consider,
for example a 2 nm Si nanoparticle subject to uniaxial compression (36 percent) on
either side. High axial stress causes structural changes within the nanoparticle. This
change is directly represented by changes in the individual cohesive energy of atoms.
In the figure below, grey atoms correspond to CE of 4.6 eV while light blue atoms
have a CE of ∼ 4.3 eV.
The particle - surface interaction energy:
Once all atoms of the particle in contact with the substrate are identified, the
32
4.3 Simulation Indicators
Figure 19: Region of contact stress in the particle (shown in red)
Figure 20: Evolution of contact stress for 2000 m/s impact
33
4.3 Simulation Indicators
Figure 21: Atoms in strained nanocluster as a function of cohesive energy − grey
atoms correspond to CE of 4.6 eV while light blue atoms have a CE of ∼ 4.3 eV.
net energy of all their bonds with atoms of the substrate is recorded. This energy is
divided by the instantaneous contact area to the contact energy per unit area.
The coordination number:
Coordination number is the total number of neighbors for a central atom. Cubic
diamond Si atoms have a coordination of 4. Other forms of Si have different coordination numbers. The β-tin phase, for example has a coordination number of 6.
Monitoring changes in this coordination number is a cue to understand the structural
changes that occur during impact.
The radial distribution function (g):
The RDF is a pair correlation function which describe how, in a structure, its
atoms are arranged radially around each other. In other words, beginning with a
random atom in the system as a centre, it is the average number of atoms one might
expect to find at a particular radial distance from it. RDFs help to easily pinpoint
the atomic structure, especially when the structure is quickly evolving. The following
formula follows reference [1]
34
4.3 Simulation Indicators
Figure 22: Atoms with coordination number 6 versus time
g(r) =
n(r)
ρ4πr2 ∆r
(4.14)
In the above equation g(r) is the RDF, n(r) is the mean number of atoms in a shell
of width ∆r at distance r, ρ is the mean atom density.
Figure 23: Radial distribution function
35
4.3 Simulation Indicators
The bond angle distribution function (BADF):
The BADF simply presents the frequency distribution of bond angles between
atoms of the particle. The BADF for cubic crystalline Si is, for example centered
around 109.5o . This, however is bound to change if the structure itself changes.
In fact, the BADF along with RDF, serves to fingerprint a particular structural
configuration.
36
5
Molecular Dynamics Results - Impact of Hydrogenated Si Nanoparticles on Bare and Hydrogenated Si Substrate
In a series of classical MD simulations we examined the microscopic details of the
collision process between a projectile Si nanosphere of 5 nm in radius (R) and a Si
substrate exposing its (001) surface. The considered impacting speeds (V0 ) were of less
than 2, 000 m/s. To account for the practical conditions of reduced surface reactivity
we considered fully H-coated surfaces [35] [34]. The covalent Si-Si bonding, the
surface chemistry, and the dynamical bonding between the nanosphere and substrate
were described with transferable potentials based on the concept of bond order [44].
This family of interatomic potentials describe very well the most stable phases of
Si in the absence and under external pressure, as well as the Si-H bonding, but
are less for describing fracture. However, the fracture of Si nanospheres was not
identified experimentally [16]. Both the nanosphere, containing 31, 075 Si atoms, and
the substrate, containing 72, 000 Si atoms, were initially equilibrated at 500 K, in
order to mimic the elevated temperature experimental conditions of Rao et al. [38].
As before [48], during collision the last two bottom layers were kept fixed in time and
the substrate temperature was controlled with Langevin dynamics applied to the next
ten atomic layers. All other atoms were followed with a velocity Verlet algorithm. A
time step of 0.8 fs was used for all atoms. Periodic boundary conditions are applied
to the horizontal X and Y directions (the Z impact direction is vertical).
Unlike under energetic impact conditions [18], our MD didn’t lead to fragmenting
37
and spreading of the nanosphere, or cratering of the substrate. Only a few H atoms
were ejected under the highest V0 . The obtained collision modes can be identified
in the panels of Fig. 24, presenting the phase space trajectories for the center of
mass of the impacting nanosphere under three values of V0 and three surface-coating
combinations. For the convenience of comparison, the instantaneous speed (V ) was
normalized by V0 .
On the approach stage the important parameter is V0 . The nanosphere is touching
down and it advances with negative acceleration until a maximum penetration point
is reached. The magnitude of V0 is reflected in the nanosphere deformation measured
when R > Z by δ = R − Z, where Z is the vertical positive displacement of the
nanosphere center of mass measured with respect to the top of the substrate. The
adhesive forces appear secondary since as in the Hertz model [26], the nanosphere
does not show accelerated motion in response to the adhesive contact forces. Additionally, under the same V0 , values for δ presented in Fig. 24 are similar for all three
surface combinations considered: H-passivated nanosphere and substrate (black), Hpassivated nanosphere and bare substrate (gray), and bare nanosphere and substrate
(light gray).
On the recoil stage the outcome depends on both V0 and the chemical reactivity of
surfaces. For V0 = 900 m/s the black and gray curves indicate that the impacting Hpassivated nanosphere is reflected by both the bare and H-passivated substrate. The
collision is practically elastic as V0 restitution is as high as 80%. Our MD showed
that the reflection mode, which eluded previous microscopic investigations [48, 9],
dominates for V0 < 1, 000 m/s when weakly interacting surfaces are involved. As
a useful reference, the light gray curve in Fig. 24(a) indicates that under the same
38
Figure 24: Phase space trajectories for three impacting speeds: (a) 900 m/s, (b)
1, 300 m/s, and 2, 000 m/s. The nanosphere approaches the substrate with positive
V , touches down at δ = 0, reaches V = 0 at maximum penetration, and recoils with
negative V
V0 but larger adhesion, the bare nanosphere follows the path of a spiral sink and it
is captured by the bare Si substrate. V and δ form a damped cyclic path towards
the final equilibrium point, which is not reached during the shown 40 ps of MD
time. Thus, energy dissipation is very slow. This mode is the previously identified
soft landing [48, 9], where the deposit maintains its crystalline structure and the
dissipation of the incident energy involves the substrate.
To our surprise, for larger V0 a different behavior occurs: The H-passivated
nanosphere impacting on the bare substrate, Fig. 24(b), and even on the H-passivated
substrate, Fig. 24(c), falls into the path of a pure sink and it is captured with no oscillations. More precisely, this behavior was obtained above the critical V0 of 1, 250 m/s
(1, 550 m/s) for the impact on the bare (H-passivated) substrate. Large δ values can
be noted, indicating that the nanosphere experiences significant deformation. The
39
noted small V0 restitution and the lack of vibrational contact demonstrate efficient
dissipation of the incident energy. It can be also seen that the soft landing mode
undergoes the same qualitative transition, which was the main finding of a previous
work [48].
The collision dynamics is further conveyed in Fig. 25. In a typical reflection the
H-passivated nanosphere experiences deformation over a finite region of circular shape
around the point of contact, as macroscopically expected. At maximum penetration,
Fig. 2(a) (middle), there is a significant potential energy (PE) increase only near the
region of contact, as indicated by the change in color to pink(gray level). This change
appears reversible as the spherical shape is regained and the energetic differences
are washed out after detachment (last frame). The capture mode, Fig. 25(b), is
characterized by a severe deformation of the approaching nanosphere, which assumes
at maximum penetration (middle) a dome-like shape with a large contact region.
There is a significant PE change in a large conical volume of height ∼ R and with the
contact region as base. The irreversible character transpires from the PE distribution
in the final configuration (last frame), and from the sphere cut out by a plane shape
of the end deposit, optimal for adhesion.
Adhesion is quantified geometrically in Figure 26(a), which presents the timeevolution of the contact radius (a) during the two collision modes. During approach
a increases and reaches its maximum at the maximum penetration instant, regardless
of the surface chemistry. The final a value depends on both V0 and the surface
chemistry. Focusing on the capture mode, we see that a decreases during the recoil,
most significantly when both the nanosphere and substrate are H-passivated. Even
in this case the final a is large, comparable with R. Adhesion was next quantified by
40
Figure 25: MD simulations of H-passivated Si nanosphere impacting onto a Hpassivated Si substrate show two collision modes: (a) reflection and (b) capture.
The middle frames show the maximum penetration instant. Only a cross-section is
shown and H atoms are not represented. The color code carries the local PE, with
blue (gray) and pink (light gray) representing atoms with PE larger and smaller than
4.4 eV/atom, respectively
41
the finite adhesion energy (γa ), defined as the bonding energy measured per contact
area, between the atoms of the nanosphere and those of the substrate. We obtained
0.07 eV/Å2 and 0.03 eV/Å2 for the H-passivated nanosphere impacting on the bare,
and H-passivated substrate, respectively. For comparison, γa = 0.1 eV/Å2 when bare
Si surfaces are involved. Thus, it appears that the capture occurs without substantial
support from the adhesive contacts. Another observation of interest is that after
approach, γa is not changing notably because of the large number of H atoms trapped
in the contact region.
Fig. 26(b) shows the mean distributed contact-stress (pm ), computed as the net
vertical force acting on the nanosphere divided by the instantaneous contact area πa2 .
Our recorded p data confirms that significant pressures are reached under hypersonic
speeds, sufficient to produce yield according to experimentation [16]. Interestingly,
pm vanishes after the collision not only in the reflection but also in the capture mode.
In the latter, it indicates the occurrence of a stress-relieving transformation.
During collision the incident energy is converted into internal degrees of freedom,
producing temperature (∆Tp ) and PE (∆Up ) changes in the nanosphere. Examination
of the energy flow confirms the large dissipation in the capture mode. In Figure 26c
the large ∆Tp shows that the particle heats up during and after the 2, 000 m/s collision
and thus a large amount of the incident energy is irreversibly transferred into thermal
agitation. Figure 26(d) demonstrates the occurrence of a plastic change since after
collision there is a significant PE excess with respect to the original crystalline state.
To identify the nature of this plastic change causing ∆Tp and ∆Up rises, we monitored the nanosphere structure by computing its pair-correlation function (g). Particularly revealing were investigations at the maximum penetration instant, presented in
42
Figure 26: Time evolution of (a) contact radius and (b) contact-stress. Variation of
nanosphere’s (c) temperature, and (d) potential energy under two impacting speeds,
V0 = 900 m/s (lower two curves) and V0 = 2, 000 m/s (upper two curves), and
two surface types, H-passivated nanosphere and substrate (black) and H-passivated
nanosphere and bare substrate (gray). Down arrows mark the particle release instants
43
Figure 27. On one hand, under V0 = 900 m/s Figure 27(a) indicates that the cubic Si
structure is maintained, as the main peaks are well centered on the neighbor position
in cubic Si, where each atom bonds to four other equivalent atoms in an undistorted
tetrahedral pattern, Figure 27(e) left. On the other hand, under V0 = 2, 000 m/s
Fig. 26(b) shows the presence of a new structural arrangement, other than cubic Si.
Indeed, there is a widening and simultaneous decrease of the main peak height, as
well as the appearance of a new peak at the interatomic distance (r) of ∼ 3 Å. For
more insight, we focused on the conical volume with high PE indicated in Fig.25(b)
middle, where we found 5, 500 sixfold coordinated Si atoms with bond lengths that
did not exceed 2.8 Å. In addition to g, we computed the bond angle (BA) distribution. These results prove that the dominant structural changes correspond to a β-tin
phase of Si, shown in Fig. 27(e) right. This new phase is derived from the cubic Si by
flattening the tetrahedral grouping to the extent that two other atoms are brought
into close proximity, which increases the coordination from 4 to 6. Indeed, in Figure
27(c) g has two sharp peaks at 2.5 and 3.1 Å, which match very well the first and
second neighbor distances of β-tin Si. All BA are 109◦ in cubic Si. However, in Figure
27(d) they are grouped around the five distinct BA of the β-tin.
On the basis of this key observation, the observed reflection to capture transition is
interpreted as follows. On the approach stage, nanosphere-substrate adhesion occurs
due to the new Si-Si covalent bonds formed in the contact region. For V0 < 1, 000 m/s,
these bonds rupture under the vigorous backwards motion caused by the largely reflected incident energy. However, for V0 > 1, 000 m/s, in the absence of dislocation,
the atomic rearrangement caused by a displacive phase change represents the next
available gateway for absorbing irreversibly a significant amount of the incident en-
44
Figure 27: Pair-correlation function of the H-passivated Si nanosphere at maximum
penetration for (a) V0 = 900 m/s and (b) V0 = 2, 000 m/s. Vertical bars mark the
neighbor position in cubic Si. (c) Pair-correlation and (d) bond-angle distributions
in the most deformed conical region. Vertical bars mark neighbor positions and bond
angles in bulk β-tin Si. (e) Unit cells for cubic-diamond (left) and β-tin Si (right)
45
ergy. Having a smaller crystallographic c/a0 ratio (0.55, with a0 = 4.73 Å) than
cubic-diamond (1.42, with a0 = 3.82 Å), the β-tin phase has geometric advantage
as it simultaneously relieves the compressional stress and augments the contact area.
Only the PE stored in elastic deformation is returned on the recoil as coherent reverse
motion of the nanosphere atoms. Capture occurs when the increased adhesion is able
to overcome the weakened recoil motion.
Further monitoring of g demonstrated that, as in the case of bare Si nanospheres [48],
almost no β-tin phase exists in the end deposit. This transition is documented in Figures 28 and 29. Figure 28 shows the RADF for the particle at the end of 40ps. As
can be seen, there is a small bulge between 3-3.5Å, which cannot be seen for bulk
crystalline Si. While amorphization is not very obvious from this figure, the Bond
Angle Distribution shown in 29 is more directly indicative of the amorphization with bond angles varying widely about the standard diamond cubic Si angle of 109.5◦
Although not themodynamically favorable, the trapped a-Si is favored here since this
non-crystalline arrangement accommodates both with the unaffected upper spherical
portion of the particle with cubic structure and the large contact, see last frame of
Figure 25(b).
In conclusion, our large-scale MD simulations showed that the poor reactivity of
surfaces prevents the sticking of projectile H-passivated Si nanospheres. However,
sticking is possible when hypersonic speeds are attained. It is accompanied by the
irreversible conversion of the incident energy into thermal agitation, and by the occurrence of a phase transition, from diamond to β-tin, the latter eventually evolving
to a-Si. These phase changes, also seen in the MD collision simulations with bare
Si surfaces [48], are not unrealistic. Although the considered impacting energies are
46
Figure 28: RADF at the end of 30 ps for impact at 2000 m/s with hydrogen on both
surfaces
Figure 29: BADF at the end of 30 ps for impact at 2000 m/s with hydrogen on both
surfaces
47
relatively low and the nanosphere-substrate contact forces are not large, because of
the nano-size extent of the contact zone the resulting pm are high enough to induce
a first-order phase change. Indeed, both experimental and theoretical examinations
of Si under pressure have revealed the existence of diverse crystalline phases, other
than cubic-diamond. The first such phase is the β-tin state, which emerges under a
pressure of 11 GPa [14]. More recently, the cubic to β-tin phase change was observed
in Si nanocrystals during anvil experiments under a pressure of 22 GPa [46], in good
agreement with our data. The subsequent amorphization of the β-tin on the recoil,
is in contrast with the recovery of a metastable BC8 phase in bulk Si [51] but in
agreement with the amorphization of the β-tin in Si nanocrystals after the release of
the hydrostatic pressure [46]. The MD obtained succession of phase changes on a ps
time scale is of interest for the fundamental understanding [23] of the transformation
kinetics of solid phases in Si nanoparticles. More practically, the identified capture
mediated by phase transition plasticity improves our comprehension of the hypersonic
nanoparticle deposition experiments.
48
6
More Molecular Dynamics Results - Impact of
Si31,075N413, C90,326 and Si12,080C12,080 Nanoparticles
on Bare Si Substrate
While the simulations in this project deal primarily with impact of hydrogenated Si
nanoparticles, preliminary simulations were done with nitrogen coated Si nanoparticles, C nanoparticles and SiC nanoparticles. All three cases present three totally
different impact scenarios, and provide important insights into the effect of chemical
adhesion and structural hardness on the outcome of the impact. The Tersoff potential
already used for all simulations is easily extendible to these elements.
6.1
Si31,075 N413 Nanoparticles
As mentioned in Chapter 2, the Tersoff potential and parameters are modified to
account for Si-N interaction. Interaction energy for Si-N is actually stronger than SiSi interaction energy. This makes nitrogen coated nanoparticles a very special case.
Coating a Si nanoparticle with N is essentially giving it a very adhesive skin. Thus,
the impact basically follows the pattern of bare Si impact which is adhesion under
all circumstances, albeit with a stronger adhesion to the substrate. The process of
coating itself is akin to the one described in Chapter 3. Initally, the nitrogen is placed
close to Si atoms on the surface with dangling bonds. Successive conjugate gradient
minimization and high temperature MD procedures are carried out to retain the N
on the surface. A time step of 0.8 fs is used through out.
At an impact speed of 2,000 m/s, the particle firmly adheres to the surface. The
49
6.1 Si31,075 N413 Nanoparticles
particle surface adhesion energy was the highest among all cases seen thus far. The
picture below plots this contact energy per unit contact area for 2000 m/s impact.
Bare Si has a unit adhesion energy of 0.1 eV/Å, whereas Si-N sees a 50% increase to
0.15 eV/Å.
Figure 30: Time evolution of contact energy
This effect only slightly seems to affect the area of contact or the structural deformation. The figures below plot the time-evolution of these 2 parameters. Ultimately,
all particles seem to settle into a nearly even contact area after 40 ps. The structural
deformation represented in Figure 32 is nearly equal for bare Si and nitrogen coated
Si.
50
6.1 Si31,075 N413 Nanoparticles
Figure 31: Time evolution of contact area for different surface chemistries
Figure 32: Time evolution of coordination number plot for SiN
51
6.2 C90,326 and Si12,080 C12,080 Nanoparticles
Finally looking at the trajectory of phase space plots for impact at 2,000 m/s,
with different surface chemistries, SiN impact-represented as a solid line in Figure
33, follows the other trajectories before diverging to settle down to a more submerged
position within the substrate.
Figure 33: Phase space trajectories at 2,000 m/s
6.2
C90,326 and Si12,080 C12,080 Nanoparticles
MD simulations were carried out to understand the impact of C nanospheres with
bare Si substrates. The carbon particles were modeled as nano-meter sized diamond
crystals using the Tersoff paramters for C [45]. The case of C is unique because of
its very high inherent strength, which in turn means there is very sparse structural
deformation. This lack of any structural deformation means an important channel for
the dissipation of internal energy is shut off and most of the impact energy is reflected
back - leading to particle bounce. The simulations were not advanced enough to keep
52
6.2 C90,326 and Si12,080 C12,080 Nanoparticles
a tab on all parameters like in the previous results. However, the phase space plot
shows the particle bouncing at all three velocities of 900 m/s, 1,600 m/s and 2,000
m/s.
Figure 34: Phase space trajectories for C90,326 nanoparticle
As can be seen from Figure 34, the restitution of impact (reflected velocity/impact
velocity) is very high for C nanoparticles. This is in sharp contrast to the case of Si,
shown in the figure above. For Si, after the onset of phase transition at 1,000 m/s,
the rebounce becomes smaller and smaller until it finally becomes zero. The lack
of palpable structural deformation is directly reflected in the C nanoparticle’s high
restitution.
Finally, when we consider SiC, we found the particle adhered to the surface on all
3 simulations that were performed. Again the simulations were not advanced enough
53
6.2 C90,326 and Si12,080 C12,080 Nanoparticles
Figure 35: Restitution for impact of Si nanoparticle with hydrogen on both surfaces
54
6.2 C90,326 and Si12,080 C12,080 Nanoparticles
to map out all system parameters, but the behavior of Si closely mimics the bare
Si behavior. We may assume that the chemical adhesion between the particle and
substrate was too high to allow for any rebounce. The phase space trajectories are
plotted below for SiC impacting at 900 m/s, 1,300 m/s and 1,600 m/s.
Figure 36: Phase space trajectories for Si12,080 C12,080 nanoparticle
55
7
Future Work
Studies in this project were limited to studying the impact of one nanoparticle. The
maximum size studied was a radius of 5 nm. These limitations were primarily because of the computational constraints of simulations involving more than 200,000
atoms. Large scale simulations of multi-particle impact have a huge potential to yield
important results. Particularly interesting would be the simulation of the impact of
newly formed nanoparticles with ones already adhered. This would ultimately give
insights into the process of sintering leading to thin film formation, for the case of
hypersonic deposition. However, carrying out the above simulations would require
for the current code to be parallelized.
Preliminary simulations done in this project with C and SiC nanoparticles can
be extended to thoroughly understand their deposition mechanism. Silicon Carbide
deposition is widely studied as a means to fabricate wear resistant coating [5], hence
simulations in this direction would be particularly useful. The modified Tersoff Potentials cannot describe Si-O interaction suitably. Given the widespread use of SiO2
in the semiconductor industry, simulations with oxygen coated nanoparticles would
be more relevant.
Finally the effect of size on particle deposition is a very interesting aspect to consider. Reference [49] studies the uniaxial compression of Si nanoparticles to quantify
phase transition within them. However, preliminary simulation with particles 1 nm or
less in radius did not find a phase transition to β−tin Si, but rather a skew displacement within the lattice. Hence the outlet for impact energy (namely phase transition)
which has been so crucial to particle adhesion in our simulations may not be equally
relevant for smaller sizes. The size threshold at which the impacting nanoparticle can
56
have an internal structural dislocation, is another important direction to pursue.
57
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