Bond Order Potentials for Group IV Semiconductors
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1 Bond Order Potentials for Group IV Semiconductors A Dissertation Presented to The Faculty of the School of Engineering and Applied Science University of Virginia In Partial Fulfillment of The Requirements for the Degree Doctor of Engineering Physics by Brian Andrew Gillespie January, 2009
Transcript of Bond Order Potentials for Group IV Semiconductors
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Bond Order Potentials for Group IV Semiconductors
A Dissertation Presented to
The Faculty of the School of Engineering and Applied Science
University of Virginia
In Partial Fulfillment of
The Requirements for the Degree
Doctor of Engineering Physics
by
Brian Andrew Gillespie
January, 2009
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Approval Sheet
This dissertation is submitted in partial fulfillment of the requirements for the degree
Doctor of Engineering Physics
Brian Andrew Gillespie
This Dissertation has been read and approved by the Examining Committee:
Leonid Zhigilei, Materials Science, Committee Chair
Vittorio Celli, Physics
James Groves, Material Science
Robert E. Johnson, Engineering Physics
Haydn N.G. Wadley, Materials Science, Advisor
Accepted for the School of Engineering and Applied Science
James H. Aylor, Dean
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Abstract
The atomic scale assembly mechanisms utilized to make semiconductor devices
are difficult to experimentally characterize. Large scale, computer simulations are
beginning to be used to identify these atomistic mechanisms. Computational modeling
of semiconductor systems is complicated by the intricacies and subtleties of the method
by which the materials form atomic bonds. The semiconductors silicon and germanium
are group IV elements which form hybrid sp3 atomic orbital’s to create a diamond cubic
crystalline lattice at standard temperature and pressure. While density functional
methods are well suited for the analysis of the atomic scale structure of these covalent
systems, computational resource limitations prohibit their application to situations
where time dependent reassembly phenomena occur. To model these phenomena,
molecular dynamics methods are used, and these must use an interatomic potential
containing terms to account for the open local atomic environment. Many empirical
interatomic potentials have been proposed over the years; most notable the Stillinger-
Weber and Tersoff potentials. The silicon and germanium parameterizations of these
potentials have been evaluated for their predictive ability for small clusters, melting
temperature, bulk properties for a wide range of crystal structures (dc, sc, fcc, bcc, -Sn,
hcp, and bc8), and the energy of low index surface reconstructions. These potentials are
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shown to give reliable estimates of the bulk properties of the dc phase, but are also
shown to be inadequate for the study of many other structures encountered during the
atomic assembly of both silicon and germanium.
A tight binding description of covalent bonding is used here to propose bond
order potentials (BOP) for the group IV semiconductors silicon and germanium. The
potential addresses both the and bonding of these sp-valent elements. A promotion
energy term associated with the formation of the hybrid orbitals is included in the
formulism. The potential is parameterized using ab-initio and experimental data. The
BOP potential’s predictions for the cohesive energy, atomic volume, and bulk modulus
of the dc, fcc, bcc, bc8, hcp, and -Sn phases of silicon and germanium compare
favorably with estimates obtained using Density Functional Theory (DFT) using local
density approximation (LDA). The BOP also gives point defect formation energies that
are in good agreement with ab initio estimates. The structure of small atomic clusters,
the melting transition temperature and the atomic structure of low index surfaces are
also used to assess the validity of the BOP potential. The functional form of the BOPs for
silicon and germanium are the same, facilitating their eventual use for the study of the
binary Si-Ge alloy system. These improved potentials have then been used for large
scale molecular dynamics simulations of a solid phase epitaxial re-growth of an
amorphous thin film (a prototypical device fabrication process) and the rate limiting
atomic scale mechanism identified.
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The low temperature vapor deposition of silicon thin films and the ion
implantation of silicon often results in the formation of amorphous silicon layers on a
crystalline silicon substrate. These amorphous layers can be crystallized by a thermally
activated solid-phase epitaxial (SPE) growth process. The transformations are rapid and
initiate at the buried amorphous to crystalline interface within the film. The initial stages
of the transformation are investigated here using a molecular dynamics simulation
approach based upon the bond order potential for silicon. The method is used first to
predict an amorphous structure for a rapidly cooled silicon melt. The radial distribution
function of this structure is shown to be similar to that observed experimentally.
Molecular dynamics simulations of its subsequent crystallization indicate that the early
stage, rate limiting mechanism appears to be removal of tetrahedrally coordinated
interstitial defects in the nominally crystalline region just behind the advancing
amorphous to crystalline transition front. The activation barriers for interstitial
migration within the crystal lattice are calculated and found to be comparable to the
activation energy of the overall solid-phase epitaxial growth process simulated here.
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Table of Contents
List of Figures . . . . . . . . . 9
List of Tables . . . . . . . . . 12
List of Terms and Abbreviations . . . . . . 14
1. Introduction . . . . . . . . 16
2. Molecular Dynamics Methods . . . . . . 28
2.1. Stillinger-Weber Potential . . . . . . 32
2.2. Tersoff Potential . . . . . . . 37
2.3. Bond Order Potential . . . . . . 42
3. SW and Tersoff Silicon Assessment . . . . . 52
3.1. Bulk Properties . . . . . . . 53
3.2. Small Clusters . . . . . . . 61
3.3. Point Defects . . . . . . . 69
3.4. Melting Temperature . . . . . . 72
3.5. Surface Reconstructions . . . . . . 74
4. Silicon BOP Assessment . . . . . . . 78
4.1. Bulk Properties . . . . . . . 78
4.2. Small Clusters . . . . . . . 82
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4.3. Melting Temperature . . . . . . 85
4.4. Point Defects . . . . . . . 86
4.5. Surface Reconstructions . . . . . . 87
5. Solid Phase Epitaxy of Silicon . . . . . . 89
5.1. Introduction . . . . . . . . 89
5.2. Simulation Details . . . . . . . 93
5.3. Amorphous Characterization . . . . . 95
5.4. Epitaxial Crystallization . . . . . . 97
5.5. Discussion . . . . . . . . 108
6. SW and Tersoff Germanium Assessment . . . . 110
6.1. Bulk Properties . . . . . . . 111
6.2. Small Clusters . . . . . . . 118
6.3. Melting Temperature . . . . . . 122
6.4. Point Defects . . . . . . . 123
6.5. Surface Reconstructions . . . . . . 125
7. Germanium BOP Assessment . . . . . . 127
7.1. Bulk Properties . . . . . . . 127
7.2. Small Clusters . . . . . . . 132
7.3. Melting Temperature . . . . . . 135
7.4. Point Defects . . . . . . . 136
7.5. Surface Reconstructions . . . . . . 138
8. Discussion . . . . . . . . 140
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8.1. Stillinger-Weber Overview . . . . . . 141
8.2. Tersoff Overview . . . . . . . 142
8.3. Improved Fidelity of the BOP Approach . . . . 144
9. Future Works: SiGe Alloy Parameterization . . . . 150
10. Conclusion . . . . . . . . 153
Appendix A: Mathematica Fitting . . . . . . 156
Appendix B. DFT Calculations . . . . . . 171
Appendix C. Qualitative Analysis Determinants . . . . 175
References . . . . . . . . . 176
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List of Figures
1. Section 1
1.1. SiGe band energy diagram . . . . . . 18
1.2. Periodic Table entries for Si and Ge . . . . . 19
1.3. sp3 orbital hybridization . . . . . . 20
1.4. SiGe film growth . . . . . . . 22
1.5. SiGe critical film thickness . . . . . . 22
1.6. Ion implantation schematic . . . . . . 24
2. Section 2
2.1. Molecular Dynamics flowchart . . . . . 30
2.2. Bond Order Potential electronic hopping paths . . . 43
3. Section 3
3.1. Crystal lattices . . . . . . . 54
3.2. SW and Tersoff Si binding energy curves . . . . 55
3.3. SW and Tersoff vibrational spectrum . . . . 62
3.4. Silicon small clusters . . . . . . . 64
3.5. Point defect structures . . . . . . 72
3.6. Si (100) 2x1 surface . . . . . . . 75
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3.7. Si (113) 3x2 surface . . . . . . . 76
3.8. Si (111) 7x7 surface . . . . . . . 77
4. Section 4
4.1. BOP Si binding energy curves . . . . . 80
4.2. BOP Si vibrational spectrum . . . . . . 82
5. Section 5
5.1. Radial distribution function for a-Si . . . . . 95
5.2. Activation barrier to SPE . . . . . . 97
5.3. SPE snapshots at 700 K . . . . . . 98
5.4. SPE snapshots at 900 K . . . . . . 99
5.5. Number of crystal atoms over time . . . . . 100
5.6. Bond angle distribution in transition region . . . . 101
5.7. Vacancy to crystalline site defect migration . . . . 103
5.8. Tetrahedral to [110]-split defect migration . . . . 104
5.9. Tetrahedral to hexagonal defect migration . . . . 106
5.10. Tetrahedral to crystalline site defect migration . . . 107
6. Section 6
6.1. SW and Tersoff Ge binding energy curves . . . . 112
6.2. SW and Tersoff Ge vibrational spectrum . . . . 117
6.3. Ge (111) c2x8 surface . . . . . . 126
7. Section 7
7.1. BOP Ge binding energy curves . . . . . 130
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7.2. BOP Ge vibrational spectrum . . . . . 131
8. Section 8
9. Section 9
9.1. SiGe phase diagram . . . . . . . 151
10. Section 10
A. Appendix A
A.1 CG-NNL.nb description . . . . . . 157
A.2 fitGSP.nb description . . . . . . 160
A.3 Variation in , and
A.4 Variation in A and
A.5 Variation in r1 and rcut . . . . . . 167
A.6 Variation in r0, rc, rc and rc
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List of Tables
2.1 Stillinger-Weber Parameters . . . . . . 35
2.2 Tersoff Parameters . . . . . . . 40
2.3 Bond Order Potential Parameters . . . . . 50
3.1 Atomic Volume as a function of lattice constants . . . 57
3.2 SW and Tersoff silicon atomic volumes . . . . 57
3.3 SW and Tersoff silicon cohesive energies . . . . 58
3.4 SW and Tersoff silicon bulk moduli . . . . . 60
3.5 SW silicon small clusters . . . . . . 65
3.6 Tersoff silicon small clusters . . . . . . 66
3.7 SW and Tersoff silicon point defects . . . . . 71
3.8 SW and Tersoff silicon surface reconstructions . . . 76
4.1 BOP silicon atomic volumes . . . . . . 81
4.2 BOP silicon cohesive energies . . . . . 81
4.3 BOP silicon bulk moduli . . . . . . 81
4.4 BOP silicon small clusters . . . . . . 84
4.5 BOP silicon point defects . . . . . . 87
4.6 BOP silicon surface reconstructions . . . . . 88
6.1 SW and Tersoff germanium atomic volumes . . . 114
6.2 SW and Tersoff germanium cohesive energies . . . 115
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6.3 SW and Tersoff germanium bulk moduli . . . . 116
6.4 SW germanium small clusters . . . . . 119
6.5 Tersoff germanium small clusters . . . . . 120
6.6 SW and Tersoff germanium small clusters . . . . 124
6.7 SW and Tersoff germanium surface reconstructions . . 126
7.1 BOP germanium atomic volumes . . . . . 128
7.2 BOP germanium cohesive energies . . . . . 128
7.3 BOP germanium bulk moduli . . . . . 129
7.4 BOP germanium small clusters . . . . . 133
7.5 BOP germanium point defects and surface reconstructions . 137
8.1 Qualitative evaluation of all three potentials . . . 146
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List of Terms and Abbreviations
In order of appearance.
dc . . diamond cubic crystal structure
sc . . simple cubic crystal structure
fcc . . face-centered cubic crystal structure
bcc . . body-centered cubic crystal structure
b-Sn . . beta-Tin crystal structure
hcp . . hexagonal close packed crystal structure
bc8 . . bc8 crystal structure
BOP . . Bond Order Potential
DFT . . Density Functional Theory
LDA . . Local Density Approximation
SPE . . Solid Phase Epitaxy
SW . . Stillinger-Weber interatomic potential
LED . . Light Emitting Diode
p-MOSFET . p-type Metal-Oxide-Semiconductor Field Effect Transistor
a-Si . . amorphous silicon
c-Si . . crystalline silicon
MD . . Molecular Dynamics
EAM . . Embedded Atom Method
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TB . . Tight Binding
IP . . Interatomic Potential
GSP . . Goodwin Skinner Pettifor pair term
HF . . Hartree Fock
T, H and X . Tetrahedral, Hexagonal and (110)-Split Interstitials
DAS . . Dimer Adatom Stacking fault for (111) 7x7 surface
EDIP . . Environment Dependent Interatomic Potential
RDF . . Radial Distribution Function
VASP . . Vienna Ab-initio Simulation Package
IBIEC . . Ion Beam Induced Epitaxial Crystallization
ZB . . Zinc Blende crystal structure
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I. Introduction
Silicon is an abundant element composing ~25% of the Earth’s crust by mass in
the form of silica and various silicates [1]. It is one of the most intensely studied
materials in the Periodic table because of its many engineering applications [2]. Silicon,
in the form of silica and silicates, is vital to the construction industry as a principle
constituent of natural stone, glass, concrete and cement [3]. Silicon is also commonly
alloyed with aluminum to produce easily cast metallic alloy parts for industry [4]. Silicon
even sees service in the children’s toy Silly Putty, which contains significant amounts of
elemental silicon (silicon binds to the silicone and allows the material to bounce 20%
higher) [5]. However, silicon’s greatest technological impact derives from its use in
microelectronic applications. Elemental silicon is the principle component of most
semiconductor devices, most importantly integrated circuits. Ultra-pure silicon can be
doped with other elements to adjust its electrical response by controlling the number
and polarity of its charged current carriers [2]. Such control is necessary for solid state
transistors, solar cells, integrated circuits, microprocessors, and other semiconductor
devices which are used in microelectronics.
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Germanium, the group IV element located directly below silicon in the periodic
table, is also a material of great scientific and engineering interest. The development of
the germanium transistor ushered in the era of modern microelectronics [6]. The
germanium transistor is considered by many to be the greatest invention of the
twentieth century [7]. Despite silicon’s superior electrical properties, germanium was
the original semiconductor of choice because silicon required much higher purity;
purities that could not be achieved commercially at the time [2]. Germanium continues
to be of interest to modern microelectronics because of its unique combinations of
electrical and other physical properties. For instance, germanium is an indirect band
gap material with a small band gap. It therefore has low light absorbance at infrared
wavelengths, and because it is easily cut and polished, it can be used for infrared lenses
and windows in the 8-14 micron wavelength range [8-9]. The small bandgap of Ge and
Ge-Si alloys also make it a useful material in solar cell applications where its high
absorbance enable the use of thinner layers of active material [10-11]. Light emitting
diodes (LEDs) have also been fabricated to take advantage of germanium’s unique
properties [12-13]. These LEDs are based on Ge-Si alloy self-assembled quantum dots,
and have exhibited a broad emission peaked at a wavelength centered upon 1.45 m
[12-13].
The performance of traditional silicon-based semiconductor technologies is also
being extended through the use of SixGe1-x alloys. These performance increases can be
tied to the differences in physical and electronic properties. For example, while
elemental silicon has a band gap of 1.12 eV [14], the inclusion of germanium reduces
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this to ~0.93 eV for Si0.5Ge0.5. A
graph showing the effect of
germanium additions upon the
band gap of SixGe1-x is given in
Figure 1.1 (the red line). In Figure
1.1 (the green line) the intrinsic
carrier resistivities are shown for
SixGe1-x [14]. This greatly increases
the number of intrinsic charge carriers with the semiconductor. For example, at 300 K,
silicon has an intrinsic carrier concentration of 1.0x1010 cm-3, germanium has 2.4x1013
cm-3, and Si0.5Ge0.5 has 1.2x1013 cm-3 [14]. Another key advantage of SiGe alloys are their
high electron hole mobility. At room temperature (300 K) silicon has a hole mobility of
450 cm2/V*s and electron mobility 3,900 cm2/V*s, while Si0.5Ge0.5 has a hole mobility of
1,175 cm2/V*s and an electron mobility of 7,700 cm2/v*s. A recent SiGe p-MOSFET
device designed by Masashi Shima displayed an increase in hole mobility of around 30%
compared with a similar device built using silicon technology alone [15]. Interest in SiGe
microelectronics has also been aided by a realization that the massive investment in
existing silicon foundry fabrication tools can be used for most SiGe alloy process steps
[16].
Both silicon and germanium are group IV metalloids with a valence state of s2p2;
the Periodic Table entries for silicon and germanium are shown in Figure 1.2 [1]. When
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two or more atoms of the same or similar electronegativities react, a complete transfer
of electrons does not occur [17]. Instead the atoms achieve noble gas valence shell
electronic configurations by sharing pairs of electrons. This form of interatomic bonding
is called covalent bonding [17]. Silicon and germanium form covalent bonds between
themselves and with each other and with the common group III and V dopants [17].
Covalent bonds are characterized by their strength and directional dependence.
In order to fill the s and p valence electron shell, silicon and germanium need to borrow
four electrons. However, in the s2p2 electronic ground state, silicon and germanium are
only capable of forming 2 covalent bonds. The s2p2 electronic orbitals hybridize into the
sp3 configuration to create four electron states capable of forming four equivalent
covalent bonds [17]. Figure 1.3 shows the structure of the sp3 configuration. The
tetrahedral arrangement of these covalent bonds leads to the equilibrium, ambient
temperature and pressure condensed phase of silicon and germanium to be the
diamond cubic structure [1].
This dissertation has been motivated by interest in the growth processes of
silicon, germanium and their alloys. The growth of silicon crystals from the
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condensation of a vapor has been thoroughly studied [18-21]. Temperature plays an
important role in vapor phase growth processes. The structure of a semiconductor film
grown at low temperature is often amorphous (i.e no long range order in the solid
phase). As the substrate temperature is increased, the mobility of atoms during the
growth process increases, and the structure becomes more crystalline; and in the case
of silicon, assumes the diamond cubic structure. An experimental study performed by
Zalm et al. at the Philips Research Laboratories in the Netherlands found that a
minimum yield of 40% crystalline 60% amorphous was obtained when depositing Si at a
rate of 0.1 nm/s at a substrate temperature of ~650 K [18]. Further experiments at
higher temperature (~745 K and 835 K) showed a significant increase in crystallinity.
These results are consistent with other works [19-21]. The growth of crystalline silicon
generally proceeds by a planar (step flow) mode as opposed to an island growth scheme
[22]. It is possible, however, to grow silicon in a columnar structure by altering the
angle of incidence of adatom flux as shown experimentally by Xie et al. [22]. They found
that columnar structures grew in the same direction as the adatom incidence angle.
They also found that the column width was dependent on the deposition temperature
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(~40 at 300 K and ~130 at 425 K, note that these temperatures are a relative
measurement as the exact surface temperature was not experimentally measured) [22].
When germanium condenses from the vapor phase the structure that is formed
depends on the temperature of the condensate [23-24]. The structure of germanium
deposited at low temperature is amorphous. As temperature is increased, the volume
fraction of the grown film that is crystalline increases [23-24]. The growth mechanism
of germanium on silicon is very similar to the homoepitaxy of Si for the first 3
monolayers [23-24]. Germanium grows in a 2-D planar method for the first three
monolayers, after which the growth proceeds via the formation island formation on the
substrate [23-24]. This is due to the 4% lattice mismatch between Si and Ge and the
resulting strain in the lattice [25-26]. A deposition temperature of at least 525 K is
needed to ensure a 95% crystalline growth region [27], and typically a ~900 K anneal is
performed to smooth out the surface.
The growth of SiGe films proceeds via a 2-D (planar) mechanism or a 3-D
(islanding) mechanism [28]. Figure 1.4 details the temperature and composition for the
two types of growth. Strain effects prevent unlimited planar growth [28]. It has been
found that at a given temperature and germanium fraction, SiGe can be grown on Si up
to a critical thickness before islanding begins [28]. For example, Si0.2Ge0.8 can be grown
in a planar fashion up to ~2500 Å, whereas Si0.5Ge0.5 can be grown up to ~100 Å before
the onset of misfit dislocation formation and islanding [29]. Figure 1.5 displays
experimental and theoretical results correlating germanium fraction to critical thickness.
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Substrate temperature for crystalline deposition varies from the temperature for
homoepitaxy of Si. The inclusion of Ge degrades the crystallinity of deposition at low
temperatures (~650 K) [18].
The crystalline quality of the silicon and germanium grown films is of great
importance because the crystalline structure generally has more desirable electronic
characteristics [2]. Fortunately the amorphous phases of both silicon and germanium
are less stable than the crystalline structures and as a result there exists driving force, in
the form of Gibbs free energy, for the rearrangement of the amorphous material to the
diamond cubic structure [30]. The spontaneous rearrangement of amorphous material
to crystalline material, also known as solid phase epitaxy when it occurs on crystal
substrates, is temperature dependent and well described by an Arrhenius equation [30].
In the case of high purity silicon films for example, the interface between the two solid
Figure 1.4 Plot of Si1-xGex film morphology
vs. growth temperature and germanium
fraction.
Figure 1.5 Plot of critical thickness of
Si1-xGex films vs. germanium fraction.
Experimental data taken from
reference 28.
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phases has been observed to move at a velocity of several m/s at 725 °C [30]. The
interface velocity is directly dependent on the temperature; at higher temperatures the
interface moves more rapidly, and at lower temperatures the interface moves more
slowly [30].
While silicon, germanium and their alloys are intrinsic semiconductors, the true
strength of their use in the semiconductor industry is derived by the inclusion of
controlled defects within the lattice [2]. These defects are usually introduced in the
form of dopant elements. These dopants greatly increase the number of charge carriers
(electrons or holes) within the semiconductor and are a necessary component for most
modern electronic devices [2]. The most commonly employed method for the
introduction of dopants into the semiconductor is through ion implantation processes
[31]. A typical ion implantation setup is diagrammed in Figure 1.6. The ion implantation
process typically involves a high purity ion source, where ions of the desired element are
produced, an accelerator, where the ions are eletrostatically accelerated to high energy,
and a target chamber containing the silicon or germanium substrate. An energetic ion
entering a solid interacts with the atoms of the solid in two principle ways. Atoms are
displaced from their lattice sites and set in motion through momentum transferring
collisions. In addition, electrons are excited by Coulomb interactions with the moving
atoms. These two types of energy losses are often referred to as nuclear and electronic
energy loss respectively. Atoms set in motion by nuclear collisions of the primary ion
may collide with other atoms and set them in motion and these, in turn, can do the
same to others [30]. This sequence is typically called a collision cascade.
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The common feature of all ion
implantation processes in a solid is the
production of lattice disorder (even for self-
implantation) [30]. The amount of disorder
generated is controlled by the ion energy,
mass, dose (the total number of ions
implanted), dose rate (the rate at which
ions are implanted) and the substrate
temperature [30]. The disorder can be so
severe that the implanted region becomes amorphous. Recovery of this damage is
required to restore the crystalline structure and to electrically activate the dopants [2].
In particular, the transition from amorphous silicon (a-Si) to crystalline silicon (c-Si) is a
process of great technological importance. In solid phase epitaxy (SPE), the
amorphous/crystalline interface propagates by a thermally induced epitaxial crystal
growth into the metastable amorphous region [30]. The SPE process has been the
subject of many research efforts in recent years [30-40]. The great challenge to
experimentalists is that the SPE process occurs within the bulk of the semiconductor
material which remains inaccessible to surface experimental probes [35]. This difficulty
has limited the usefulness of experimental methods to investigate the SPE process to
measurements of macroscopic quantities such as the activation energy for growth.
While the growth from the vapor phase of silicon, germanium and their alloys is
well studied; a better understanding of the atomic scale assembly mechanisms that
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occur during vapor deposition and the atomic mechanisms responsible for the
rearrangement of atoms at the amorphous/crystalline interface in solid phase epitaxy
would be helpful to development of new techniques for the growth of semiconductor
thin films. A fundamental study of the ion impact process and atomic reassembly
induced by it would also be of benefit.
Molecular dynamics (MD) techniques have demonstrated the ability to identify
and capture the atomistic mechanisms responsible for experimental phenomenon [41-
45]. Molecular dynamics simulations make use of interatomic force calculations
obtained using interatomic potentials. These potentials coarse grain the quantum
mechanical behavior of the many electrons into a force law that relates the interatomic
force to the local atom configuration. As a result, any fundamental insights that can be
derived from the examination of the electronic ground state are sacrificed. In return,
interatomic potentials allow for the simulation of a sufficient number of atoms to
realistically approximate atomic assembly processes requiring thousands of atoms over
a nanosecond time scale. The investigation of these atomistic assembly mechanisms
inherently involves considerations of interatomic forces, therefore kinetic Monte Carlo
simulation methods which can accommodate many more atoms are untenable.
Molecular dynamics has been used successfully in studies of the surfaces of
metals during their assembly from the vapor phase [41-45]. The approach is most
precisely implemented with fcc metals where the bond forces are dependent only on
the nearest neighbor distance (r) and are not dependent on the angle () formed by an
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atom with its nearest neighbors. The simulated growth of close packed metallic
multilayers has utilized spherically symmetric embedded atom method (EAM) potentials
to examine hot atom and ion assisted deposition and surfactant mediated growth [43-
44]. In the case of ion assisted growth it was shown that inert gas ion beams could be
used to aid in the growth of Cu/Ni/Cu multilayers by smoothing out the interfaces [44].
It was discovered that even low energy (~3 eV) inert gas impacts could cause atomic
exchange at the Cu/Ni interface [44]. It was shown that by modulating the ion beam
assistance it is possible to significantly reduce the deleterious intermixing, while at the
same time enabling ion assisted flattening [44]. These observations were subsequently
corroborated experimentally [45]. Despite these successes, the EAM potentials are not
suitable for modeling the complex covalent bonding observed in silicon and germanium
systems because it treats only omnidirectional bonding.
Numerous groups have attempted to incorporate the physical concepts
underlying covalent bonding in many-body interatomic potentials [46-54]. This has led
to semi-empirical sets of equations that attempt to approximate the phenomenological
nature of the bond. An assessment for the GaAs system has shown that this approach
results in mixed success [55]. In an alternative approach, Pettifor et al. have shown that
it is possible to derive an analytic, many-body interatomic potential by coarse graining
the electronic structure within the orthogonal two-center tight-binding (TB)
representation of covalent bonding [56-59]. These analytic bond order potentials
(BOPs) explicitly link the bond order (and therefore the bond energy) to the positions of
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atomic neighbors. Applications of the approach to the GaAs system and to
hydrocarbons systems have given encouraging results [60].
This dissertation explores the application of the BOP formalism developed by
Pettifor and co-workers to elemental silicon and germanium. The resulting interatomic
potentials are assessed by comparison of their predictions with those of the widely used
Stillinger-Weber (SW) and Tersoff potentials as well as experimental and ab initio data.
Each potential is assessed for its ability to predict cohesive energies, elastic moduli,
atomic volumes, small clusters, defect formations energies, melting temperature and
surface reconstructions. The potentials for both Si and Ge have been formulated in a
manner that will simplify the future development of SiGe potentials. The utility of the
silicon potential is investigated by examining the atomistic mechanisms responsible for
the rearrangement of atoms at the amorphous/crystalline interface during the early
stages of solid phase epitaxy.
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II. Molecular Dynamics Methods
If the interatomic forces between the atoms of an ensemble can be written as a
function of atom separation, the solution to Newton’s equations of motion for the
ensemble provide a detailed description of the ensembles atomic coordinates, the
vibrational states (and the excursions of these that result in atom migration), and the
time varying forces acting upon all the atoms in the system. This approach is known as
molecular dynamics (MD) and it was first introduced by Alder and Wainwright in the
1950’s [61, 62]. System properties such as pressure [63], the internal energy [63], the
viscosity [64], diffusion rates [64], and the specific heat [65] of atomic ensembles can be
well predicted using this method. The approach numerically integrates Newton’s
equations of motion to track the particle positions as a function of time. The emergence
of powerful modern computers enables large and/or complicated systems to be
simulated. One such example is the simulation of supersonic crack propagation in two-
dimensional crystal lattices by Farid Abraham et al. at IBM [66]. Another is the work of
V. V. Zhakhovskii et al. [67] in which extreme overheating is applied to a planar structure
in vacuum, consisting of between 105 and 106 atoms, causing it to expand.
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The theory behind MD can be traced back to Gibbs in 1902 [68]. In his landmark
work “Elementary Principles in Statistical Mechanics”, Gibb’s introduces the idea of a
canonical ensemble. A canonical ensemble refers to an infinite number of systems of a
constant volume V, at a constant temperature T, each containing N atoms. In such an
ensemble, the properties are allowed to change over time, and can vary from one
system to another system. Despite the statistical variation it is possible to calculate
observable properties via two different methods. The first is to calculate the property at
a fixed time for all the systems. The average value would be the observable property of
the ensemble. The second method takes the average of the property over a long period
of time for a single system in the ensemble. Given an evolution in time that sampled all
potential structures the second method gives the same value as the first. MD is an ideal
approach for studying short time dynamics. It can also be used (using the methods
described above) to calculate the average (observable) properties of a system.
The general method involved in molecular dynamics is displayed in a flowchart in
Figure 2.1. The first step is to create the system under consideration. This involves
setting the initial position and velocities for all the atoms in the system. Next the forces
between the atoms are calculated using an interatomic potential describing the forces
between atoms in terms of their local configuration. In the third step the atoms are
moved by integrating the equations of motion over the time step chosen for the system.
A periodic boundary condition is then applied to move atoms back into the central
30
simulation cell. A desired property is then calculated from the new atom positions. The
program iterates this process until the total number of time steps is equal to the time
limit set.
In principle, any analytic or numerical method for calculating the forces between
atoms allowing the equations of motion to be solved can be employed in MD
simulations. Many such methods have been proposed, including density functional
theory [69], tight binding [70], and interatomic potentials [71]. The computational
expense of each method versus the level of approximation is often the deciding factor in
the selection of a modeling regime. Because of the need to simulate a moderate
number of atoms (in the thousands) in a dynamic fashion, this dissertation has
employed interatomic potentials.
Figure 2.1 Flowchart displaying the basic logical pathways in the MD process.
Setup initial
configuration
Calculate force
on each atom
Simulation time
exceeded?
Average time-dependent property values
to obtain accurate property values YES
NO
Calculate the desired
properties
Integrate
equations
of motion
31
Interatomic potentials abandon the time consuming calculation of the electronic
ground state in favor of empirically generated equations to capture the interaction of
atoms [71]. These potentials consist of a set of empirical functions dependent upon
atomic positions and “fitting” parameters. These functions and parameters can give a
complete description of system energy and forces acting on each particle (excluding
energetic interactions within individual atoms).
Recent work on non-empirical, physically motivated bond order potentials (BOP)
by David Pettifor and his collaborators has expanded the nature of the classical
interatomic potential [56-60]. The functional form of the BOP has been derived from
orthogonal TB and draws heavily on the molecular orbital bonding there. The BOP
format includes extrinsic terms to account for both s and p contributions to covalent
bonding as well as a term to account for the orbital hybridization. This format holds
great promise for atomistic modeling due to its ability to operate on a timescale close to
that of empirical IPs yet still retaining an element of the electronic interactions which
lend predictive validity to TB methods. It must be noted that the predictive validity of
these interatomic potentials is inextricably tied to the quality of their parameterization.
The parameterization process, or potential fitting, is by no means an easy process. In
fact the bulk of the time spent during the course of this research has been dedicated to
the fitting of the silicon and germanium BOP potentials. Great care must be employed
to ensure that modeling predictions are reasonable. This requires rigorous evaluations
of an IP’s transferability across a wide range of properties.
32
2.1 Stillinger-Weber Potential
The Stillinger-Weber (SW) many-body interatomic potential was proposed in
1985 in response to the growing interest in the utilization of computer simulation
modeling techniques. Previously developed pair potentials, such as the Lennard-Jones
potential [72], are incapable of stabilizing the equilibrium diamond cubic structure of
silicon under ambient conditions [73]. These pair potentials invariably prefer close
packed crystalline phases [73]. The silicon system presented interesting challenges,
most notably, unlike most other elemental systems, silicon is observed to increase in
density when it melts [1]. Diffraction experiments have shown that the melting process
causes the crystal structure to collapse, substantially increasing the coordination from 4
to an average value in excess of 6 [74-76]. This process is also accompanied by a shift in
the electronic properties of the element from semiconductor to metal [77]. The SW
potential (or family of potentials) was introduced to model these condensed phases of
silicon with the desire to predict details of the change in local order for tetrahedral
semiconductors as they melt [46].
The SW potential remains one of the more widely used interatomic potential
formats for the simulation of open structure materials. This is due to its combination of
ease of implementation and reasonable predictive validity. The potential employs a
common approximation approach for the interaction energy of a system of N identical
particles in which the energy is expressed in terms of sums of all possible one-body (i),
33
two-body (i, j), three-body (i, j, k), etc. interactions terms. Thus, the potential energy is
represented by:
2.1
This representation depends heavily on the rapid convergence of the energy with
increasing interaction order. The SW potential truncates the form of Eq. 2.1 to include
only the two and three-body terms [46]. Some modifications/extensions of the SW
format include the four-body term; however these potentials will not be discussed here
[78].
The two-body term, f2, is required to be a function only of radial distance, and is
written:
2.2
Where the multiplicative combination of parameters a and set the range of the
potential. B is an adjusted value that determines the location of the potential minimum,
and A is set such that the value of f2 at the potential minimum is -1. The parameter is
a scaling parameter. The values chosen for simulation of silicon are A = 7.04956, B =
0.60222, = 2.0951 Å, = 2.1702 eV, a = 1.8, p = 4, and q = 0 [46] (values with no unit
are unitless). The potential naturally approaches a value of 0 at r = a·, thus avoiding
Nji
N
jii
NjifjififU
),(),()( 21
0
exp2)(2
ar
rrB
A
rf
qp
ar
ar
34
the need for a separate cutoff function. The smooth approach to 0 aids the potential in
avoiding the generation of artifacts in the simulation [46].
The three-body term, f3, is required to have rotational and translational
symmetry [46]. It is expressed as a symmetrized sum:
2.3
Where h(ijk) is dependent on the atomic separation distances rij and rjk, and the angle
ijk. Given both radii are less than a·, h has the following form:
2.4
If either rij or rjk are greater than a·, h = 0. The angular term is strongly preferential in
favor of the tetrahedral configuration yet does not preclude the possibility of alternative
geometries [46]. The SW potential assigns values of 21 and 1.2 to the parameters and
respectively [46] (both are unitless).
The SW potential for Si was parameterized using a least mean squares fitting
routine with an emphasis on several criteria [46]. First and foremost was the
requirement that the diamond cubic structure be the most energetically stable at zero
temperature and pressure [46]. This is of course the nature of real silicon [1]. Secondly
the potential was required to predict in “reasonable accord” the melting point and
liquid structure [46]. In these two criteria the SW potential is quite successful in
reproducing the desired experimental properties of silicon [46]. The potential, however,
)()()()(3 jkihjikhijkhijkf
23
1cosexp),,(
ijk
jkij
ijkjkijarar
rrh
35
does not predict a -Sn transition at high
pressure [63,79]. This raises concerns
regarding the transferability quality of the
structural and mechanical energy calculations.
The SW potential has been further employed
to examine many features of the elemental Si
system, and a few of these are discussed here.
The SW parameters for silicon and germanium
can be found in Table 2.1.
Feuston, Kalia, and Vashishta performed a detailed molecular dynamics study of
the silicon microclusters using the SW potential [80]. Their results indicated the
presence of unusually stable “magic number” clusters of size 4, 6, and 10. This result is
mirrored in photofragmentation experiments performed by Bloomfield et al. [81].
While the energetic stability of the magic number clusters matches that found in
experiment, the geometric structure of small silicon clusters of size n = 3-6 are
inaccurate [82,83]. The geometric structures preferred by the SW potential for clusters
of size 3, 4, 5, and 6 are the equilateral triangle, square, pentagon, and two equilateral
triangles directly on top of each other respectively [80]. Experiment shows the proper
geometric structures to be a bent chain, rhombus, trigonal bipyramid and distorted
octahedron [82,83].
Table 2.1 Parameters for silicon and
germanium for the Stillinger-Weber
potential
Si Ge
A 7.04956 7.04956
B 0.60222 0.60222
Å 2.0951 2.181
eV 2.1702 1.93
a 1.8 1.8
p 4 4
q 0 0
21 31
1.2 1.2
36
Khor and Das Sarma employed the SW potential to examine low index surfaces
of silicon [84,85]. Of particular interest was the investigation of the (100) surface [84]
and the dispute (at the time) between buckled and symmetric dimers [86-89]. Also of
interest was the Pandey defect reconstruction model which had been observed using
scanning tunneling microscopy [84]. The SW potential was found to provide an accurate
description of the symmetric dimers on the (100) surface and that the Pandey-type
defect reconstruction may be favorable in energy compared to the dimer reconstruction
[84]. The SW potential was unable to predict the presence of buckled dimers even as
higher energy metastable structures [84].
Ding and Andersen extended the SW potential format to germanium and
employed it to examine amorphous germanium [54]. They attempted to parameterize
the SW format to accommodate three separate phases of germanium: the liquid, the
diamond lattice crystalline solid, and the amorphous solid. However, they found no
parameter set that simultaneously gave a good description for all three phases of
interest [54]. As a result they settled on a good description of the amorphous and
crystalline solid phases. The resulting parameter set for Ge is: A = 7.04956, B = 0.60222,
p = 4, q = 0, a = 1.8, = 31, = 1.2, = 1.93, and = 2.181 [54]. This potential is shown
to accurately reproduce radial distributions in the amorphous and crystalline phase of
germanium as well as a reasonable prediction (within 10%) of the phonon dispersion
curves [54].
37
2.2 Tersoff Potential
The Tersoff interatomic potential was proposed in 1986 with the intent to
develop a potential with which it was feasible to calculate the structure and energetics
of complex covalently bonded systems [47-49]. Tersoff recognized that previous
interatomic potentials, SW included, did not attempt to describe accurately the
properties of nontetrahedral forms of silicon [47]. The potential format that Tersoff
proposed was motivated by intuitive ideas about the dependence of bond order upon
the local environment. The Tersoff potential represents the first such potential to
incorporate the bond order in the functional format, however empirically [47]. The
Tersoff potential is modeled off a Morse-type pair potential, which allows a physical
interpretation of the potential parameters [47-49].
As one of the earliest interatomic potentials to become readily adopted, the
Tersoff potential had a fairly simple functional format. Over the course of a few years,
Tersoff continued to make minor adjustments to the functional format [47-49]; the
format employed in this research and presented here is that published in Ref. 49. If one
were to consider a system of atoms, the total potential energy of the system, U, could
be captured as the sum of the energy of the individual bonds in the system. The Tersoff
potential [49] represents this empirically with:
ji
ijAijijRijc rfbrfrfU )()()(2
1
2.5
38
The function fc is a cutoff function designed to restrict the range of the potential. The
first term in square brackets in Eq. 2.5 is interpreted to represent the repulsive
electrostatic force that two atoms encounter when brought within close proximity to
each other. The second term is interpreted to represent the bonding energy between
two atoms. This term is adjusted by bij which includes considerations of the bond order
and the local environment.
The functions fR and fA represent repulsive and attractive pair potentials. The
choice of exponential functional forms for these, as in a Morse potential, was based on
the “universal” bonding behavior discussed by Ferrante, Smith, and Rose [90, 91]. They
had shown that a large number of calculated binding energy curves could be mapped
onto a single dimensionless curve which could be well described by the assumption of a
Morse-type pair potential [90, 91]. This lead to fR and fA to have the form:
ijijijijR rArf exp)(, 2.6
ijijijijA rBrf exp)( , 2.7
where Aij and Bij are the geometric average of the fitted parameters A and B for atoms i
and j. The parameters ij and ij are the average of the fitted parameters and for
atoms i and j.
The function bij represents the only deviation from a potential that is otherwise
wholly pair-wise [47-49]. It represents a measure of the interatomic bond order and has
been written:
39
iii nn
ij
n
iijijb2
1
)1(
, 2.8
)()(,
ijkikik
jik
cij grf
, 2.9
22
2
2
2
))cos((1)(
ijkii
i
i
i
ijkhd
c
d
cg
. 2.10
where ijkis the bond angle between bonds ij and ik. The parameter ij has a value of
unity if between two atoms of the same type, for a bond between silicon and
germanium this term has a value of 1.00061. This small adjustment was included to
properly simulate the heat of formation of the zinc-blende structure [49]. Similarly the
parameter ij also has a value of unity, and was included in the potential for possible
future flexibility. The parameters , n, c, d, and h are all dependent only on atom i.
Values for these parameters for the silicon and germanium systems are listed in Table
2.2 [49].
Unlike the Stillinger-Weber potential, the Tersoff functional format includes an
explicit cutoff term [47-49]. This step function is designed to create a smooth transition
to between the separation ranges of R and S.
ijij
ijijij
ijij
ijij
ijij
ijC
rS
SrRRS
Rr
Rr
rf
0
)cos(2
1
2
1
1
)( .
40
The Tersoff parameters for Si-Si,
Ge-Ge, and Si-Ge interactions are listed
in Table 2.2. The 11 unknowns per
material (parameters) are fitted to 7
different materials properties: cohesive
energies of real and hypothetical bulk
crystals (exact structures are not
specified in the publication [49]), bulk
modulus, diamond bond length, and
elastic constants c11, c12 and c44
constrained to within 20%. Tersoff justified the use of a small database during the
fitting by subsequently comparing the resulting potential against a much larger database
to verify its suitability [47-49].
The Tersoff potential has been widely used for molecular dynamics research due
to the ease of its implementation and its physical motivation (albeit empirically
formulated). For example, Dyson and Smith utilized the Tersoff potential in the
examination of the (001) and (111) 2x1 surface reconstructions of diamond [92]. They
compared the Tersoff predictions to other interatomic potentials; the Stillinger-Weber
and two proposed modifications to the Tersoff potential by Brenner [53, 92]. They
concluded that while the Tersoff potential gave better results than the SW potential, it
was still not in qualitative agreement with other work. The Tersoff-Brenner potentials
Table 2.2 Parameters for silicon and
germanium for the Tersoff potential.
Si Ge
A (eV) 1830.8 1769
B (eV) 471.18 419.23
Å) 2.4799 2.4451
Å) 1.7322 1.7047
1.1 x 10-6 9.0166 x 10-7
n 0.78734 0.75627
c 1.0039 x 105 1.0643 x 105
d 16.217 15.652
h -0.59825 -0.43884
R Å) 2.7 2.8
S Å) 3.0 3.1
41
[53] performed significantly better in surface property prediction, and they cautioned
the use of empirical potentials fitted to bulk properties when examining surfaces [92].
Other research, such as that performed by Mura et al., has shown more
promising results [93]. Mura et al. employed the Tersoff potential to examine the
structural trends in SiC amorphous alloys [93]. They were able to characterize the local
coordination found in amorphous Si1-xCx as a function of the alloy composition. They
concluded that for high carbon content the disorder is limited by the distortion of the
silicon sublattice [93]. Tarus and Nordlund used the Tersoff potential in the simulation
of germanium surface segregation during deposition of Si on Ge/Si (001) surfaces [94].
Their molecular dynamics research uncovered a surface segregation mechanism
involving the movement of two atoms at the surface. This mechanism, they observed, is
thermally activated with a fairly short time scale for the process (~1 ns) [94].
Of particular interest to this research is the research of Motooka of Kyushu
University in Japan [31, 95]. Motooka investigated amorphization and crystallization
processes in ion implanted silicon [31, 95]]. Using the Tersoff potential and cross-
sectional transmission electron microscopy he determined that amorphization in a
crystalline lattice can be induced when the concentration of the di-vacancy and di-
interstitial pairs exceeds a certain threshold value [31, 95]. He also observed that
epitaxial growth mechanisms at the amorphous/crystalline interface are quite similar to
crystal growth from the melt [31, 95]. The solid phase epitaxial growth of silicon will be
discussed later in section V along with calculations of the process using the BOP.
42
2.3 Bond Order Potential
If an interatomic potential for a covalently bonded material is to be used to
simulate assembly of the condensed state, it must reliably model the radial and angular
dependence of the dynamic interatomic forces present during the formation and
breaking of atomic bonds. Germanium has 2 s and 2 p electrons and its covalent
bonding within the condensed phases can be described using sp-hybridized atomic
orbitals whose overlap results in bonding and anti-bonding molecular orbitals with or
symmetry [17]. The strength of a covalent bond (its bond order) can be characterized
as one-half the difference between the number of electrons in bonding and anti-
bonding molecular orbitals within the bond [96]. Bond orders, when multiplied by bond
(hopping) integrals, then yield bonding energies [97]. The bond integrals are related to
the probability that an electron will hop between atoms, and so depends upon the
interatomic separation and angle and the overlapping orbital type [98]. Figure 2.2
illustrates the specific hopping paths that are directly accounted for within the BOP.
A general analytic expression for the reduced (separate and bond-order
contributions) BOP derived from two-center orthogonal tight binding theory has been
detailed by Pettifor and his collaborators [56-59]. While this analytic bond order
potential might at first appear complicated, it is composed of several terms with straight
forward physical interpretations. The bond order potential seeks to describe the real
space potential energy function E(r,) (where r is the interatomic separation and an
angular dependence) for a collections of atoms. If the atoms are bound by sp hybridized
43
electron states, this potential can be written as the sum of three terms; an electrostatic
repulsive term (Urep), an attractive bonding term (Ubond), and a promotion energy term
(Uprom) associated with creation of the sp3 hybridized orbitals:
prombondrep UUUE 2.12
The atoms in the condensed phases of group IV elements such as germanium can
form bonds with their neighbors that have either (primary covalent) or (secondary
covalent) bonding characteristics. The bonding term in Eq. 2.12 (Ubond) is therefore a
sum of the separate and contributions to the bonding energy:
i ijj
ijijijijbondU,
,,,, 222
1 , 2.13
where and are the and bond (hopping) integrals between atoms i and j. The
coefficients and are the bond orders of the and bonds between atoms i and j.
Figure 2.2 Hopping integration paths of length 2 (a and d), and length 4 (b, c and e)
that contribute to the potential energy of the bond between atoms I and j. The
position of atoms k and k’ determine the local environment of the bond.
44
The bond integrals are radially dependant functions of the atomic separation between
atoms i and j [56]. The bond orders are dependent on the local environment of the ij
bond [56]. This methodology connects well with the single, double, and triple bond
terminology from chemistry [17]. It also results in maximum allowable values for the s
and p orbital contributions to the bond order. These are 1 for bonds, and 2 for
bonds.
The repulsive potential energy component of Eq. 1, Urep, approximates the
electrostatic repulsion between the atomic cores using a simple pair potential :
i ijj
ijrepU,2
1 , 2.14
The and bond integrals, and , found in Eq. 2.13, and the repulsive pair potential
found in Eq. 2.14 can be writing in a similar fashion:
n
rf0, 2.15
n
rf0, , 2.16
mrf0 2.17
where f(r) is a radially dependent Goodwin-Skinner-Pettifor (GSP) function [99], 0,
0,and 0 are fitted parameters that control the magnitude of the bond energy, and
nand n are fitted parameters that control the rate at which and respectively
approach zero as r increases. The value m/n is a measure of the hardness of the
45
repulsive potential. It is fully soft in the limit m/n = 1 and totally hard for m/n = ∞
[100]. For group IV sp valent materials such as silicon and germanium, a m/n value of
~2 is expected [100]. The germanium potential described here has a fitted m/n ratio of
1.88.
The GSP function is a pairwise function derived from the TB expression for the
binding energy [99] of the form:
cc n
c
n
c r
r
r
r
r
rrf 00 exp , 2.18
where r is the interatomic spacing between bonded atoms, r0 is the spacing at which f(r)
= 1, rc is a characteristic radius, and nc is a characteristic exponent. We have assumed
that the parameters r0, rc, and nc have identical values for the three pair functions in Eqs.
2.15-2.17. This simple constraint significantly eased the subsequent parameterization
process used to fit the potential framework to the condensed states of Ge and did not
decrease the potential’s performance.
The bond order, in Eq. 2.13, has been derived for half-full valence shell
systems such as germanium and later generalized to include systems of generalized
valence band occupancy [56-59]. The half-full valence shell bond-order has been
employed here and has the derived form:
2/1
2
4
42242, ~
1
~2
~~2
1
jiij
ij
R 2.19
46
where
, 2.20
. 2.21
The 2 term in Eq. 2.19 describes the self-returning, second moment two-hop
contributions to a bond between atoms i and j, and is illustrated in Fig. 2.2a. This term
has the form:
jik ij
ik
jikij g,
2
,
,2
,,2
2.22
where g is a function introducing angular dependant contributions to the bond order
resulting from orbital overlap. It has the form:
1cos1, jikjik pg 2.23
where ijk is the angle, centered on atom i, between the atom pairs ij and ik. The term
g has a single free parameter, p. This parameter has physical significance. For s
orbital bonding, because of radial symmetry, there is no angular dependence (p = 0);
for p orbitals, because of the orthogonalality, the energy minimum is found at 90°
angles and therefore the function g has a cos form (p = 1) [100]. This parameter
ji
224
4
4
~
ji
ji
ji
224
22
22
~~
47
allows for the potential to control the hybridized atomic orbital overlap dependency
[100]. The angular term can be optimized for the sp3 diamond cubic structure by
choosing p = 0.75
The remaining two terms in Eq. 2.19 represent hopping paths of length four. The
4term describes the self-returning hopping paths of length four linking atoms i and j
while the R4 term represents a 4-atom ring-type interference path linking atoms i and j.
These terms are illustrated in Fig. 2.2b-c respectively and are written in the form:
2.24
,
jikk ij
jk
ij
kk
ij
ik
kijjkkkikjikij ggggR,, ,
,
,
,
,
,
,,,,4,
2.25
The analytic expression of the bond order for a half full valence shell with hops
of length two and four can be written as:
ij, 2.26
where
jik ij
ki
ij
ik
ijkkkijik
jik ij
ik
jik gggg,
2
,
,
2
,
,
,,,
,
4
,
,2
,4
jikk ij
kk
ij
ik
kikjik gg,,
2
,
,
2
,
,2
,,
48
21
,4
,2,2
21
ij
jiij
2.27
The 2 terms are defined as:
jikk ij
ik
jikikij
,,
2
,
,2
,,2 2sinˆ
2.28
where the angle jik is described in Fig. 2.2d and
2
,
,
2
,
,
,ˆ
ij
ik
ij
ik
ik p
2.29
The fitting parameter p is species dependant only on the central atom i (for
multicomponent systems). The f4 term includes four-body dihedral angles, fkk’, that
are influential in p type bonding.
kjijik
jikk
kiikij
22
,,
2
,
2
,,4 sinsinˆˆ
kijjikkjik 222
,
2
, sinsinˆˆ 2.30
kjiijkjkki 222
,
2
, sinsinˆˆ
4
2cossinsinˆˆ 222
,
2
,kk
kijijkkjjk
where the angles are defined in Fig. 2.2e and the dihedral angle is defined as
49
1sinsin
coscoscos22cos
22
2
kjijik
jikkjikki
kk
2.31
The promotion energy term in Eq. 2.12 represents the energy penalty incurred by
elevating a s level electron into a p level atomic orbital and reconfiguring the atomic
orbitals into the hybrid sp3 configuration found in the diamond structure. Following
reference 56, the promotion energy can be approximated by:
212
,11
ij
ij
prom AU
2.32
where and A are fitting parameters.
To summarize, the format of the BOP employed herein uses a potential energy
function (Eq. 2.12) that is expressed in terms of bonding (Eq. 2.13), repulsive (Eq. 2.14)
and promotion energy (Eq. 2.32) components. The repulsive component is purely
pairwise in nature whereas the bonding component has been further subdivided into
and terms. The and bonding elements are the products of bond integrals (Eq. 2.15
and 2.16) and bond orders (Eq. 2.19 and 2.26) respectively. The bond order (Eq. 2.19)
contains hopping paths of length 2 and 4 including a 4-atom ring-type interference path
(Eq. 2.25). The bond order (Eq. 2.26) contains hopping paths of length 4 and includes
dihedral angle effects.
There are a total of 15 free parameters that need to be fitted for the BOP
defined above. The best estimates of these 15 parameters are given in Table 2.3. There
50
are 6 GSP parameters (r0, rc, m, n, n and nc), 3 prefactors (repulsive 0, bonding ,0
and ,0), 2 angular terms (p and p), 2 promotion energy parameters ( and A) and 2
cutoff parameters (r1 and rcut). These have been determined for the BOP using a
method adapted from the work of Albe et al. [101].
A systematic three-step process for free parameter fitting first determined the
pair function (the bond energy of cubic structures and dimers can be expressed solely as
a function of bond length) and then in a second step incorporated the angular function
parameters. Because the promotion energy term as described in previous publications
Table 2.3 Parameters for silicon and germanium for the Bond Order Potential. The
silicon potential uses term sensitive values for the rc and nc parameters.
Symbol Quantity Value
Ge Si
r0 GSP reference radius (Å) 2.32 2.349
rc GSP characteristic radius (Å) 3.634 : 14.932 : 11.675 : 2.771
m GSP repulsive exponent 6.187 8.742
n GSP attractive exponent 3.405 2.944
n GSP attractive exponent 1.012 4.040
nc GSP decay exponent 10.0 : 2.523 : 6.533 : 29.482
r1 Spline start radius (Å) 3.2 3.1
rcut Spline cutoff radius (Å) 3.7 3.6
0 Repulsive energy prefactor (eV) 2.415 1.012
,0 bond integral prefactor (eV) 2.171 2.224
,0 bond integral prefactor (eV) 0.299 0.196
p bond 3-body angular term 0.75 0.732
p p bond 3-body angular term 0.1 0.1
Promotion energy term (eV) 6.605 10.179
A Promotion energy term 0.798 1.852
51
[56-59] cannot be expressed in the form of a pair function, it could not be employed
with the first two steps of the fitting. The third step therefore introduced the
promotion energy term. The methodology for the first two steps of this approach is
discussed in detail in Appendix A of reference 102. The promotion energy contributions
to the various structures were determined using a least mean squares fitting routine.
This enabled incorporation of a physically meaningful promotion energy term into the
potential following an iterative process of parameter optimization. A detailed look at
the methodology of this fitting process is presented in Appendix A.
There exist a great many different methods of employing DFT and each method
often predicts slightly different energies for similar systems. Because of this, it was
necessary to construct a consistent database of bulk material properties (including
cohesive energy (Ec in eV), atomic volume (Va in Å3), lattice constants (a and c in Å), and
bulk moduli (B in GPa)) of various equilibrium and metastable silicon and germanium
phases. These calculations were performed with the local density approximation (LDA)
of DFT using the VASP DFT package. A description of the details of these VASP
calculations can be found in Appendix B. In the fitting of silicon and germanium higher
priority (weights) were given to the atomic volume and cohesive energy of the diamond
cubic and -Sn phases since these are well established silicon and germanium phases.
Accurate structural and energetic predictions of surface energies and defect formation
energies were considered important. An accurate simulation of theoretical bulk phases
(such as fcc and bcc) and the configuration (but not the energy) of clusters were given a
lower weight, and as a result are less well predicted.
52
III. SW and Tersoff Silicon Assessment
It would generally be considered unwise to employ an interatomic potential to
study the atomistic mechanisms of a material system if it is known that the potential’s
predictions of those properties are poor. It is therefore necessary to perform a robust
assessment of an interatomic potential to determine which properties a potential is well
suited to model. This research is motivated by an interest in atomic assembly; therefore
the material properties assessed are all ones which can be tied in some way to atomic
assembly processes. The following properties have been examined for a wide range of
structures with varying atomic coordination (sc, fcc, bcc, dc, -Sn, hcp and bc8): atomic
volume, cohesive energy and bulk modulus. The following bulk properties of the
diamond cubic phase have also been calculated: the elastic constants (c11, c12 and c44),
the phonon vibration spectrum, the Cauchy pressure and the melting temperature (Tm).
The structure and energetics of small silicon clusters (Sin with n=2-6), defect formation
energies and the energy of low index surface reconstructions have also been examined.
This chapter details the simulation methods employed to calculate the
properties listed above using the Stillinger-Weber and Tersoff potentials for the silicon
53
material system. Subsequent chapters will forgo explanations of calculation methods
and merely present data and comments in a similar format. Examination of the SW and
Tersoff Ge predictions can be found in chapter V, while the BOP predictions of these
properties can be found in chapters IV and VI for Si and Ge respectively.
3.1 Bulk Properties
An essential requirement of an interatomic potential is a proper prediction of
the equilibrium phase under ambient conditions. Failure to meet this critical
requirement will jeopardize the confidence in predictive validity that is an essential
component of modeling research. Furthermore, the ability of a potential to model the
local bonding environment can be inferred by examination of the structure of various
other phases where the atomic coordination ranges from 4 to 12. The structures that
have been selected are as follows (abbreviations and atomic coordination are listed in
parenthesis): simple cubic (sc, 6), face centered cubic (fcc, 12), body centered cubic (bcc,
8), diamond cubic (dc, 4), -Sn (4), hexagonal close packed (hcp, 12) and bc8 (4). In the
case of silicon, the lowest energy equilibrium phase at room temperature and
atmospheric pressure is the dc structure. The unit cell for each of these structures and
the DFT predicted lattice constants can be found in Figure 3.1.
The bulk properties (atoms volume, cohesive energy and bulk modulus) were all
calculated concurrently. Crystal files (labeled “r” files) were generated using a
54
Mathematica program developed by Dewey Murdick called CG-NNL (this program will
be discussed in more detail in a chapter IX) for each of the seven examined crystal
structures. These crystals were then minimized in energy by inducing strains on the
crystal to find the lattice constant a (a and c in the case of -Sn and hcp). In this way we
constructed the binding energy curves, useful graphical tools for determining the fidelity
of a potential. From these curves the atomic volume, cohesive energy and bulk modulus
g)-Sn
55
Figure 3.2 Binding energy vs. relative volume (with respect to equilibrium dc
volume) for a range of Si bulk structures. a) Stillinger-Weber b) Tersoff c) DFT.
dc
-Sn
bc8
sc
bcc
fcc
hcp
56
can be extracted. The binding energy curves for silicon as predicted by SW, Tersoff and
DFT can be found in Figure 3.2.
The binding energy curves are a quick way of examining a potential’s viability
over a wide range of structures. As such, they are the first property that has been
examined for each potential. The binding energy curves give a wide spectrum visual
overview of a potential’s predictions of energy, volume and moduli of varying phases
simultaneously. As can be observed from Figure 3.2 the SW potential for silicon predicts
with reasonable accuracy the relative energy differences between the bulk lattice
phases (the minimums of the curves lie approximately where DFT calculations indicate
they should). The SW potential, however, does not predict proper volume relations for
the phases. The relative volumes of the curves differ significantly from the DFT
calculations. The binding energy curves also reveal that the SW potential incorrectly
places -Sn and bc8 phases in reversed order [79]. As pressure increases (relative
volume V/V0 decreases) the bc8 phase curve should cross the dc phase curve first
followed by the -Sn phase curve [63]. The SW potential predicts the opposite. The
Tersoff potential slightly overestimates the energy differences between structures. The
most notable failure of the Tersoff potential binding energy curves is that the bc8 phase
curve does not at any point become the lowest energy. Instead the sc phase finds itself
located between the dc and -Sn phase curves. This suggests that the Tersoff potential
is unlikely to reproduce the -Sn to bc8 phase transition properly.
57
The volume of an atom in a given crystal
structure can be written as a function of the lattice
constants of that structure. Table 3.1 lists these
functions for each of the crystal structures examined.
The SW and Tersoff predictions of the atomic volumes
are listed in Table 3.2. Also included are the
percentage differences from the DFT calculations. The
average percent difference in atomic volume predicted
by the SW potential is ~17%. Only the dc and bc8
phases are predicted within 10% of the DFT
calculations. The Tersoff predictions are notably
superior, with all but the hcp phase predicted within
10%. These results are not encouraging for the
potential’s transferability to those phases.
The cohesive energy of an atom is defined as the energy required to remove an
atom from the bulk and move it beyond interaction range (i.e. infinity) [98]. This can be
Table 3.2 Atomic Volume Predictions (Å3)
Structure dc sc fcc bcc bc8 -Sn hcp
DFT 19.79 0.789 0.704 0.717 0.869 0.749
SW 20.013 0.889 0.892 0.854 0.902 0.861 0.936
% diff 1.13 13.91 28.04 20.42 4.92 16.10
Tersoff 20.013 0.818 0.739 0.735 0.918 0.764 0.937
% diff 1.13 4.78 6.09 3.61 6.80 3.09
Table 3.1 Atomic Volume as a
function of lattice constants
structure Va
sc 3a
fcc 4
3a
bcc 2
3a
dc 8
3a
-Sn 4
2ca
hcp ca 2
4
3
bc8 16
3a
58
found by taking the total sum of the energy of a system of simulated bulk atoms and
dividing by the total number of atoms in the system. Table 3.3 contains the cohesive
energies for the DFT, SW and Tersoff predictions, as well as the percentage differences
between the DFT and interatomic predictions. DFT calculations indicate an energy-
structure trend from minimum to maximum of dc, bc8, -Sn, sc, hcp, bcc and fcc. The
SW potential captures the energy trends well; the order of structures from minimum to
maximum energy is the same as the DFT predictions with the exception of the bcc phase
and hcp phase switching places. The Tersoff potential switches the bcc and hcp phases
as well, and more importantly also switches the sc and b-Sn phases. Of particular note
is the SW prediction of the dc phase cohesive energy. The SW potential predicts a
cohesive energy of -4.34 eV for the dc phase of silicon. Experimental measurements
have found that the energy is -4.63 eV [1]. Stillinger and Weber address this issue in
their original publication and recognize that a rescaling of certain parameters (A and B)
would allow for a proper energy prediction [46]. They instead chose to prioritize the
solid-liquid transition temperature. They also indicate that augments to the potential in
Table 3.3 Cohesive Energy Predictions (eV)
Structure Dc sc fcc bcc bc8 -Sn hcp
DFT -4.63 0.292 0.468 0.451 0.122 0.213
SW -4.34 0.274 0.396 0.281 0.191 0.20 0.305
% diff 6.26 6.27 5.24 2.87 7.96 6.27
Tersoff -4.63 0.319 0.761 0.432 0.247 0.329 0.522
% diff 0.0 0.62 7.04 0.44 2.77 2.63
59
the form of single particle, position independent terms could be used to correct the
cohesive energy [46].
The bulk modulus can be calculated by considering the sound derivative of the
total energy with respect to strain near the equilibrium. The bulk modulus can be
defined [98] as
2
2
9
1
tot
c
E
VB 3.1
where Vc is the unit cell volume and a strain parameter. A similar method is used to
obtain the elastic constants c11 and c44 where
2
2
11
1
tot
c
E
Vc and 3.2
2
2
444
1
tot
c
E
Vc . 3.3
The elastic constant c12 can be calculated from B and c11 (c12 = (3B-c11)/2) [98]. A more
in-depth description of the method used can be found in a book by M.W. Finnis entitled
“Interatomic Forces in Condensed Matter” [98].
The bulk moduli predictions can be found in Table 3.4. Of the bulk properties
examined, the bulk moduli are the poorest predicted by the SW and Tersoff potentials.
Both the SW and Tersoff potential’s predictions of the dc bulk modulus are in
reasonable accord with the DFT calculation. The other structures, however, do not fare
60
as well. The average percent difference of the prediction from the DFT is 240% and
182% for the SW and Tersoff respectively. It should be noted however that the Tersoff
average is skewed heavily by an aberrantly large prediction of the fcc bulk modulus. In
general, the Tersoff predictions are in good agreement with DFT.
The three independent elastic constants (c11, c12 and c44) have also been
calculated for the dc structure. Silicon has values of 165.78, 63.94 and 79.62 GPa for c11,
c12 and c44 (relaxed) respectively [8]. The SW potential predicts corresponding values of
90.32, 107.69 and 39.55 GPa. The Tersoff potential predicts values of 139.77, 77.26 and
29.01 GPa. These predictions lie within 50% of the elastic moduli, a result which is
within the acceptable range limitations found within Tight Binding models. The Cauchy
pressure (c12 – c44) of the strong covalently bonded material silicon is negative (-16 GPa)
[8]. This is because the high hardness and stiffness of silicon arise not from a high bulk
modulus, but from a high shear modulus due to strong resistance to interatomic
bending [98]. The SW and Tersoff predictions are 68.14 and 48.25 GPa respectively.
The prediction of a positive Cauchy pressure is a common feature among many
empirical interatomic potentials [98].
Table 3.4 Bulk Modulus Predictions (GPa)
Structure dc sc fcc bcc bc8 -Sn hcp
DFT 105.4 105.6 93.54 111.3 128.1 155.9
SW 101.9 190.6 714.5 742.7 74.8 410.2 322.4
% diff 3.32 80.49 663.8 567.3 41.6 163.1
Tersoff 98.1 132.6 1180.9 153.3 103.9 143.4 139.9
% diff 6.92 25.57 1162.45 37.74 18.89 8.02
61
The phonon spectrum of the diamond cubic phase was also calculated. This
probes the performance of the potential at near equilibrium bond spacing conditions. A
diamond cubic crystal of 512 atoms was annealed at a simulated temperature of 300 K
and the velocities of a randomly selected 50 atom sample were tracked and used to
calculate the velocity-velocity autocorrelation function. The vibrational spectrum for
the system was then calculated by taking the Fourier transform of this correlation
function. The resulting vibrational spectrum is shown in Figure 3.3. Silicon exhibits a
single strong peak at 520 cm-1 [103]. The SW and Tersoff potentials predictions both
exhibit strong peaks in the 500-540 cm-1 range, however the Tersoff potential shows a
primary peak at 110 cm-1.
3.2 Small Clusters
The calculation of the structure and energetics of small silicon clusters is
straightforward. The xyz position data for many high symmetry configurations of each
cluster size was input into the MD code and a molecular statics minimization was
performed. The resulting structure and bond lengths are recorded and the total energy
of the system is the binding energy of the small cluster. The investigation of silicon
clusters helps characterize the nature of atomic bonding as predicted by the
corresponding potential because the structure and bond energies of small clusters are
heavily reliant on both the angular and radial component of the interatomic potential.
Thus examination of cluster properties is a useful gauge of a potential’s transferability.
62
Examined herein are the structure of various silicon small clusters containing up to six
atoms by means of the SW and Tersoff potentials.
A wide range of high symmetry small clusters have been examined for silicon.
These are: the dimer, the trimer: chain (D∞h), bent chain (C2v), equilateral triangle
(D3h), tetramer: chain (D∞h), square (D4h), rhombus (C2v), flagged triangle (c2v),
tetrahedron (Td), pentamer: pentagon (d5h), pyramid (c4v), trigonal bipyramid (d3h),
hexamer: octahedron (Oh), edge-capped trigonal bipyramid (c2v), face-capped trigonal
63
bipyramid (c2v), and bent-chair hexagon (c2v). The structures examined are the same
structures examined in references 104 and 105. These structures are pictured in Figure
3.4 along with the Hartree-Fock predicted bond lengths [104, 105]. The Hartree-Fock
method (HF) is an approximate method for the determination of the ground state wave
function and energy of a many-body system. It is often used as the starting point in
many ab initio studies of molecules. The SW and Tersoff potential predictions for the
structure and energies of these small clusters are listed in Tables 3.5 and 3.6 for the SW
results and Tersoff results respectively.
The dimer is the simplest of the silicon clusters examined (and in fact the
simplest silicon cluster possible). It consists solely, by definition, of 2 silicon atoms
bound together. The angular dependence terms found within an interatomic potential
have no influence on the determination of the dimer properties. Experimental studies
have shown the bond length of the silicon dimer to be 2.246 Å and the binding energy to
be -2.41 eV. HF calculations predict a bond length of 2.265 Å and binding energy of -
3.06 eV [104, 105]. The SW potential predicts a bond length of 2.35 Å and a binding
energy of -2.17 eV. The Tersoff potential predicts a bond length of 2.313 Å and a
binding energy of -2.62 eV. The SW potential overestimates the dimer bond length by
4.7% and underestimates its energy by 10% (although much of the energy difference is
due to the scaling of the energy parameters to better match the melting temperature as
previously mentioned). The Tersoff potential overestimates the dimer bond length by
2.9% and overestimates the binding energy by 8.7%. These results are representative of
64
65
Table 3.5 SW Si Small Clusters.
Structure
Point
Group Bond
Bond
Length (Å)
Binding Energy
(eV)
SW HF
dimer dimer D∞h 1-2 2.352 -2.17 -3.06
trimers
linear chain D∞h 1-2 2.423 -3.81 ~
bent chain C2v 1-2 2.352 -4.34 -3.04
triangle D3h 1-2 2.563 -4.44 ~
tetramers
linear chain D∞h 1-2
2-3
2.413
2.512 -5.49 -7.29
square D4h 1-2 2.388 -8.15 -8.84
rhombus D2h ~ ~ ~ -10.71
tetrahedron Td 1-2 2.714 -6.68 -8.09
flagged triangle C2v
1-2
1-3
2-4
2.618
2.553
2.419
-6.11 -8.86
pentamers
pentagon D5h 1-2 2.352 -10.85 -10.08
trigonal bipyramid D3h
1-2
1-5
2-3
2.497
3.289
3.254
-10.32 -13.92
pyramid C4v 1-2
2-3
2.853
2.456 -10.74 -13.47
hexamers
edge capped
trigonal bipyramid C2v
1-2
1-3
2-3
3-4
3-5
2.435
2.563
3.333
3.273
2.435
-14.01 -18.26
face capped trigonal
bipyramid C2v
1-2
1-3
1-5
3-4
3-5
2.374
3.604
2.371
3.305
2.426
-14.15 ~
octahedron Oh 1-2 2.733 -12.86 ~
bent chair hexagon D3d 1-2
1-3 2.352 -13.02 -13.32
66
Table 3.6 Tersoff Si Small Clusters
Structure Point
Group Bond
Bond
Length (Å)
Binding Energy
(eV)
Tersoff HF
dimer dimer D∞h 1-2 2.3 -2.66 -3.06
trimers
linear chain D∞h 1-2 2.313 -5.25 ~
bent chain C2v 1-2 2.313 -5.25 -3.04
triangle D3h 1-2 2.313 -7.87 ~
tetramers
linear chain D∞h 1-2
2-3 2.319 -7.98 -7.29
square D4h 1-2 2.38 -8.64 -8.84
rhombus D2h ~ ~ ~ -10.71
tetrahedron Td 1-2 2.623 -7.09 -8.09
flagged triangle C2v
1-2
1-3
2-4
2.305
2.305
2.181
-6.65 -8.86
pentamers
pentagon D5h 1-2 2.323 -12.43 -10.08
trigonal bipyramid D3h
1-2
1-5
2-3
2.451
2.997
3.361
-10.95 -13.92
pyramid C4v 1-2
2-3
2.654
2.53 -10.34 -13.47
hexamers
edge capped
trigonal bipyramid C2v
1-2
1-3
2-3
3-4
3-5
2.389
3.002
2.53
2.88
2.483
-11.37 -18.26
face capped trigonal
bipyramid C2v
1-2
1-3
1-5
3-4
3-5
2.599
2.701
2.604
3.083
2.552
-13.23 ~
octahedron Oh 1-2
2-3
2.563
2.945 -12.38 ~
bent chair hexagon D3d 1-2
1-3
2.30
3.984 -15.79 -13.32
67
a reasonable level of potential predictive ability given that the silicon dimer (or any
other cluster property) was not considered during the parameterization of these
potentials.
Of the three different trimer configurations examined (the linear chain, the bent
chain and the equilateral triangle) ab initio calculations consistently determine that the
lowest energy configuration is the bent chain [104, 105]. The HF predictions have
determined the bond lengths of the bent chain to be 2.165 Å with an apex angle of 77.8°
[104, 105]. Neither the SW nor the Tersoff potential are able to stabilize a structure
with that apex angle, instead preferring to minimize into the equilateral triangle. Both
the SW and Tersoff potentials predict the equilateral triangle as the lowest energy
trimer structure. The SW potential predicts the bent chain structure to be only 0.1 eV
higher in energy than the equilateral triangle. The bent chain structure is very
competitive in energy because it is optimized at an angle of 109°. This angle
corresponds to the tetrahedral angle found in the diamond cubic lattice and is the
optimum angle for the SW three-body term [46]. The Tersoff potential predicts a clear
energetic advantage for the equilateral triangle. The linear chain and bent chain
structures are predicted by the Tersoff potential to have identical bond lengths and
energies; however, the apex bond angle of the bent chain structure is 132°.
We have examined five geometric arrangements as possible ground state
configurations of the silicon tetramer. These are: the linear chain, the square, the
rhombus, the tetrahedron and the flagged triangle. Ab initio studies indicate that the
68
lowest energy silicon tetramer can be found in the rhombus configuration [104, 105].
The SW and Tersoff potentials are unable to stabilize the rhombus structure. Instead
the higher symmetry square structure is preferred. The highest symmetry structure, the
tetrahedron, is not energetically competitive despite having a greater number of first
nearest neighbor bonds. This can be attributed to the large amount of bond bending
found in the structure.
Three pentamer geometries have been investigated here: the pentagon, the
trigonal bipyramid and the square pyramid. Two initial configurations for the trigonal
bipyramid and the square pyramid were considered. For the trigonal bipyramid the first
configurations had a large distance between atoms 1 and 5 (atom numbers are defined
in Figure 3.4) and a compact central triangle between atoms 2, 3 and 4. The alternative
configuration was distorted such that atoms 1 and 5 were closer together and the
central triangle was large. The second configuration has been predicted by ab initio
calculations to have a small energy advantage over the first (0.9 eV) and is the HF
predicted lowest energy pentamer configuration [104, 105]. The two square pyramid
configurations consist of a tight square with long bond lengths to atom 1 and a large
square with short bond lengths to atom 1. The SW potential predicts that all three
pentamer geometries are close in energy (only a 0.53 eV total energy range between
the structures) with the planar pentagon as the lowest in energy. The 108° angle found
in the planar pentagon is very close to the 109° optimal angle found in the SW potential,
therefore little bond bending is occurring. As a result, despite having fewer bonds than
the other configurations (5 compared to 8 in the square pyramid and 9 in the trigonal
69
bipyramid) the planar pentagon is predicted the lowest in energy. The Tersoff potential
predicts a clear energetic advantage for the planar pentagon structure for similar
reasons.
Four of the possible configurations for the silicon hexamer have been
investigated. These are: the edge-capped trigonal bipyramid (Ecap), the face-capped
trigonal bipyramid (Fcap), the octahedron (tetragonal bipyramid), and the chair-bent
hexagon. Ab initio studies have shown that the Ecap structure is the lowest energy
[104, 105]. The SW potential predicts the two capped trigonal bipyramid structures to
be very close in energy, with a small energy advantage for the Fcap structure (0.14 eV
lower in energy). The Tersoff potential predicts the chair-bent hexagon as the lowest
energy structure. This result is interesting in that it is the only cluster size for which a
diamond cubic lattice fragment is predicted by the Tersoff to be lowest in energy.
3.3 Point Defects
Point defects enhance diffusion rates in materials and are therefore important to
control during the synthesis of semiconductor devices [106]. For example, ion
implantation of dopants into a semiconductor substrate results in a super-saturation of
point defects [107]. After annealing at high temperature the diffusion rate for these
defects is anomalously high; a transient effect dependent on intrinsic carrier
concentration at the annealing temperature [108]. Some progress has been made in
examining the contributions of point defects to self-diffusion through the use of ab initio
70
calculations, such as local density approximation [109, 110], and, to some extent,
empirical descriptions for the energy [111, 112]. However the defect migration
pathways and diffusivities are still not well established. In order to address these issues
a potential that gives a reasonable approximation for defect formation energies is
required.
The defect formation energy, Ef, can be defined using the approach proposed
by Finnis [98] as
)0,(),(lim NEN
NxNEE d
Nf 3.4
where E(N,0) is the total energy of a perfect crystal with N occupied lattice sites, E(N,a)
is the total energy of a crystal with N lattice sites and x defects, and Nd is the number of
atoms in the defected crystal. Four point defects were considered: the vacancy (V), the
tetrahedral interstitial (T), the hexagonal interstitial (H) and the (110)-split interstitial
(X). Computational crystals were assembled with 1536 lattice sites (N) and the Finnis
method used to estimate defect energies. We note that this number of atoms was
sufficient to have converged to a good estimate of the energy.
The defect formation energies predicted by the SW and Tersoff potentials are of
the same order as those calculated by DFT methods [113-119]. The SW and Tersoff
predictions of defect formation energies are listed in Table 3.7. The lowest energy
interstitial configuration has been determined by DFT to be the (110)-split interstitial.
This interstitial, as well as the tetrahedral and hexagonal are illustrated in Figure 3.5.
The SW potential correctly predicts the X interstitial as the lowest energy and correctly
71
determines that the tetrahedral and
hexagonal are the second and third
in energy respectively. The SW,
however, fails in predicting the
magnitude its estimates. The SW
overestimates the formation energy of the X interstitial by 1.2 eV, the T by 0.8 eV and
the H by 2.0 eV. The SW potential is also unable to stabilize the H interstitial, instead
upon molecular statics minimization rearranges to a T interstitial (the transition
pathway from H-to-T is kinetically easier than the H-to-X, this will be discussed in section
V). The Tersoff potential, in contrast to the SW, is able to successfully stabilize the H
interstitial. The Tersoff potential, however, predicts the T interstitial as the lowest
energy configuration. Despite this, the Tersoff potential predictions are very close to
the DFT estimates.
The silicon vacancy has also been considered. This defect plays a key role in
diffusion mechanics [4]. Ab initio measurements indicate that the formation energy of
the silicon vacancy lies in the range of 3.3-4.3 eV [118]. These estimates have also
reported that vacancy formation is accompanied by a relaxation of the surrounding
lattice inwards towards the vacancy thereby decreasing the vacancy volume [119]. The
SW potential fails to capture this decrease in volume upon relaxation of the lattice. The
SW potential does not exhibit any lattice relaxation inwards or outwards; the atoms
surrounding the vacancy remain in their bulk lattice positions. As a result of this, the
defect formation energy of the silicon vacancy as predicted by the SW potential is
Table 3.7 Point Defect Formation Energies (eV)
Defect SW Tersoff DFT
V 4.34 3.73 3.3 - 4.3
X 4.46 4.45 3.3
T 4.98 3.52 3.7 – 4.8
H 6.57 4.67 4.3 – 5.0
72
identical to the cohesive energy (-4.34 eV). In contrast to ab initio estimates [119], the
Tersoff potential predicts a non-physical increase in volume surrounding the vacancy.
The magnitude of the volume increase estimated by the Tersoff potential is 24%.
3.4 Melting Temperature
Prediction of a materials melting temperature (Tm) is a good test of an
interatomic potential. Near the melting temperature, the interatomic spacing is
significantly larger than the separation at room temperature to which the potential was
fitted. The melting temperature therefore provides insight about the strength of the
interatomic bond and the shape of the interatomic potential at large interatomic
Figure 3.5 The diamond cubic lattice (a) and the three low energy
interstitial configurations: b) tetrahedral, c) hexagonal, and d) [110]-split.
73
separation (where the spline smoothing function may affect the interactions). Good
estimates of the melting temperature are often correlated with the accurate prediction
of surface structures, surface evaporation, and the rate of surface diffusion.
Experimental testing has shown that silicon undergoes a phase transition from the solid
to liquid at a temperature of 1691 K at ambient pressure.
The melting temperature was estimated for the potential following an approach
of Morris et al. in which a half-liquid/half-solid supercell is allowed to achieve an
equilibrium temperature under constant pressure [120]. A large supercell (2160 atoms,
60 plans of 36 atoms each) was used and two temperature control regions were applied,
one well above Tm and the other well below. The system was allowed to equilibrate for
20 picoseconds at which point the supercell was a half melted and half crystalline. The
temperature control regions were then removed and the system allowed to achieve
equilibrium (this was assumed to occur within 500 ps). If the resulting atomic system
retained a combination of crystalline and liquid material the uniform temperature of the
system was the melting temperature. The calculation was repeated several times and
predicted uncertainty was ±50 K.
The SW potential was parameterized with a strong focus of obtaining a model for
the simulation of liquid silicon properties [46]. As such, an accurate approximation of
the melting temperature was a weighty consideration. The SW potential predicts a
melting temperature 1750 K. The predictive uncertainty of ±50 K places this value very
close to the experimentally observed value of 1691 K [1]. The Tersoff potential does not
74
predict the melting temperature with accuracy. The Tersoff potential estimates the
melting temperature of silicon at 2800 K, significantly above the experimental value.
Little attempt is made in the original publications to justify this value accept to say that
given the short length of time of simulation, overheating may play a significant role [47-
49].
3.5 Surface Reconstructions
A robust description of surface morphology is an important part of the
simulation of vapor phase deposition. The most common surfaces that are used for
crystal growth in silicon are the (100) and (111) surfaces [2]. The widely observed
surface reconstructions for those surfaces are the (2×1) dimer row [121, 122] and the
(7×7) dimer adatom stacking fault (DAS) [123] respectively. At low temperatures the
(100) surface is often seen to have a c(2×4) buckled dimer configuration [124]. It is
thought that at higher temperatures the buckled dimers oscillate at a high frequency
such that they appear to be symmetrical [122, 125, 126]. Here we examine the (100)
(2×1) [121, 122], the (113) (3×2) [127-130], and the (111) (7×7) DAS [123]. These
surfaces are illustrated in Figures 3.6 to 3.8. The energies of surfaces were calculated
from the surface area, number of atoms, bulk cohesive energy, and total energy of the
computational supercell of the reconstructed surface. Each supercell had between
75
1000-2200 atoms with reconstructed top and bottom surfaces. The calculated surface
energies relative to the unreconstructed surfaces are compared with the ab initio/TB
data in Table 3.8.
The SW predicted values of the (100) 2x1 symmetric dimer rows is quite
accurate. The SW, however, fails to accurately capture the energy advantage of the
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(111) 7x7 reconstructed surface, predicting an energy underestimation by a factor of 2.
The (113) 3x2 surface reconstruction is
overestimated by nearly a factor of 2 as
well. DFT calculations, however, have
not obtained a clear minimum energy
reconstruction for the (113) surface
Table 3.8 Surface Reconstruction Energies
(eV/Å2)
Surface SW Tersoff DFT
(100) 2x1 -0.052 -0.061 -0.054
(111) 7x7 -0.231 0.586 -0.403
(113) 3x2 -0.067 -0.075 -0.036
77
rendering an evaluation of a potential’s
predictions difficult [128-130]. The Tersoff potential also successfully captures the
simple dimer rows of the (100) 2x1 reconstruction. The Tersoff potential notably
predicts a positive surface energy for the (111) 7x7 reconstruction. This would imply
that the surface reconstruction is not stable. As a result, any simulation that employs
the Tersoff potential to examine this surface is called into question. The Tersoff
potential also overestimates the surface energy of the (113) 3x2 reconstruction;
however for reasons already stated this potential prediction cannot be adequately
critiqued.
78
IV. Silicon BOP Assessment
The previous chapter presented a detailed assessment of the Stillinger-Weber
[46] and Tersoff [47-49] silicon potentials. In this chapter, the recently published silicon
bond order potential [131] is presented and assessed in the same style as the SW and
Tersoff potentials in chapter III. Namely, the bulk properties for a range of experimental
and theoretical crystal structures, the energy and structure of small clusters, the melting
temperature, the defect formation energies, and surface reconstructions are all
examined. This potential has been developed because of a perceived weakness on the
part of other currently available interatomic potentials to explicitly account for the
physical concepts involved in covalent bonding in an analytic, non-empirical fashion.
The BOP format, which has been detailed in chapter II, incorporates additional physical
concepts such as -bonding and promotion energy. The inclusion of these bonding
concepts allows the BOP to capture additional features of silicon.
4.1 Bulk Properties
At standard temperature and pressure (273 K and 1 atm) the lowest energy
crystalline phase of Si has the diamond cubic structure [8]. The dc structure has an
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atomic volume of ~20 Å3/atom and a cohesive energy of -4.63 eV/atom [8]. If sufficient
pressure (~12.5 GPa) is applied to Si, it will distort from the four-fold coordinated dc
structure to a six-fold coordinated Sn phase [79]. This transition is also accompanied
by an electronic shift from the semiconducting to metallic state [79]. When pressure is
removed from Sn phase silicon, it does not return to the dc phase. Instead it follows
a lower energy kinetic path to the bc8 phase at a pressure of ~8 GPa [79]. The structural
changes can be qualitatively observed in the DFT calculated binding energy curves for
silicon shown in Figure 4.1. The bulk energetics of the Si BOP have been explored by
producing binding energy vs. atomic volume curves for the structures; these are also
graphed in Figure 4.1. It is apparent that at a very high pressure the transition from dc
to Sn is properly modeled by the BOP. The BOP also correctly places the bc8 phase as
a stable phase at an intermediate pressure. Comparison between the DFT calculations
and the BOP predictions indicate that in general the BOP reproduces well the volume
dependent relative energies of different silicon phases.
The atomic volumes, cohesive energies, and bulk moduli of crystalline structures
predicted by the Si BOP are in reasonable agreement with DFT calculations. The atomic
volumes predicted by the BOP shown in Table 4.1 are within ±3% of the VASP DFT
estimates, with the exception of the bcc and bc8 structures (both within ±8%). The
cohesive energies summarized in Table 4.2, are within ±6% of DFT results. The bulk
moduli, summarized in Table 4.3, are within 50% of DFT results. This level of bulk
modulus prediction is within the acceptable predictive range limitations [60] found
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within the tight binding model from which the BOP is derived and represents an
improvement over the SW and Tersoff potentials.
The elastic properties (c11, c12 and c44) are a noteworthy success of the Si BOP.
The BOP predicts c11, c12 and c44 to have values of 134.89 (∆19%), 70.98 (∆11%) and
84.03 (∆5%) GPa respectively. These predictions are very close to the experimentally
determined values of 165.78, 63.94 and 79.62 GPa respectively [8]. Of particular note is
the Cauchy pressure prediction. The Cauchy pressure (c12 – c44relaxed) was calculated to
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be -13.05 GPa, in good agreement with the experimental value of -16.0 GPa [8]. The
realistic modeling of the Cauchy pressure makes the silicon BOP unusual amongst other
available interatomic potentials and indicates that the silicon BOP is uniquely capable of
reliably modeling the elastic behavior of silicon. The successful prediction of a negative
Cauchy pressure has been attributed to the inclusion of an environment dependent
repulsive promotion energy term [131].
Table 4.1 Atomic Volume Predictions (Å3). Comparable to Table 3.2.
Structure dc sc fcc bcc bc8 -Sn
DFT 19.79 0.789 0.704 0.717 0.869 0.749
BOP 20.01 0.804 0.696 0.647 0.924 0.719
% diff 1.1 3.1 0.04 8.7 7.6 2.8
Table 4.2 Cohesive Energy Predictions (eV). Comparable to Table 3.3.
Structure dc sc fcc bcc bc8 -Sn
DFT -4.63 0.292 0.468 0.451 0.122 0.213
BOP -4.639 0.434 0.536 0.539 0.134 0.224
% diff 0.2 3.1 1.3 1.9 0.07 0.04
Table 4.3 Bulk Modulus Predictions (GPa). Comparable to Table 3.4.
Structure dc sc fcc bcc bc8 -Sn
DFT 105.4 105.6 93.54 111.3 128.1 155.9
BOP 100.76 224.36 86.86 119.02 163.23 90.31
% diff 4.4 112.5 7.1 6.9 27.4 42.1
82
Examining the bulk phonon spectrum of the diamond cubic phase is also a useful
method of evaluating the performance of the potential near equilibrium. This
calculation is performed by annealing a sample crystal of sufficient size (512 atoms were
employed) and calculating the velocity-velocity autocorrelation function of the system.
The vibrational spectrum for the system can then be calculated by taking the Fourier
transform of the correlation function. The resulting vibrational spectrum is illustrated in
Figure 4.2. The highest peak calculated is at 550 cm-1 which is within ~5% of the
experimentally observed highest intensity peak at 520 cm-1 [103].
4.2 Small Clusters
Silicon dimer properties (energy and separation) were employed as target values
during the parameterization of the silicon BOP. It is therefore not possible to tout the
83
accuracy of their predictions as outputs of the potential. The experimental values for
the silicon dimer’s binding energy and separation distance are -2.41 eV and 2.34 Å [104,
105]. The BOP predictions are -2.61 eV and 2.336 Å. These data, as well as the BOP
predictions for all other cluster structures and energies are found in Table 4.4.
Three possible configurations for the Si trimer, the linear chain, the bent chain,
and the equilateral triangle, were examined. The BOP Si potential predicted that the
equilateral triangle had the lowest energy. Data for the linear chain and equilateral
triangle were not available for HF calculations [104, 105]. HF calculations predicted that
the bent chain has the minimum energy at an apex angle of ~80°. The apex angle that
minimizes the energy of the bent chain was found to ~111.6° using the BOP, matching
the minimum of the angular term.
Four Si tetramer structures were examined including the linear chain, the square,
the trigonal pyramid, and the corner-capped triangle. The BOP Si potential predicted
that the square had the lowest energy. The HF calculations indicated the square and the
capped triangle had the lowest energy (around -8.85 eV) [104, 105]. Both the bond
length and energy of the square predictions of the BOP agree well with the HF
calculations. However, there are some deviations for the other higher energy
structures. For example the Si4 linear chain structure has 2 separate bond lengths, as
shown in Figure 3.4. The BOP predicted that the 2-3 bond is significantly (~0.4 Å) longer
than the 1-2 bond, whereas the HF method predicted that the 2-3 bond is shorter (~0.1
84
Å) than the 1-2 bond [104, 105]. This may indicate an overestimation of the -bonding
nature in the 1-2 bond by the BOP.
The BOP predictions for the Si pentamer and hexamer structures are comparable
to the HF results [104, 105]. With BOP, the lowest energy pentamer is the planar
pentagon, however this was not the most stable structure with HF [104, 105]. The
Table 4.4 BOP Si Small Clusters
Structure
Point
Group Bond
Bond
Length (Å)
Binding Energy
(eV)
BOP HF
dimer dimer D∞h 1-2 2.336 -2.61 -3.06
trimers
linear chain D∞h 1-2 2.37 -4.84 ~
bent chain C2v 1-2 2.39 -4.64 -3.04
triangle D3h 1-2 2.47 -4.91 ~
tetramers
linear chain D∞h 1-2
2-3
2.433
2.815 -8.19 -7.29
square D4h 1-2 2.43 -8.87 -8.84
rhombus D2h ~ ~ ~ -10.71
tetrahedron Td 1-2 2.581 -7.26 -8.09
flagged triangle C2v
1-2
1-3
2-4
2.498
2.487
2.342
-7.18 -8.86
pentamers
pentagon D5h 1-2 2.384 -11.61 -10.08
trigonal bipyramid D3h
1-2
1-5
2-3
2.515
3.124
3.415
-11.38 -13.92
pyramid C4v 1-2
2-3
2.617
2.50 -11.29 -13.47
hexamers
edge capped
trigonal bipyramid C2v
1-2
1-3
2-3
3-4
3-5
2.466
2.506
3.544
3.062
2.466
-15.11 -18.26
octahedron Oh 1-2 2.554 -15.01 ~
85
planar pentagonal structure favored by the BOP does not provide a significant energetic
advantage over the competing high symmetry cluster structure’s predicted energies in
BOP predictions. The planar pentagon structure is preferred by BOP due to the minimal
amount of bond bending required to obtain the structure (bond angles of 108°). The
edge-capped trigonal bipyramid was found by the BOP to be the minimum energy
hexamer structure, in agreement with the HF predictions [104, 105].
In general the BOP Si potential prefers larger atomic spacing in clusters than the
HF calculations [104, 105]. The source for this discrepancy arises from the selection of
the target dimer separation in the BOP parameterization (2.336 Å). The HF
underestimates the dimer separation at 2.227 Å (~0.11 Å shorter than the experiment)
[8]. Even though the BOP does not predict the same low energy structures for Si
clusters as those predicted by the HF calculation (with the exception of the Si hexamer),
the improvement of the predictions of the overall binding energies for different clusters
compared to other potentials is anticipated to improve the simulations of complex
atomic arrangement processes.
4.3 Melting temperature
The predicted melting temperature (Tm) can affect surface structures,
evaporation rates, and surface diffusion. Near the melting temperature the interatomic
spacing is significantly larger than the separation at room temperature to which the
potential has been fitted. Therefore the melting temperature can be a combinational
86
measure of the strength of the interatomic bond (depth of the interatomic potential
well) and the shape of the interatomic potential at large separation. During deposition
large interatomic separation distances are frequently encountered. It is therefore
important to examine the melting temperature.
The Tm estimate was determined following the approach by Morris et al. in which
a half-liquid/half-solid supercell is allowed to achieve an equilibrium temperature under
constant pressure [120]. A large supercell (2160 atoms, 60 plans of 36 atoms each) was
used and two temperature control regions were applied, one well above Tm and one
well below. After 20 picoseconds, the supercell was ½ melted and ½ crystalline. The
temperature control regions were then removed and the system allowed to reach
thermal equilibrium (assumed to occur within 500 ps). At equilibrium the boundary
between the liquid and solid phases will have stopped moving, and the temperature of
this equilibrated region is taken as the melting temperature. The Tm obtained in this
manner has an uncertainty of ±50 K (obtained from variations in repeated runs).
The BOP silicon potential predicts a melting temperature of 1650 ±50 K. This
temperature range includes the experimentally observed value of Tm = 1687 K [1]. It
indicates that the BOP potential models well the interaction of silicon at large
separation distances.
4.4 Point defects
Precise defect formation energies were not expected because defects were not
incorporated in the fitting process. As a result, the relative order of the defect energies
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predicted by the BOP and ab initio calculations is
not exactly the same. These results can be found
in Table 4.5. For example, the BOP predicts that
the T interstitial has the lowest energy, whereas
the DFT predicts that the X-split interstitial has the
lowest energy [113-119]. Nonetheless the BOP predicted interstitial energies are
generally close to previous DFT calculations. The vacancy formation energy of 2.759 eV
predicted by the BOP also well matches the past DFT value of 3.17 eV [118]. The
vacancy volume was seen to shrink during energy minimization. This again matches the
observations from ab initio calculations [119].
4.5 Surface reconstructions
A robust description of surface morphology is an important part of the
simulation of vapor phase deposition. The most common surfaces that are used for
crystal growth in silicon are the (100) and (111) surfaces. The widely observed surface
reconstructions for those surfaces are the 2×1 dimer row [121, 122] and the 7×7 dimer
adatom stacking fault (DAS) [123] respectively. At low temperatures the (100) surface is
often seen to have a c(2×4) buckled dimer configuration [124]. It is thought that at
higher temperatures the buckled dimers oscillate at a high frequency such that they
appear to be symmetrical [122, 125, 126]. The BOP potential was fitted approximately
to the (100) 2×1 surface energy. Here we examine the (100) 2×1 [121, 122], the (113)
Table 4.5 Point Defect Formation Energies (eV)
Defect BOP DFT
V 2.76 3.3 - 4.3
X 3.37 3.3
T 2.63 3.7 – 4.8
H 3.85 4.3 – 5.0
88
3×2 [128-130], and the (111) 7×7 DAS [123].
These surfaces have been illustrated in Figures
3.6-3.8. The energies of surfaces were calculated
from the surface area, number of atoms, bulk
cohesive energy, and total energy of the
computational supercell of the reconstructed surface. Each supercell had between
1000-2200 atoms with reconstructed top and bottom surfaces. The calculated surface
energies relative to the unreconstructed surfaces are compared with the ab initio/TB
data in Table 4.6.
It can be seen from Table 4.6 that the surface free energy relative to the
unreconstructed surface is within 0.06 eV/Å2 for each surface except the (113) 3×2
surface. However, this is not an issue because there is no clear minimum energy
reconstruction for the (113) surface in the literature [128-130]. Most importantly, the
highly complex (111) 7×7 surface reconstruction is found to be stable with nearly the
same relative free energy as tight binding calculation [132]. The BOP shows a marked
improvement in the calculation of surface energies over other available silicon
interatomic potentials [63].
Table 4.6 Surface Reconstruction Energies (eV/Å2)
Surface BOP DFT
(100) 2x1 -0.046 -0.054
(111) 7x7 -0.379 -0.403
(113) 3x2 -0.139 -0.036
89
V. Solid Phase Epitaxy of Silicon
5.1 Introduction
The synthesis of epitaxial silicon thin films by the low temperature condensation
of vapor [133], followed by the subsequent solid state transformation of these
amorphous layers [30] is widely utilized during the manufacture of microelectronic
devices. Amorphous layers can also be formed during ion implantation and are
recrystallized by thermal treatments [31]. The solid phase epitaxial growth (SPEG)
process that occurs during the annealing of these amorphous layers proceeds by the
thermally induced epitaxial growth of a crystal seed into the metastable amorphous
region [30]. The need to control point defect populations, dislocation types and
densities and stacking fault concentrations in films grown by these processes has
stimulated significant interest in the mechanisms of atomic reassembly at the epitaxial
interface.
Experimental thermal annealing studies indicate the growth rate of the
crystalline phase into an amorphous silicon system is well characterized by an Arrhenius
relation with a single activation energy of 2.7 eV which is thought to consist of a defect
formation energy of 2.4 eV and a defect migration energy of 0.4eV [30]. These
90
experimental studies also indicate that the transformations rates can be rapid. For
example, a 2500 Å thick a-Si film transforms fully to a crystalline structure in 2.5 seconds
at a temperature of 725 °C (a growth rate of 0.1 m/s) [30]. The SPEG rate can be
effected by self ion bombardment of the films [30]. The ion bombardment of
amorphous films results in a lowering of the activation energy for the SPEG process to
0.18-0.4 eV [32, 33]. Bernstein, Aziz, and Kaxiras argue that this low activation energy
reduction results from the ion impact assisted formation of the rate limiting defects. The
transformation from the amorphous to crystalline state of the ion irradiated structure is
then only controlled by the migration of these defects with an activation energy in the
0.4eV range [34].
A detailed understanding of the atomistic mechanisms involved in SPEG has
been impeded by the difficulties of high resolution imaging of the moving (sometimes
very rapidly) buried interface [35]. Computational modeling has therefore been used to
investigate the SPEG process [34-40]. The use of computational modeling techniques is
restricted by the relatively large number of atoms (of order 103) that must be used to
characterize each phase [34]. Additionally these systems must be simulated for an
extended time (>1ns) to observe even initial movement of the a-c interface [34]. The
use of high fidelity, unbiased parameter-free quantum mechanical calculations such as
density functional theory (DFT), is therefore prohibited [35]. A molecular dynamics
(MD) approach appears the most promising approach provided it employs an
interatomic potential that adequately captures the radial and angular dependences of
the interatomic interactions.
91
A computational study by Motooka et al. employing on the order of 1000 atoms
over a nanosecond timescale and utilizing the Tersoff potential [48] found two
temperature driven growth regimes in contrast to the single experimentally observed
temperature dependency [35]. Their MD results indicated that at low temperature
SPEG proceeded via a 2-D planar growth mechanism with an activation energy of 2.6 eV.
At higher temperatures however, {111} facets were formed at the interface and the
activation energy for growth decreased to 1.2 eV. Bernstein et al. also identified two
temperature driven growth regimes in a study employing an environmentally
dependent interatomic potential (EDIP) [34, 134]. However, they report an activation
energy of only 0.4 ± 0.2 eV at low temperature and an energy of 2.0 ± 0.5 eV in the high
temperature regime. These activation energies are clearly in conflict with the Tersoff
potential predictions. While both studies argue that removal of lattice defects at the a-c
interface is rate limiting, the defect whose migration controls the transformation rate
remained unclear.
The most recent computational studies of SPEG performed by Garter and Weber
[36-40] employ both the Tersoff and the Stillinger Weber potential [36-40, 46, 135].
They examined the morphology of the a-c interface and observed that the interface is
not sharp, but rather extends over a 6-8 atomic monolayer thick region. They argue
that the rearrangement of atomic defects in the transition region is the limiting atomic
mechanism in SPEG. They also show that the concentration of the defects predicted by
the Tersoff potential is about double that predicted by the Stillinger Weber approach
[135].
92
Here, we use the recently developed BOP for silicon and a molecular dynamics
simulation method to examine the initial stages of the amorphous to crystalline
transformation of silicon. Computational resource limitations constrain our simulations
using the BOP approach to systems of 1000 atoms and for short time scales (~1 ns). As a
result, the simulation is capable of only resolving the initial atomic reassembly
processes. We note at the outset that the initial growth rates observed below and in
previous studies of the simulations of the SPEG process in silicon are several orders of
magnitude faster than that observed experimentally [34-40]. We are confident that this
result does not indicate flaws in our work because previous studies [34-40] have
obtained similar results when examining the initial growth period. We suspect there
exists a SPE mechanism that is activated after the initial stage examined here is
complete. The discovery of this mechanism awaits much larger timescale MD
simulations.
The atomic scale structure of the amorphous film created using the BOP has
been examined, and is found to contain a high concentration of both 3 and 5
coordinated atoms. This indicates that the BOP predicts a highly defected amorphous
film similar to that observed in ion implanted films. These defects result in fast diffusion
pathways and a correlation is drawn between removal of interstitial defects at the a-c
interface and rapid epitaxial growth. The BOP based simulations are then employed to
investigate the activation energy barriers to the migration of interstitial defects within
the bulk. These energy barriers are observed to be similar to the calculated overall
activation energy for solid phase epitaxial growth process.
93
5.2 Simulation Details
There are numerous ways to synthesize amorphous silicon including quenching
from the melt [136], low temperature vapor deposition [137-139], and ion
bombardment [140, 141]. The a-Si films generated by each method have different
atomic scale structures [30]. Amorphous silicon generated by rapidly cooling the liquid
phase results in the formation of a network of tetrahedrally coordinated atoms with no
long range order (an “ideal” amorphous film) [136]. Low temperature vapor deposition
grown films are amorphous but also sometimes contain low density regions, or voids
[30]. These are a consequence of self-shadowing during the deposition process
combined with low atom mobility on the film surface [30]. When the deposited atoms
are unable to significantly migrate across the surface, the atoms assemble into a
random network with no long range periodicity. Ion implantation causes atoms to be
displaced from their lattice sites by primary and secondary collision processes [30].
Continuous ion bombardment results in the overlap of the damage zone of individual
impacts eventually leading to a continuous amorphous structure. Amorphous silicon
films created by ion bombardment contain a large concentration of three and five-fold
bonding defects [142].
To prepare a computational amorphous/crystalline (a-c) interface a 23.0375 Å
[10 1 ] by 43.44 Å [010] by 23.0375 Å [101] volume single crystal was created. This crystal
was made up of 1152 atoms distributed in 32 (010) layers. Periodic boundary conditions
were employed in the [10 1 ] and [101] directions. The top 24 monolayers were then
94
melted elevating their temperature to 2000 K for 50 ps (for 25000 time steps with a
time step, t = 2 fs), while the bottom 8 monolayers were thermally constrained to 500
K with the lowest 2 layers rigidly fixed in space. This resulted in a thin crystalline
substrate with a layer of liquid silicon on top. This system was then quenched to the
desired temperature over a period of 50 ps to create a computational sample containing
a region of crystalline and amorphous material separated by an amorphous to
crystalline interface. This system was then thermally annealed for 5 ns and the resulting
epitaxial growth rate and rate limiting defect migration energy barriers investigated.
The epitaxial growth rate was measured by tracking the rate of change of the
number of crystalline atoms present in the system by identifying the bonding
environment of an atom and that of its neighbors. In order for an atom to be classified
as part of a crystalline region it was required to maintain, within a tolerance of 10°, the
109o tetrahedral arrangement of its bonds with its four nearest neighbors. The increase
in number of atoms that satisfy this condition gives the epitaxial growth rate in atoms
per unit time interval.
The interstitials present in the crystallized region were also identified and their
diffusion pathways were also examined using molecular statics minimization techniques
[143]. Starting from a given interstitial configuration, the interstitial atom was
incrementally moved in the direction of minimum energy along a path to another
interstitial position. After each incremental movement, the entire system was allowed
to relax in energy around the constrained interstitial atom. This then allowed the
95
energy along the migration path to be computed and the energy barrier to interstitial
migration within the crystal to be determined.
5.3 Amorphous Characterization
The radial distribution function (RDF) for the a-Si film produced by the rapid
solidification simulation is shown in Figure 5.1. The simplest view of amorphous silicon
is that it consists of a continuous random network of tetrahedrally coordinated atoms
[30]. Therefore one would expect that the RDF would display a large slender peak
centered near the bulk silicon equilibrium nearest neighbor distance, followed by
broadening secondary and tertiary peaks. The RDF obtained by analyzing the quenched
Figure 5.1 Radial distribution function g(r) for a-Si compared to experimental a-Si
data. Inset: the atomic coordination distribution of silicon atoms in the a-Si region
graphed as a percentage of frequency as predicted by simulation.
96
structure generally agrees with this interpretation. The data obtained from the
simulation also agree well with experiment [144]. While this is an encouraging
affirmation of accurate modeling of a-Si by the BOP approach, it is not conclusive [145].
Many of the other interatomic potentials poorly predict the random tetrahedral
network of a-Si, predicting a large number of 3 and 5 coordinated atoms [145]. The
inset in Figure 5.1 shows the distribution of atomic coordinations in the amorphous
region predicted by the BOP analysis. The BOP predicts a broad distribution of atomic
coordination with a maximum at 4 and an average coordination of 4.16. Generally
amorphous silicon films are considered to be a uniform random network of coordination
4 atoms, however the various experimental methods for generating amorphous thin
films result in different amorphous states [30]. For example, the amorphous film
generated by ion implantation techniques contains many atoms that are not four-fold
coordinated [30]. Spectroscopic studies of self-implanted a-Si, have shown a large
concentration of dangling bonds associated with three-fold coordinated atoms and
floating bonds resulting from five-fold coordinated atoms [142]. In simulation, the many
atoms that do not have four-fold coordination suggest the amorphous region generated
by the fast quenching technique is highly defected. The high concentration of defects in
the very rapidly cooled BOP a-Si structure suggests that the simulated structure is
similar to ion beam irradiated structures where defects are introduced to the system by
a non-thermal ion collision process.
97
5.4 Epitaxial Crystallization
The quenched computational system was annealed at numerous temperatures
between 700 and 950K and the growth rate of the a-c interface during the SPEG process
was determined. This is plotted as a function of inverse absolute temperature in Figure
5.2. The growth rate data is reasonably well fitted by an Arrhenius relation with an
activation energy barrier of 0.87 eV. This activation energy is about twice that
experimentally reported for ion beam irradiated silicon [31]. Earlier simulation studies
have reported a wide range of activation energies [34-40]. Using their environmentally
dependent modeling approach, Bernstein et al [34] have shown that the low
temperature activation barrier (0.2 eV) observed in their simulations corresponds to the
migration of defects while the larger
value (2.0 eV) seen at higher
temperatures corresponds to a more
complicated process involving defect
formation and diffusion. The a-Si film
formed by quenching the Si-BOP structure
contained a large concentration of
“frozen in” defects and we suspect this is
responsible for our observation of an
activation barrier lying between the
Bernstein et al limits.
98
To investigate the atomic scale details of the a-c transformation, time resolved
atomic structures near the a-c interface during simulations of the SPEG process at
annealing temperatures of 700 and 900 K are shown in Figures 5.3 and 5.4. They show
the advancing amorphous to crystalline front moving through the amorphous region. It
can be seen that the most recently crystallized region in each time resolved consists of a
99
predominantly crystalline lattice containing a high concentration of interstitial defects.
Detailed examination of the transition region indicates that overlap of the lattice strains
of these interstitial defects eventually gives rise to the continuous random network of
the a-Si layer as one moves upward through the simulated region.
To investigate the influence of these remnant defects upon the advance of the a-
c interface we have plotted the number of crystallized atoms in the simulated structure
100
against transformation time for a transformation at 800K, Figure 5.5. It can be seen that
the growth of the crystalline phase is unsteady with sudden jumps in transformation
rate interspersed by periods of significantly slower transformation. The overall growth
rate of the crystalline phase is therefore limited by the atomistic rearrangements
occurring during these periods of low crystalline growth. An examination of the atomic
structure of the system before and just after a shift from a slow to fast growth mode
reveals that the velocity jump occurs upon elimination of an interstitial atom in the
crystalline phase just behind the transition region. This has been to occur at all of the
simulated temperatures. An example of such a rapid crystallization after the elimination
of a near interface defect can be clearly seen in the time resolved atomic structures of
Figures 5.4(c) to 5.4(d). This result is in qualitative agreement with the conclusions of Lu
et al. who argued that removal of defects residing at the a-c interface was the rate
controlling mechanism for SPEG [142].
101
To characterize the nature of the defects within the partially crystallized system,
we have calculated bond angle distributions functions, g(), for each region, Figure 5.6.
The angular distribution function for the transition region, Figure 5.6(b), shows a broad
peak centered on the tetrahedral bond angle (109o) with a shoulder extending towards
~75°. A small secondary peak is also evident at a bond angle of ~50°. We note that a
[110]-split type interstitial defect has a bond angle of ~50°, while a tetrahedral defect
has bonds with an angle of ~70°. The average coordination of these interstitials was
obtained by visually identifying 100 interstitial atoms in the various simulations and
determining their coordinations. This resulted in an average coordination of 4.05 for
102
the transition region interstitials. The large angular distribution concentrations at 50°
and 75° within the transition region indicate the presence of a high concentration of
defects with configurations that include those bonding angles, such as the [110]-split
and tetrahedral interstitial types. Previous calculations, presented in section 4, have
determined the formation energy of these point defects to be 3.37 eV and 2.63 eV
respectively.
The diffusion pathways associated with these two interstitials have been
investigated using a molecular statics approach. We have employed notational
shorthand for the point defects as follows: silicon vacancy (V), tetrahedral interstitial (T),
hexagonal interstitial (H), and the [110]-split interstitial (X). The specific defect
migration pathways that have been examined are presented in Figures 5.7-5.10. In no
particular order these are the X-to-C, T-to-C, V-to-C, T-to-X, and T-to-H. It is important
to remember that these are intended to be approximations of the motions encountered
at the a-c interface. These combinations were chosen because each involves a single
defect migrating to either another defect location or to a crystal lattice site. The energy
barriers to motion range from 0.49 – 1.88 eV.
Of the examined atomistic motions, migration of a vacancy into an adjacent
lattice site has the lowest activation energy. Figure 5.7 details atomistic motions that
occur as the vacancy is moved. The formation energy of the silicon vacancy has been
previously predicted by the BOP to be 2.76 eV. The Si-BOP predicts a volume decrease
103
of ~40% for the silicon vacancy as the adjacent atoms relax inwards. The migration
pathway for the silicon vacancy is simple; an adjacent atom switches places with the
vacancy. The energy barrier to this motion is 0.49 eV. In more detail, the adjacent
atom, labeled 1 in Figure 5.7, moves in a [111] direction towards the vacancy site. At
104
the midpoint of this motion, the third configuration in Figure 5.7(c), bonds are formed
with 3 new atoms. Atom 1 after this point has 6 atoms to which it is bonded. As the
atom moves closer to the vacancy site, the original three bonds are subjected to
significant bonding strains (and subsequently the atoms bonded to those atoms as well)
and shift inwards toward atom 1. The original three heavily strained bonds finally break
when atom 1 occupies the former vacancy site. The non-symmetric nature of the
energy curve in Figure 5.7(f) reflects this atomic mechanism. The observed drop in
energy when those bonds finally break corresponds to the relaxation of the stored strain
energy in the original bonds.
105
The second lowest energy migration pathway involves the switch between a
tetrahedral interstitial and a [110]-split interstitial. This calculation has been performed
in two ways. The first, the T-to-X has atoms T and C, as labeled in Figure 5.8, fixed in
space and moved incrementally to the minimum energy configuration of the X
interstitial. The second method, the X-to-C, has atom X1, as labeled in Figure 5.8, fixed
in space and incrementally moved into its associated lattice site; atom X2 is allowed to
move freely and as a result minimizes into a tetrahedral site. In both of these methods
all other atoms are allowed to reach their minimum energy configuration at each
increment. Both of these methods simulate the motion between a T and X interstitial
configuration. The energy barrier to this migration is 0.75 – 0.97 eV. Figure 5.8 shows
the high energy configuration for this atomic motion. Significant lattice distortions
occur in the neighboring atoms. Because of the very small energy barrier to migration in
the X-to-T direction (0.14 eV) it is unlikely that X type interstitials will have a long
lifespan in simulation.
The third defect migration of interest is the migration of a tetrahedral site to
another tetrahedral site shown in Figure 5.9. It should be noted that the midpoint
between any two tetrahedral sites is always occupied by a hexagonal site. Therefore, by
symmetry, the energetics of the migration can be fully considered by examining the T-
to-H pathway. The hexagonal interstitial is predicted by the Si-BOP to be metastable
with an energy of formation of 3.85 eV. Any small thermally induced distortion from
perfect symmetry will result in the hexagonal defect transitioning into a tetrahedral site.
106
As a result the energy barrier to migration from one tetrahedral site to another is
approximately the same as the difference between their formation energies, 1.10 eV.
The final and highest energy defect migration examined is for a tetrahedral atom
to directly displace a neighboring lattice site atom. This atomic motion is labeled T-to-C.
The tetrahedral interstitial atom is incrementally moved towards a neighboring lattice
atom site. This pushes the crystalline atom out of its lattice site towards a nearby low
energy tetrahedral site (not the same tetrahedral site that is already occupied). The
high energy configuration of this motion is shown in Figure 5.10. Considerable lattice
distortion is encountered in this motion due to the motion requiring the breaking and
reforming of 6 strong atomic bonds. This atomistic mechanism encounTersoff an energy
barrier of 1.88 eV, considerably higher than any of the other motions examined. Due to
107
the considerably higher energy involved, it is unlikely that this motion is encountered
with any frequency in simulation.
If we examine the possible energy pathways for one tetrahedral interstitial to
migrate to a nearby tetrahedral interstitial site there are a number of ways of going
about it. The tetrahedral interstitial atom could directly displace an adjacent crystal
atom, pushing that atom into a nearby tetrahedral site. This motion, the T-to-C
mechanism, has been shown to have a very large energy of activation, 1.88 eV. Another
path the tetrahedral interstitial could take would be to migrate through a hexagonal
interstitial site. This pathway has been found to have an activation barrier of 1.1 eV,
much preferable to the T-to-C mechanism but still higher than the T-to-X pathway. The
108
lowest energy pathway for tetrahedral interstitial migration is to first overcome a 0.75-
0.97 EV energy barrier to form a X interstitial. Secondly, the X interstitial then must
overcome a 0.14 eV barrier to transform back into a T interstitial in a new tetrahedral
site. Because this last motion has the lowest energy, it is the mechanism that is most
likely to occur.
5.5 Discussion
We have studied the solid phase epitaxial growth of silicon using the recently
developed bond order potential. The solid phase epitaxial process involves the
spontaneous, thermally activated rearrangement of atomic bonds at the
amorphous/crystal interface [30]. For an ion-implanted amorphous surface,
experimental studies have shown that this process results in the motion of a sharp a-c
interface towards the free surface [30]. The growth rate of the crystalline region (or the
velocity of the a-c interface) can be well modeled by an Arrhenius relation with
activation energy of 2.7 eV [30]. Ion bombardment introduces defects into the
amorphous film which enhance the growth rate and reduce the activation energy to
0.18-0.4 eV [34]. The rapid quenching from the liquid phase method used to obtain the
amorphous film used in the present simulations results in a highly defected amorphous
film. As a result the a-Si found in the present simulations is similar to an unrelaxed
amorphous film under ion bombardment. The BOP predicts activation energy of 0.87 eV
for the SPE process.
109
Previous atomistic modeling research has shown that defects play a key role in
SPEG [34-40]. The exact nature of that role has differed depending on which
interatomic potential was employed. The BOP also indicated that defects are an integral
piece of the atomistic mechanism that limits the SPE process. The rapidly advancing
crystalline front is slowed by the need to remove trapped interstitials at the interface.
These interstitials distort the surrounding crystal lattice and prevent further
crystallization until they are annealed out. These interstitials have been found to be
four-fold coordinated, indicating that they are predominantly of the (110)-split (X) and
tetrahedral (T) type. The bonding environment of the transition region in which these
interstitials are found is predominantly crystalline. We therefore believe that the
activation energy for the movement of these interstitials through a crystalline lattice will
be comparable to the activation energy of their movement in the transition region. The
X-to-T transition was found to have an energy barrier between 0.75 and 0.97 eV. This
energy is comparable to the 0.87 eV activation energy for SPE as predicted by the BOP.
This result suggests that the annihilation of interstitial defects at the a-c interface is the
rate limiting mechanism for solid phase epitaxial growth.
110
VI. SW and Tersoff Germanium
Assessment
Germanium is a covalently bonded material of great scientific and engineering
interest because of its unique combinations of electrical and other physical properties.
For instance, germanium is an indirect band gap material with a small band gap. It
therefore has low light absorbance at infrared wavelengths, and because it easily cut
and polished, it can be used for infrared lenses and windows in the 8-14 micron
wavelength range [8, 9]. The small bandgap of Ge and Ge-Si alloys also make it a useful
material in solar cell applications where its high absorbance enable the use of thinner
layers of active material [10, 11]. Light emitting diodes (LEDs) have also been fabricated
to take advantage of germanium’s unique properties [12, 13]. These LEDs are based on
Ge-Si alloy self-assembled quantum dots and have exhibited a broad emission peaked at
a wavelength centered upon 1.45 m [12, 13].
The electronic structure of germanium is very similar to silicon due to the fact
that both are group IV elements [1]. As mentioned in the introduction, the ground state
equilibrium structure of germanium is the diamond cubic phase [8]. The nearest
neighbor bond length in this phase is 2.446 Å which is approximately 4% larger than for
111
silicon [8]. Germanium also forms weaker bonds than silicon; the cohesive energy of the
diamond cubic phase of germanium is -3.82 eV compared to the -4.63 eV for silicon [8].
This arises because germanium has a weaker electronegativity than silicon, and
subsequently forms a shallower energy well [17]. These differences necessitate a
reparameterization of the existing potentials to accommodate germanium. In this
chapter we present the SW and Tersoff predictions for germanium properties using the
published parameter sets [49, 54]. In chapter VII, we shall present the predictions of the
recently developed Ge BOP.
6.1 Bulk Properties
Binding energy curves were constructed for germanium using the VASP DFT
software package [146-149] and the Stillinger-Weber [46, 54] and Tersoff [48-50]
potentials. These results can be seen in Figure 6.1. The SW binding energy curve
prediction for germanium shows a significant weakness in the potential to predict the
energetics of alternative bulk structures. The Ding and Andersen fit of Ge [54]
significantly overestimates the energy differences between the phases; the sole
exception being the bc8 phase. It is not surprising that the bc8 phase is modeled
reasonably given that the atoms in the bc8 structure possess a similar local bonding
environment to those found in the dc structure. The close packed phases (sc, fcc and
bcc) as well as -Sn are significantly more open (i.e. higher relative volume) than what is
expected from the DFT estimates; these structures are seen to approach a relative
112
Figure 6.1 Binding energy vs. relative volume (with respect to equilibrium dc
volume) for a range of Si bulk structures. a) Stillinger-Weber b) Tersoff c) DFT.
dc
-Sn
bc8
sc
bcc
fcc
hcp
113
volume of unity with respect to the diamond cubic phase. Experimental studies of
germanium phases observe an atomic rearrangement to the -Sn structure at high
pressure [150]. In contrast to this, the SW predicted binding energy curves suggest that
the dc phase remains the equilibrium structure even under high pressure.
The Tersoff binding energy curves indicate that the Tersoff potentials provides a
reasonably approximation of the energies of many of the other examined structures.
Two of the phases (fcc and bcc) have their energy minimums at significantly lower
relative volumes than expects however. This results in the aberrant prediction of the
bcc phase becoming the minimum energy phase at high pressure. Also absent from the
Tersoff predictions is a region of slightly elevated pressure for which the minimum
energy phase is bc8 (in DFT this occurs at ~0.81 V/V0). Another odd presentation of the
Tersoff binding energy curves is the curvature of the fcc phase. Normally it is difficult to
estimate a value for the bulk modulus visually, the fcc curve appears abnormally sharp
(calculations show that the predicted modulus for the fcc phase is 914.3 GPa quite in
excess of the DFT estimate of 63.5 GPa).
DFT calculations have predicted an atomic volume trend for germanium of (in
decreasing order): dc, bc8, sc, -Sn, fcc, bcc and hcp. This trend is similar to the DFT
estimates obtained for silicon. In those calculations the first four structures were in the
same order. The DFT, SW and Tersoff atomic volume predictions are presented in Table
6.1. Stillinger-Weber predictions for atomic volume are poor. The SW predicts the
structure with the greatest atomic volume to be the hcp structure. This result is
114
contradictory to the DFT results. Additionally, the second highest atomic volume
structure as predicted by DFT, the bc8 phase, is found by the SW to be ranked 5th in
terms of atomic volume. The Ge SW potential systematically overestimates the atomic
volume of the close packed structures. The Tersoff potential handles the atomic volume
predictions much better. The only failure observed by the Tersoff potential is the
overestimation of the hcp phase atomic volume. The Tersoff also switches the order of
the bcc and fcc phases, however, the two phases are very close in atomic volume and
this is a minor issue.
Cohesive energy trends have been calculated for germanium and this data is
found in Table 6.2. The DFT calculations have determined an energy trend of (in
decreasing order): dc, bc8, -Sn, sc, fcc, hcp and bcc. The SW predicts the dc and bc8
phase energies with an accuracy of 2%, however the remaining phases are poorly
predicted. The average percentage difference for the remaining phases (-Sn, sc, fcc,
Table 6.1 Atomic Volumes as estimated by DFT and by the SW and Tersoff
potentials. The atomic volume of the dc structure is given in cubic angstroms while
all other values are given relative to the dc volume. For example the fcc Tersoff
atomic volume can be obtained by multiplying the dc value by the number listed;
22.54 * 0.72 = 16.23.
Structure dc sc fcc bcc bc8 -Sn hcp
DFT 22.54 0.83 0.81 0.80 0.92 0.81 0.80
SW 22.54 0.99 0.97 0.92 0.93 0.92 1.01
% diff 0.0 18.96 19.56 15.44 0.84 13.16 26.07
Tersoff 22.54 0.83 0.72 0.73 0.93 0.78 0.82
% diff 0.0 0.46 10.68 9.08 0.84 4.06 3.15
115
bcc and hcp) is approximately 16%. It should be noted the SW does predict the order of
the four lowest energy phases properly despite the considerable energy difference. The
Tersoff potential predicts the energies of the germanium phases much more accurately
than the SW. The Tersoff potential does not predict the b-Sn phase to be the third
lowest in energy however. Rather unusually, the Tersoff predicts the sc, bcc, -Sn and
hcp phases to have nearly identical energies (all within ±0.015 eV of each other).
Similar to the SW and Tersoff predictions for silicon, the predictions of the bulk
moduli for the examined phases of germanium are poor. These predictions are
presented in Table 6.3. Both the SW and Tersoff are in reasonable agreement for the dc
phase bulk moduli (32% and 19% differences respectively), however the remaining
phases are not as well modeled. Particularly noteworthy are the extreme predictions of
the fcc bulk modulus which the SW and Tersoff overestimate by 856% and 1340%
respectively. These results strongly argue against the use of these potentials in
simulations where these structures might be present.
Table 6.2 Cohesive energy estimates by DFT and the SW and Tersoff potentials. The
dc values are listed in terms of eV and all other values are given relative to the dc
value. i.e. the DFT estimation of the sc cohesive energy can be found by taking the
dc value and adding the 0.24 to obtain -3.58 eV.
Structure dc sc fcc bcc bc8 -Sn hcp
DFT -3.82 0.24 0.34 0.35 0.13 0.24 0.34
SW -3.86 0.81 1.12 0.99 0.25 0.73 0.84
% diff 1.05 14.72 21.18 17.38 2.06 12.62 13.07
Tersoff -3.85 0.32 0.53 0.35 0.26 0.33 0.32
% diff 0.78 6.89 4.71 0.75 2.79 1.7 1.35
116
The three independent elastic constants (c11, c12 and c44) have also been
calculated for the dc structure. Experimental measurements of germanium have
determined that these values are 131.5, 49.5 and 68.4 GPa for c11, c12 and c44
respectively [8]. A related property, the Cauchy pressure (c12 – c44) for germanium is -
19 GPa. The SW potential predicts values of 90.3, 75.3 and 39.6 GPa for c11, c12 and
c44 respectively and a Cauchy pressure of 35.7 GPa. The Tersoff potential predicts
values of 138.4, 78.2 and 23.3 GPa for c11, c12 and c44 respectively and a Cauchy
pressure of 54.9 GPa. A negative Cauchy pressure is indicative of a material in which
high hardness and stiffness arise not from a high bulk modulus, but from a high shear
modulus [98]. The prediction of a positive Cauchy pressure indicates that the atomic
bonds are more likely to undergo bending rather than stretching [98]. This non-physical
prediction of a fundamental germanium property is distressing.
The final examination of the dc bulk properties is the phonon spectrum.
Germanium exhibits a single strong vibration spectrum peak at 390 ± 10 cm-1 [103]. The
Table 6.3 Bulk modulus estimates by DFT and the SW and Tersoff potentials. All
values are listed in GPa.
Structure dc sc fcc bcc bc8 -Sn hcp
DFT 60.72 67.38 63.49 63.59 64.45 68.69 70.45
SW 80.31 155.71 607.37 554.01 74.79 300.34 239.83
% diff 32.27 131.09 856.64 771.22 16.04 337.24 240.42
Tersoff 72.21 98.98 914.29 122.88 78.41 104.19 196.03
% diff 18.92 46.89 1340.05 93.24 21.66 51.68 178.25
117
SW and Tersoff predictions are illustrated in Figure 6.2. The SW potential predicts a
single strong peak at 315 ± 10 cm-1. The Tersoff potential predicts a single strong peak
at 300 ± 10 cm-1. The vibration spectrum is a measure of how fast the lattice is
vibrating, a lower primary peak indicates that the oscillation of atoms in their lattice
sites is occurring at an increased rate. This may adversely influence other potential
predictions such as thermodynamic properties (i.e. melting temperature).
118
6.2 Germanium Small Clusters
Extensive ab initio data was not readily available for germanium small clusters,
therefore the construction of a germanium equivalent of Figure 3.4 was not possible.
Despite this, similar configurations were employed as the initial atomic configurations in
the MS minimizations (albeit with bond lengths increased by 4%). Published data on
germanium small clusters reveals the minimum energy configuration and the binding
energy of those configurations; this data is listed in Tables 6.4 and 6.5 together with the
SW and Tersoff cluster predictions. The absence of ab initio data for germanium
clusters precludes the inclusion of such data in Tables 6.4 and 6.5.
Experimental studies have determined that the germanium dimer has a
separation distance of ~2.3 Å with a binding energy of ~2.64 eV [151]. Both the SW and
Tersoff potentials overestimate the dimer separation distance (2.45 Å and 2.43 Å for SW
and Tersoff respectively). These two potentials obtain a minimum energy separation
closer to the nearest neighbor distance of the diamond cubic structure (2.446 Å). Both
potentials also significantly underestimate the binding energy of the Ge dimer.
Ab initio calculations have determined that a bent 3 member chain is the lowest
energy structure for the Ge trimer [151, 152]. In contrast to their predictions for silicon,
both the SW and Tersoff germanium potentials are able to correctly predict this. The
SW potential predicts an energy preference for the bent chain structure over the
equilateral triangle of 0.45 eV. Not surprisingly, the apex angle of the ent chain is
predicted to be 109° by the SW potential. This is the result of the potential’s preference
119
Table 6.4 Stillinger-Weber germanium cluster predictions
Structure Point
Group Bond
Bond
Length (Å)
Binding
Energy (eV)
dimer dimer D∞h 1-2 2.448 -1.93
trimers
linear chain D∞h 1-2 2.557 -3.19
bent chain C2v 1-2 2.448 -3.86
triangle D3h 1-2 2.752 -3.41
tetramers
linear chain D∞h 1-2
2-3
2.529
2.716 -4.55
square D4h 1-2 2.515 -6.99
rhombus D2h ~
tetrahedron Td 1-2 2.924 -5.65
tetrahedral fragment C3v 1-2 2.462 -4.85
pentamers
pentagon D5h 1-2 2.448 -9.64
trigonal bipyramid D3h
1-2
1-5
2-3
2.656
3.461
3.490
-8.65
pyramid C4v 1-2
2-3
3.125
2.561 -8.11
hexamers
edge capped
trigonal bipyramid C2v
1-2
1-3
2-3
3-4
3-5
2.552
2.766
3.616
3.453
2.552
-11.34
face capped trigonal
bipyramid C2v
1-2
1-3
1-5
3-4
3-5
2.482
3.771
2.478
3.514
2.566
-12.12
square bipyramid C4v 1-2
2-3
2.846
3.162 -9.39
bent chair hexagon D3d 1-2
1-3
2.448
3.998 -11.58
120
Table 6.5 Tersoff germanium cluster predictions
Structure Point
Group Bond
Bond
Length (Å)
Binding
Energy (eV)
dimer dimer D∞h 1-2 2.432 -2.01
trimers
linear chain D∞h 1-2 2.482 -3.60
bent chain C2v 1-2 2.452 -4.01
triangle D3h 1-2 2.622 -3.79
tetramers
linear chain D∞h 1-2
2-3
2.429
2.761 -5.07
square D4h 1-2 2.497 -6.86
rhombus D2h ~ ~ ~
tetrahedron Td 1-2 2.733 -5.78
flagged triangle C2v
1-2
1-3
2-4
2.622
2.622
2.432
-5.48
pentamers
pentagon D5h 1-2 2.442 -9.79
trigonal bipyramid D3h
1-2
1-5
2-3
2.561
3.098
3.521
-8.86
pyramid C4v 1-2
2-3
2.758
2.638 -8.48
hexamers
edge capped
trigonal bipyramid C2v
1-2
1-3
2-3
3-4
3-5
2.652
2.708
2.813
3.098
2.559
-10.79
face capped trigonal
bipyramid C2v
1-2
1-3
1-5
3-4
3-5
2.694
2.801
2.706
3.308
2.657
-10.59
octahedron Oh 1-2
2-3 2.754 -10.98
bent chair hexagon D3d 1-2
1-3
2.431
4.125 -12.06
121
for the tetrahedral bonding angle. The Tersoff potential predicts an apex angle of 115°.
Both interatomic potentials underestimate the bonding energy of the trimer by slightly
less than 3 eV.
Similar to their predictions of the silicon tetramer, HF based calculations have
determined that the germanium tetramer has its lowest energy in a rhombus
configuration [151, 152]. Neither the SW nor Tersoff potential are able to stabilize a
rhombus structure. When molecular statics minimization is performed both the SW and
Tersoff potential’s predict a shift to a square configuration. The square configuration is
predicted by both the SW and Tersoff potentials to be the lowest in energy.
Interestingly, the SW potential was also unable to stabilize the flagged triangle
structure. Instead when a MS minimization was applied the resulting structure was a
tetrahedral fragment (the 1-3 bond was broken, and atom 2 was raised out of the plane
created by the other 3 atoms).
Ab initio data on the structure and energetics of germanium clusters of sizes 5
and 6 that are consistent with the values for clusters of size 2-4 are not readily available.
As a result, no direct comparison between the SW and Tersoff potentials and highly
accurate DFT calculations can be performed. The supposition that germanium clusters
of size 5 and 6 will behave similarly to silicon clusters of size 5 and 6 is not unreasonable,
however it must be recognized as an assumption. Both silicon and germanium display
similar characteristics for clusters of size 3 and 4 as well as having very similar electron
valence. Regardless of which high symmetry structure has the lowest binding energy, it
122
is fairly reasonable to assume that it isn’t the planar pentagon structure (which is the
least stable of the pentamers examined by silicon HF calculations). The SW and Tersoff
potentials both predict the planar pentagon as the lowest in energy.
The SW and Tersoff potentials predict differing structures as the lowest energy.
The SW potential prefers the face-capped trigonal bipyramid and the Tersoff potential
prefers the chair-bent hexagon. The SW potential predicts considerable distortions
from the initial f-cap structure, however. This includes an increased in the 1-3 bond (as
defined in figure 3.4) from 2.79 Å to 3.77 Å. Also the 3-4 bond increases from 2.97 to
3.51 Å. These lengths are approaching, but not exceeding, the SW cutoff distance for
germanium of 3.92 Å. As a result the final structure is similar to a five member ring with
a single atom replaced by a dimer pair. The chair-bent hexagon structure preferred by
the Tersoff potential is similar to a tetrahedral fragment. The 1-2 bond length is very
close to the nearest neighbor distance encountered in the bulk diamond cubic lattice.
6.3 Melting Temperature
Prediction of a materials melting temperature (Tm) is a good test of an
interatomic potential. Near the melting temperature, the interatomic spacing is
significantly larger than the separation at room temperature to which the potential was
fitted. The melting temperature therefore provides insight about the strength of the
interatomic bond and the shape of the interatomic potential at large interatomic
separation (where the spline smoothing function may affect the interactions). Good
123
estimates of the melting temperature are often correlated with the accurate prediction
of surface structures, surface evaporation, and the rate of surface diffusion. The
melting temperature was estimated for germanium in the same fashion used to
calculate the melting temperature of silicon.
The SW germanium potential predicts a melting temperature of 1350±50 K, a
temperature range above the experimentally observed Tm of 1211 K [1]. This result is a
significant variance from the experimental melting temperature, however it does reflect
the priorities assigned in the parameterization of the Ge SW potential. Ding and
Andersen [54] were unable to find a set of parameters for the SW potential that gave a
good description of all three phases of germanium: diamond crystal, amorphous solid,
and liquid. As their primary interest was in the development of a germanium potential
to model the amorphous structure, and because they recognized the importance of
properly capturing the structure of the diamond crystal, they chose to sacrifice liquid
property predictive accuracy [54]. The Tersoff germanium potential, similar to its silicon
melting temperature predictions, is a significant overestimation at 2400±50 K. It is
possible that the short cutoff distance of the Tersoff potential (3.1 Å) hampers its ability
to accurately model the interaction of germanium atoms at large separation distances,
such as encountered during melting simulation.
6.4 Point Defects
The formation energy of point defects has been examined for germanium in the
same fashion as for silicon. Table 6.6 lists the SW and Tersoff defect formation energy
124
predictions. Similar to its prediction
for the silicon vacancy, the SW
germanium potential also predicts
the vacancy formation energy to be
the same magnitude as the cohesive
energy. This is accompanied by the lack of any lattice relaxation around the vacancy.
The Tersoff potential significantly overestimates the formation energy of the vacancy,
placing it at 3.6 eV (nearly double the DFT estimation). The DFT calculations of the
germanium vacancy have shown a distinct relaxation of the surrounding lattice atoms
inwards towards the vacancy, resulting in a 25-40% volume decrease for the vacancy
[153]. The Tersoff potential predicts an increase in vacancy volume of 4.7%.
The SW germanium potential has very poor interstitial formation energy
predictions. The SW overestimates the formation energies by a factor of ~3. The source
of this prediction may be traced back to the change in the parameter. The germanium
and silicon SW parameterizations are identical with the exception of the energy and
length scale and the parameter (which was increased from 21 to 31 for germanium).
This parameter represents the reduced strength of the three-body interaction. The
increase in this term increases the scale of the energy penalty for deviations from the
tetrahedral angle. As a result, the decidedly non-tetrahedral angles found in the local
environment of the interstitials are found to be erroneously high in energy. The Tersoff
potential presents a reasonable set of predictions for interstitial formation energies,
Table 6.6 Point Defect Formation Energies (eV)
Defect SW Tersoff DFT
V 3.86 3.60 1.9
X 9.16 3.59 3.19, 3.84
T 7.52 4.44 2.29, 3.55
H 9.83 4.86 2.94, 3.99
125
although does display a preference for the T interstitial over the X interstitial in contrast
to some DFT predictions [153, 154].
6.5 Surface Reconstructions
The local bonding environment found at the surface differs significantly from the
bonding structure of the bulk lattice phases employed in the parameterization.
Therefore the examination of surface energies is a stringent test of a potential’s
transferability. We compute the surface energies for several of the low energy
reconstructions on low index germanium surfaces ((100) and (111)). The surfaces
examined are illustrated in Figures 3.6 and 6.3 (there are no structural differences
between the Si (100) 2x1 and the Ge (100) 2x1 surfaces). The energies of surfaces were
calculated from the surface area, number of atoms, bulk cohesive energy, and total
energy of the computational supercell of the reconstructed surface. Each supercell had
between 1000-2200 atoms with reconstructed top and bottom surfaces.
Ab initio simulations observe a buckling effect for the (100) 2x1 surface dimer
rows as the free energy of a (100) germanium surface is minimized [155]. The SW and
Tersoff germanium parameterizations reveal the buckling of surface dimers to be
energetically unstable. The room temperature stable surface reconstruction for the Ge
(111) surface is the c(2x8) reconstruction [156]. This configuration, in contrast to the Si
(111) 7x7 surface, does not require stacking faults or dimers, and is described by a
simple adatom model. Each c(2x8) surface unit contains 2 adatoms that stabilize the
126
structure by saturating the surface dangling bonds and through a charge transfer from
the adatoms to the surface [156]. The calculated surface energies relative to the
unreconstructed surfaces are compared with the ab initio/TB data in Table 6.7. Both the
SW and Tersoff potentials provide predictions for the (100) 2x1 surface that are
reasonable, however their
predictions of the (111) c(2x8)
reconstruction significantly
overestimate the surface energy.
Table 6.7 Surface Reconstruction Energies (eV/Å2)
Surface SW Tersoff DFT
(100) 2x1 -0.082 -0.072 -0.031
(111) c2x8 -0.231 -0.237 -0.012
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VII. Germanium BOP Assessment
7.1 Bulk Properties
DFT calculations indicate the atomic volume increases as the structure of
germanium changes from: bcc, hcp, -Sn, fcc, sc, bc8, to dc. The BOP predicted atomic
volume trend is slightly different, increasing in the sequence: bcc, fcc, sc, -Sn, hcp, bc8,
and dc. While the atomic volume trend for the more closely packed phases differs
slightly in order, the atomic volumes for these structures are all within 7% of the DFT
predictions. These results can be found in Table 7.1.
The bond order potential proposed here correctly predicts the lowest energy
structure to be diamond cubic. The DFT calculations reveal an energy trend in
germanium crystal structures from lowest to highest energy of: dc, bc8, -Sn, sc, fcc,
bcc, and hcp. The BOP predicts cohesive energy trends from lowest to highest energy
of: dc, bc8, -Sn, sc, bcc, hcp and fcc. The BOP predictions for the cohesive energy are
listed in Table 7.2. The BOP notably captures the significant cohesive energy trend of
the experimentally observed, open structures (dc, bc8, and -Sn) with high fidelity
(within 1%). The cohesive energies of the remaining structures are within 3-10% of the
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corresponding DFT values. The level of accuracy obtained for cohesive energies of the
experimentally observed phases of germanium justifies the sacrifice of accurate
predictions of the more open structure phases.
Bulk moduli data from the BOP and DFT-LDA calculations are compared in Table
7.3. The bulk moduli were calculated without internal relaxation of the selected
structures. This had a negligible impact for cubic structures, but can become more
significant for non-cubic phases [102]. The BOP predicted bulk moduli for the
equilibrium dc phase is found to be within 6%. This demonstrates that the BOP
Table 7.2 The cohesive energies, Ec (in eV), in decreasing order, of seven condensed phases of germanium as calculated by the BOP and DFT. The accurate prediction of the dc phase was a critical weight of the parameterization process so there is no variance between the DFT and BOP predictions.
Structure dc bc8 -Sn sc fcc hcp bcc
DFT -3.82 -3.69 -3.58 -3.57 -3.48 -3.48 -3.47
BOP -3.82 -3.73 -3.53 -3.45 -3.11 -3.13 -3.35
% diff 0.0 1.1 1.4 3.6 10.6 10.0 3.4
Table 7.1 The atomic volumes, V (in Å3), in decreasing order, of seven condensed phases of germanium as calculated by the BOP and DFT. The BOP predicted c/a values
for the -Sn and hcp phases are 0.46 and 1.79 respectively compared to the DFT predictions of 0.51 and 1.65 respectively. An accurate prediction of the volume of the dc phase was a large weight of the parameterization process so there is no variance between the DFT and BOP predictions.
Structure dc bc8 sc -Sn fcc bcc hcp
DFT 22.54 20.74 18.71 18.26 18.25 18.03 18.02
BOP 22.54 20.51 18.26 17.13 17.12 16.68 16.90
% diff 0.0 1.1 2.4 6.2 6.2 7.5 6.25
129
reasonably models the stretching and compression of the tetrahedrally coordinated dc
atomic bonds. The predicted bulk moduli for the range of structures examined is within
~10% for the sc and bc8 phases, within ~35% for the hcp phase and within ~50% for the
fcc, bcc and -Sn phases. The larger discrepancies are partially a result of the
parameterization process in which the atomic volume and cohesive energy were given a
higher weight during fitting. The discrepancies in bulk moduli observed for the higher
energy structures are similar to those encountered using other potentials [46-54]. A
visual comparison of the BOP and DFT predictions of the cohesive energy, atomic
volume (through relative volumes), and bulk moduli (through curvature) can be found in
the binding energy curves shown in Figure 7.1. The trends described in Tables 7.1-7.3
are clearly visible in Figure 7.1.
The three independent elastic constants (c11, c12, and c44) have also been
calculated for the dc structure. The predicted values are 60.81, 24.94, and 25.79 GPa for
c11, c12, and c44 respectively. These results underestimate the experimental values of
131.5, 49.5, and 68.4 GPa for c11, c12, and c44 respectively [8]. The experimental Cauchy
Table 7.3 The bulk moduli, B (in MPa), in decreasing order with dc listed first, of seven condensed phases of germanium as calculated by the BOP and DFT. A precise prediction of the dc bulk moduli was a low priority of the parameterization process; results within ~50% were considered acceptable.
Structure dc hcp -Sn sc bc8 bcc fcc
DFT 60.72 70.45 68.69 67.38 64.45 63.59 63.49
BOP 64.34 108.46 132.82 71.11 72.39 123.69 137.11
% diff 5.62 35.0 48.3 5.2 10.9 53.7 53.7
130
pressure (c12 – c44) for germanium is -19 GPa whereas the BOP predicts a value of -0.86
GPa. These BOP predictions are an output of the potential, i.e. the elastic constants
were not employed in the fitting process. Other interatomic potentials, such as the SW
and Tersoff shown in section 6, do not predict a negative Cauchy pressure. The
implementation of an explicit representation of the promotion energy enabled the
prediction of a negative value of the Cauchy pressure [60].
The phonon spectrum of the diamond cubic phase was also calculated. This
probes the performance of the potential at near equilibrium bond spacing conditions. A
diamond cubic crystal of 512 atoms was annealed at a simulated temperature of 300 K
and the velocities of a randomly selected 50 atom sample were tracked and used to
calculate the velocity-velocity autocorrelation function [157]. The vibrational spectrum
for the system was then calculated by taking the Fourier transform of this correlation
131
function. The resulting vibrational spectrum is shown in Figure 7.2. The primary
spectral peak was located at ~410 cm-1, which is reasonably close to the experimentally
reported value of 390±10 cm-1 [103].
In summary, the atomic volume and cohesive energy of germanium are well
reproduced by the BOP and compare favorably with the self consistent DFT data. The
bulk moduli predictions for the various structures examined by the BOP are comparable
in accuracy to those found using other available interatomic potentials [46-54]. The
BOP gives a negative Cauchy pressure in qualitative agreement with experiments and it
models the vibrational spectrum well.
132
7.2 Small Clusters
Germanium clusters can be made using laser photofragmentation methods
[158], and have been analyzed by mass spectrometry [159], ultraviolet photoelectron
spectrometry [160], and infrared spectrometry [161]. Germanium clusters have
structures that are quite different to those of the bulk material. The structure and bond
energies of small clusters are heavily reliant upon both the angular and radial
components of the interatomic potential. Examination of predicted cluster properties is
therefore a useful means of testing an interatomic potential. A conjugate gradient
method [162] was used to calculate the relaxed structure and binding energies for
numerous high symmetry configurations, and the results are listed in Table 7.4 together
with ab initio electronic structure calculations (where available) [151, 152]. We note
that the properties of germanium dimers were included in the fitting process and so
cannot be used to independently test the potential. The BOP predicted interatomic
spacing is fairly close to the expected atomic separation (2.394 Å versus ~2.3 Å
experiment). Their cohesive energy is slightly overestimated (-1.36 eV/atom versus -
1.32 eV/atom experiment) [151].
Three trimer configurations, the linear chain, the tetrahedral fragment, and the
equilateral triangle are summarized in Table 7.4. Electronic structure calculations have
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Table 7.4 Germanium small cluster energetics and structure as predicted by the BOP.
Structure Bond Bond Length
(Å)
Cohesive Energy
(eV/atom)
BOP DFTa
dimer
1-2 2.394 -1.36 -1.32
trimers
1-2 2.43 -1.509 ~
1-2
1-3
2.441
4.287 -1.508 -2.24
1-2 2.568 -1.60 ~
tetramers
1-2
2-3
2.432
2.464 -1.620 ~
1-2
1-4
2.449
3.465 -2.054 ~
1-2 2.671 -1.794 ~
1-2 2.444 -1.587 ~
1-2
2-3
3-4
2.452
3.374
3.558
-2.055 -2.6
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consistently found an isosceles triangle (i.e. a bent chain) to be the minimum energy
configuration with bond lengths (from the central atom) of 2.28 to 2.35 Å and a bond
angle of 81 to 86° [151, 152]. Using the BOP the linear chain and tetrahedral fragment
were observed to have nearly the same energy. The tetrahedral fragment distorted to
an angle of 122.78° from 109.7° with bond lengths of 2.441 Å from 2.446 Å with a
decrease in energy of ~0.005 eV. The equilateral triangle was found to be ~0.1 eV/atom
lower in energy than the two competing configurations, and was the minimum energy
configuration found using the BOP. The BOP was unable to minimize the structure into
the DFT predicted isosceles triangle, preferring instead the higher symmetry equilateral
triangle configuration.
Five high symmetry germanium tetramer configurations were also investigated.
They include the linear chain, the square, the tetrahedral fragment, the rhombus, and
the tetrahedron, Table 7.4. Electronic structure calculations have found the rhombus
configuration to be the lowest in energy with predicted bond lengths of 2.432 Å to
2.515 Å and 2.555 Å to 2.659 Å for the nearest neighbor (bond 1-2 in Fig. 3.4) and short
diagonal (bond 2-3 in Figure 3.4) distances respectively [151, 152]. The BOP successfully
predicts a rhombus configuration as the lowest energy configuration (although the
higher symmetry square is a close competitor). It is interesting to note that the
tetrahedral fragment is the highest in energy, illustrating the remarkable difference
between the bonding environments and preferences of small clusters and the bulk
lattice.
135
Overall the germanium BOP predictions compare well with ab initio calculations
performed previously by others [151, 152]. The ab initio data on germanium small
clusters available in literature was not comprehensive enough to provide a rigorous
comparison between the ab initio and BOP predictions for the range of structures
examined. However, with the exception of the germanium dimer, the cluster
predictions here are outputs of the potential, indicating that the BOP parameterization
is able to reasonably predict both bulk and cluster properties with a single parameter
set.
7.3 Melting Temperature
Prediction of a materials melting temperature (Tm) is a good test of an
interatomic potential. Near the melting temperature, the interatomic spacing is
significantly larger than the separation at room temperature to which the potential was
fitted. The melting temperature therefore provides insight about the strength of the
interatomic bond and the shape of the interatomic potential at large interatomic
separation (where the spline smoothing function may affect the interactions). Good
estimates of the melting temperature are often correlated with the accurate prediction
of surface structures, surface evaporation, and the rate of surface diffusion.
The melting temperature was estimated for the potential following an approach
of Morris et al. in which a half-liquid/half-solid supercell is allowed to achieve an
equilibrium temperature under constant pressure [120]. A large supercell (2160 atoms,
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60 plans of 36 atoms each) was used, and two temperature control regions were
applied, one well above Tm and the other well below. The system was allowed to
equilibrate for 20 picoseconds at which point the supercell was a half melted and half
crystalline. The temperature control regions were then removed and the system
allowed to reach equilibrium (this was assumed to occur within 500 ps). If the resulting
atomic system retained a combination of crystalline and liquid material the uniform
temperature of the system was the melting temperature. The calculation was repeated
several times and predicted uncertainty was ±50 K.
The germanium potential predicts a melting temperature of 1115±50 K, a
temperature range below the experimentally observed Tm of 1211 K [163]. This
indicates that the BOP’s predictions of the bond strength at large separation distances is
slightly off, although not to a significant extent. The small discrepancy in melting
temperature may be related to the potential cutoff function, as believed to be the case
in other potentials [102, 164].
7.4 Point Defects
Point defects enhance diffusion rates in materials and are therefore important to
control during the synthesis of semiconductor devices [106]. For example, ion
implantation of dopants into a semiconductor substrate results in a super-saturatution
of point defects [107]. After annealing at high temperature the diffusion rate for these
defects is anomalously high; a transient effect dependent on intrinsic carrier
137
concentration at the annealing temperature [108]. Some progress has been made in
examining the contributions of point defects to self-diffusion through the use of ab initio
calculations, such as local density approximation [109, 110], and, to some extent,
empirical descriptions for the energy [111, 112]. However the defect migration
pathways and diffusivities are still not well established. In order to address these issues
a potential that gives a reasonable approximation for defect formation energies is
required.
The BOP predicted defect formations energies are compared to those estimated
from DFT calculations in Table 7.5. The formation energies predicted by the BOP fall
within the ranges calculated by the DFT methods (with the exception of the
tetrahedral). The lowest energy interstitial configuration has been found by DFT
calculations to be the (110)-split interstitial, followed by either the hexagonal or
tetrahedral interstitial as next
lowest in energy depending on
the exact method employed
[153, 154]. The BOP predicts the
lowest energy interstitial to be
tetrahedrally coordinated, with
the (110)-split configuration less
favorable by 0.31 eV. The
hexagonal interstitial is found to
Table 7.5 Point defect formation energy (in eV) and surface energy (in eV/ Å2) predictions by the BOP.
BOP DFT / ab initio
Defects Ef (eV)
V 1.97 1.9
IT 2.34 3.19-3.84
IX 2.65 2.29-3.55
IH 3.37 2.94-3.99
Surfaces (eV/ Å2)
(100) - 2x1 -0.051 -0.031
(111) – c2x8 -0.039 -0.012
138
be metastable; small distortions caused by thermal fluctuations were sufficient to
displace the interstitial atom from the hexagonal site. The DFT calculations of the
germanium vacancy have shown a distinct relaxation of the surrounding lattice atoms
inwards towards the vacancy, resulting in a 25-40% volume decrease for the vacancy
[153]. The BOP predicts the unrelaxed lattice with a vacancy to be metastable, however
with a small amount of thermal energy (1 ps anneal at 200 K) the system was able to
relax inwards. The BOP predicts the volume decrease of the vacancy to be ~50%, a
value that is comparable to the DFT calculations.
7.5 Surface Reconstructions
The local bonding environment found at the surface differs significantly from the
bonding structure of the bulk lattice phases employed in the parameterization.
Therefore the examination of surface energies is a stringent test of a potential’s
transferability. We compute the surface energies for the 2x1 surface dimer rows of the
(100) surface and the c2x8 reconstruction on the (111) surface. The surfaces examined
are illustrated in Figures 3.6 and 6.3. The energies of surfaces were calculated from the
surface area, number of atoms, bulk cohesive energy, and total energy of the
computational supercell of the reconstructed surface. Each supercell had between
1000-2200 atoms with reconstructed top and bottom surfaces.
Ab initio simulations observe a buckling effect for the (100) 2x1 surface dimer
rows as the free energy of a (100) germanium surface is minimized [155]. The Ge BOP,
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like other interatomic potentials reveals the buckling of surface dimers to be
energetically unstable [73]. The room temperature stable surface reconstruction for the
Ge (111) surface is the c(2x8) reconstruction [156]. This configuration, in contrast to the
Si (111) 7x7 surface, does not require stacking faults or dimers, and is described by a
simple adatom model. Each c(2x8) surface unit contains 2 adatoms that stabilize the
structure by saturating the surface dangling bonds and through a charge transfer from
the adatoms to the surface [156]. The calculated surface energies relative to the
unreconstructed surfaces are compared with the ab initio/TB data in Table 7.5. The Ge
BOP does not explicitly account for charge transfer at a reconstructed surface and as a
result surface predictions are simplified, however the overall trends are favorable. For
increased fidelity (and increased computational cost) a charge transfer model can be
employed with the BOP. This model has been developed and can be found in reference
165.
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VIII. Discussion
Like most avenues of scientific inquiry, the evolution of interatomic potentials
starts with an original basic concept, revolutionary for its time, which is developed in
incremental stages over time. The original interatomic potential employed a step
function form; if two atoms were within the cutoff distance they had a given potential
energy, if they were outside the cutoff they had no potential energy [157]. Over time,
as computational power improved, the models employed became more sophisticated.
While this dissertation makes no attempt to catalogue the full history of interatomic
potentials, it is interesting to note how far they have come. Two of the most commonly
employed interatomic potentials in use today for covalently bonded systems are the
Stillinger-Weber [46] and Tersoff [47-49] potentials. These potentials are popular
choices for silicon and germanium modeling because they are relatively simple to
implement, are accurate within certain constraints, and there exists a broad spectrum of
available literature on them [46-54]. We have examined these potential’s predictive
ability for a wide range of material properties and compared the results to experiment
and ab initio estimates. Furthermore, we have parameterized a new interatomic
potential which promises an increase in predictive validity through the use of an
141
analytically derived format; this potential is the Bond Order Potential, or BOP for short.
The BOP has been subjected to the same level of scrutiny as the SW and Tersoff
potentials. In this section we will give an overview of the performance of each
potential, clearly outline where the BOP predictions improve or fail to improve upon the
predictions of the SW and Tersoff, and discuss potential uses of the BOP for future
modeling.
8.1 Stillinger-Weber Overview
Our examination of a broad range of material properties reveals certain key
weaknesses in the Stillinger-Weber potential. While the potential was shown to be
proficient in the prediction of the bulk lattice properties of the diamond cubic phase,
the potential has significant difficulty in accurately modeling other lattice structures.
The inability to accurately model these non-equilibrium lattice structures is indicative of
poor potential transferability. Also troublesome for the SW potential are the energy
predictions of complicated surface reconstructions. This result leads us to believe that
the SW potential is not suitable to calculations involving free silicon or germanium
surfaces. Finally, the SW potential does not model point defects with high fidelity.
Lattice relaxation inwards at a vacancy site is observed by ab initio estimates; however
the SW potential predicts no relaxation of the lattice inwards or outwards at a vacancy
site. Additionally, the interstitial formation energies for germanium are extremely
142
inaccurate. These results indicate that the use of the SW potential in simulations of
defected systems will result in questionable outcomes.
The Stillinger-Weber potential does a number of things right however. It
predicts reasonable values for the elastic constants, despite not capturing a negative
Cauchy pressure, and provides a good estimate of small cluster energies. The structure
of small clusters is preferential to the tetrahedral angle, a feature not observed in ab
initio estimates. The SW potential also provides a reasonable prediction of the melting
temperature. The silicon (100) 2x1 surface reconstruction energy and structure are also
well described. Over all, the SW potential performs well in situations where the local
atom environment is tetrahedral. The SW potential does not seem to be appropriate for
use in simulations in which a significant number of non-tetrahedral atoms are present.
8.2 Tersoff Potential Overview
The Tersoff potential manages to improve upon the SW potential in a few key
areas: the energy and structure predictions of non-dc crystals of silicon and germanium,
and the defect formation energy of interstitials. Despite this, the Tersoff potential still
has a number of predictive difficulties. The first of these is the fidelity of the bulk
moduli of non-dc crystals, most notably the fcc structure. The Tersoff potential also
significantly overestimates the binding energy of the silicon trimer and predicts the
lowest energy trimer structure as the equilateral triangle. These two failures, however,
are not terribly debilitating to the potential’s usefulness. Far more troublesome for the
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Tersoff potential is the prediction of a positive energy for the (111) 7x7 DAS surface
reconstruction. This would suggest that the Tersoff potential is unsuitable for
simulations that include a (111) free surface. Also problematic is the predicted melting
temperature being so high. The Tersoff potential overestimates the melting
temperature of both silicon and germanium by over one thousand Kelvin. This is a
serious issue because the accurate prediction of melting temperature indicates that the
potential reasonably models the interaction of atoms at large separation distances. It
has been suggested that temperature in simulation can be equated to real temperature
by scaling the simulated temperature to the real melting temperature outside the
simulation. In other words, if the temperature in simulation was 1500 K and the
simulated melting temperature was 2000 K while the real melting temperature 1800 K,
the real temperature you were simulating would be 1350 K. This solution is
unsatisfactory because it does not address the fact that each individual atom would
contain the equivalent energy of a particle at a much higher temperature than that
required to melt the material.
The Tersoff potential does provide an accurate prediction of the silicon and
germanium diamond cubic bulk properties. The great improvement in the predictions
of alternative bulk structures is one of the strengths of the potential. Small clusters are
neither a strength nor a weakness of the potential; however it is unlikely that the
Tersoff potential could be used to reliably predict cluster properties. Point defect
formation energies are reasonably well approximated, however it should be noted that
the Tersoff potential predicts that the lattice relaxation around a vacancy is outward,
144
not inward like ab initio estimates. The Tersoff potential does provide a good
description of the (100) 2x1 surface reconstruction for both silicon and germanium,
however more complicated reconstructions like the Ge (111) c2x8 are not well modeled.
Overall the Tersoff potential is adept at bulk predictions (except moduli) and the
prediction of interstitial formation energies. The failure of accurate prediction of high
temperature properties does limit its usefulness. This suggests that the Tersoff
potential is most suitable to simulations of atomistic mechanisms that occur at low
temperature within the bulk.
8.3 Improved Fidelity of the BOP Approach
The bond order potential (BOP) represents a fundamental shift in approach to
the formation of an interatomic potential. Prior potentials, such as the Stillinger-Weber
and Tersoff potentials presented here, were developed empirically. The bond order
approach differs by utilizing a functional form that has been derived from the highly
accurate tight binding methodology. This method has been discussed in detail in section
2, however it bears repeating here because of its great significance. It is well
understood by scientists and theoreticians that the predictions generated from
computer models do not necessarily reflect reality. It is of great importance to the
researching scientist who wishes to use computer modeling to be able to believe the
results he or she obtains from that computer modeling. The BOP represents an
appealing approach because it uses a format that is directly derived from the way that
145
atomic bonds form. Because of this, the BOP is inherently superior in terms of
predictive believability.
The BOP theory is, however, limited by the quality of the parameterization. A
parameterization of the BOP for silicon and germanium has been presented and
evaluated in the preceding sections. Table 8.1 provides a qualitative overview of each of
the three potentials’ predictions on a scale of 1 (very poor) to 5 (very good). The
method by which these numerical evaluations are assigned is discussed in appendix C.
As can be seen in this table, the BOP improves on the predictions of the SW and Tersoff
potentials in many categories. The BOP captures the bulk properties of the dc phase
very well; most notably the BOP is the only potential examined that predicts a negative
Cauchy pressure. The Ge parameterization of the BOP would have received a higher
rating for elastic constant predictions because of this but the potential is slightly soft,
i.e. the values for c11, c12, and c44 are low (although no worse than either the SW or
Tersoff). The BOP also remains highly accurate even for non-dc crystal structures, a
great improvement over the SW potential. The BOP mirrors ab initio estimations of
lattice relaxation inwards at a vacancy site, and reasonably captures the defect
formation energies of the examined interstitial configurations. The BOP accurately
predicts the melting temperature of both silicon and germanium; a considerably
improvement over the Tersoff potential. And lastly, the BOP represents a significant
step forward in predictive accuracy for complicated surface reconstructions as
evidenced by the (111) 7x7 DAS which is modeled very well by the BOP.
146
Table 8.1 Qualitative evaluation of the three interatomic potentials predictions for all
the properties examined in this dissertation. The properties are graded on a scale of
1-5 where 5 is very good and 1 is very poor. Appendix C details the value ranges used
to determine these qualitative assessments.
SW Tersoff BOP
Property Si Ge Si Ge Si Ge
Bulk Properties
• dc
Va 4 5 4 5 4 5
Ec 3 4 5 4 5 5
B 5 3 4 4 5 5
c11, c12, c44 3 3 3 3 4 3
phonon 4 3 3 3 4 4
• other crystals
Va 2 2 3 3 3 3
Ec 2 1 3 3 4 3
B 1 1 1 1 3 3
Small Clusters
Dimer 3 4 4 4 4 5
Trimers 3 3 1 4 3 3
Tetramers 3 3 3 3 3 4
Pentamers 3 3 3 3 4 4
Hexamers 3 3 3 3 4 4
Point Defects
Vacancy 2 2 2 2 4 5
Interstitials 3 1 4 3 4 4
Melting Temp. 4 3 1 1 5 5
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Table 8.1 Continued.
SW Tersoff BOP
Property Si Ge Si Ge Si Ge
Surface Recon.
• Si surfaces
(100) 2x1 5 5 5
(113) 3x2 3 3 3
(111) 7x7 2 1 5
• Ge surfaces
(100) 2x1 3 3 4
(111) c2x8 2 2 4
These improvements do not come without a cost however. One way in which
the BOP is inferior to both the SW and Tersoff potentials is in computation time.
Simulations using the BOP consume more computational resources due to the increased
complexity of the functional format. As a result, the length of time for any simulation
using the BOP will be longer than an equivalent simulation using either the SW or
Tersoff potentials. This is not to say that the amount of time for BOP computations is
prohibitive, by no means is that the case. Molecular dynamics simulations employing
the BOP are still considerably faster than electronic structure calculations (such as DFT).
A rough estimate of the factor by which the BOP is slower than SW or Tersoff is 2-4.
While it is true that the BOP does not improve upon earlier interatomic
potentials in every way (certain features are well captured by the SW and Tersoff), the
148
BOP does successfully capture a much broader scope of material properties. The BOP
parameterization for silicon and germanium presented here is a much more transferable
potential than has been previously seen. As a result the applications to which it can be
put to use are many and varied. Within this dissertation is an example of one such use;
the examination if the solid phase epitaxy of silicon. This problem represents many
distinct challenges to a potential. The simulation of solid phase epitaxy involves the
modeling of complex atomic arrangements in the amorphous film and at the
amorphous/crystalline interface. The rearrangement of atoms at the a/c interface
involves the forming and breaking of atomic bonds, and so an interatomic potential
must accurately model the interaction of atoms at both long and short distances. Solid
phase epitaxy is a thermally driven process; therefore the interatomic potential must
respond and behave appropriately at a given temperature. Because the BOP is
successfully able to model these material properties it is well suited to the study of SPE.
In section 5 we presented a study of solid phase epitaxy using the BOP. We, like
others before us, observed an accelerated growth rate of the crystalline region. Where
the experimentally observed growth rate of the crystalline region can be measured in
m/s, the simulated growth rate is found to be orders of magnitude faster. Also
observed was a different activation energy for growth than that observed
experimentally; the value predicted by the BOP was closer to the activation energy of a
process related to SPE, ion beam induced epitaxial crystallization (IBIEC). Because of our
confidence in the validity of the BOP’s predictive ability, we concluded that what we
observed in simulation was a nanoscale atomistic mechanism that tells only a single
149
aspect of the larger story that is the SPE process. There must exist an atomistic
rearrangement mechanism that occurs on a longer timescale that limits the rate of
movement of the a/c interface. Sadly the investigation of an atomic system of such a
scale is beyond the computing resources at our disposal. This may well be resolved in
the future by continuing advances in microchip design, and the implementation of a
parallel processing version of the molecular dynamics code for the BOP.
Other avenues of investigation are open to the BOP. The investigation of rapid
crack propagation in silicon was proposed by Clint Geller at Bettis. This would require
the simulation of ~50,000 atoms, a number that is easily within reach using modern
computation techniques. Given the promising performance of the BOP in the prediction
of surface reconstructions, a wide range of surface simulations could be performed.
Also of great interest would be extending the BOP material database to include
semiconductor dopant materials such as phosphorous, boron, or arsenic. This would
require a separate parameterization for each material and the parameterization of the
binary terms. In a similar vein, it would be of great interest to parameterize the silicon-
germanium binary. The parameterization of the BOP is by no means a simple matter. It
needs to be approached in a careful manner, and often involves making choices which
material predications are more critical. In section 9 we present an examination in detail
of the parameterization process, and discuss how one would go about constructed the
Si-Ge binary potential (parallels can easily be drawn to how to develop a Si-P binary or
any other).
150
IX. Future Work: SiGe Alloy Parameterization
A critical feature of the developed BOP parameter sets for Si and Ge is that both
use an identical form of the bond order potential. This allows for the two material
systems to be employed together directly with the addition of a parameter set for SiGe
interactions. The development of a SiGe alloy BOP is the logical extension of the
research presented here. A BOP for SiGe alloys would allow for the simulations of many
modern complex SiGe multilayers semiconductor structures. This section will briefly
introduce the complexities involved in the parameterization of the SiGe alloy material
system.
The parameterization of a binary system presents a number of significant
challenges that are not present when fitting an elemental system. One such challenge
for the SiGe alloy system is that the Si0.5Ge0.5 alloy does not form a zinc-blende structure
like one would initially suppose (there have been reports of a stress induced ordered
Si0.5Ge0.5 lattice grown, however the ordered microstructure was not predominant in
the film and was difficult to reproduce [133]). Instead, the arrangement of silicon and
151
germanium atoms within the diamond cubic structure is random. Put another way,
each dc lattice site has a chance of containing either a silicon or germanium atom with a
probability directly related to their alloy concentrations. This is a direct result of the
total solid solubility of SiGe alloys. Figure 9.1 presents the phase diagram of silicon
germanium alloys. The lack of a regular crystalline structure presents a challenge for the
parameterization process.
Because fitting requires the parameters to be adjusted such that certain atomic
configurations have specific energies, one would need to provide data for specific
atomic configurations. The dearth of SiGe crystal structures forces the use of a fitting
database composed entirely of theoretical SiGe crystal configurations. Crystal
structures that likely should be employed in the fitting of the SiGe binary are: the zinc-
blende structure (ZB), the NaCl structure, and the SiGe dimer. Also of interest is the
ordered SiGe structure observed in limited deposition experiments. Energy and
Figure 9.1 Phase diagram for the silicon-germanium binary system.
152
structure information for these configurations should first be obtained from DFT
calculations such as those performed using VASP.
Because both the silicon and germanium BOP parameterizations employ the
same formats they can be integrated directly. As seen in Figure 9.1, silicon and
germanium exhibit total solid solubility. It is therefore reasonable to assume that the
parameters for silicon-germanium interactions can be derived from the two elemental
parameter sets. Precedence exists for this supposition; the Tersoff potential uses either
the average or geometric average of the silicon and germanium parameters for the alloy
parameters [47-49]. A similar approach may be possible with the BOP. The average of
the silicon and germanium BOP parameters should be a very reasonable first guess in
the parameterization of the silicon-germanium binary system. Appendix A discusses in
greater detail the Mathematica program employed in the fitting of the elemental
systems presented here, and would be an aid for anyone first approaching the complex
task of parameter fitting.
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X. Conclusion
The modeling of large scale time dependent reassembly phenomena using highly
accurate density functional methods is computationally prohibitive. Because of this,
approximations of the covalent bonding of group IV semiconductors are necessary. To
model the complex reassembly mechanisms often seen in these materials, a molecular
dynamics approach has been employed. The predictive output of this method is highly
dependent on the quality of the interatomic potential employed. Many empirical
interatomic potentials have been proposed over the years; most notable are the
Stillinger-Weber and Tersoff potentials. The silicon and germanium parameterizations
of these potentials have been evaluated for their predictive ability for small clusters,
melting temperature, bulk properties for a wide range of crystal structures (dc, sc, fcc,
bcc, -Sn, hcp, and bc8), and the energy of low index surface reconstructions. These
potentials are shown to give reliable estimates of the bulk properties of the dc phase,
but are inadequate for the study of many other structures encountered during the
atomic assembly of both silicon and germanium.
154
Pettifor et al. have developed a bond order potential (BOP), derived from the
tight binding description of covalent bonding, to model atomistic systems. This
potential format is used here to propose a BOP for the group IV semiconductors silicon
and germanium. The potential addresses both the and bonding of these sp-valent
elements. A promotion energy term associated with the formation of the hybrid
orbitals is included in the formulism. The potential was parameterized using a self-
consistent database of DFT estimates and experimental data. The BOP potential’s
predictions for the cohesive energy, atomic volume, and bulk modulus of the dc, fcc,
bcc, bc8, hcp, and -Sn phases of silicon and germanium compare favorably with DFT
estimates. The BOP also gives point defect formation energies that are in good
agreement with ab initio estimates. The structure of small atomic clusters, the melting
transition temperature and the atomic structure of low index surfaces are also used to
assess the validity of the BOP potential. The functional form of the BOP’s for silicon and
germanium are the same, facilitating their eventual use for the study of the binary Si-Ge
alloy system.
In conclusion, this dissertation presents new interatomic potentials for the highly
important group IV semiconductors silicon and germanium. These potentials are
rigorously evaluated to ensure they accurately model the energy of atomic bonding in a
wide array of bonding environments. While many of the structures that were examined
are not observed experimentally, it is possible that during atomic reassembly processes
an atom may encounter a bonding environment similar to those non-experimental
phases. It is therefore a significant success to obtain interatomic potentials that are
155
flexible enough to simultaneously accurately model multiple phases. The BOP has been
further employed to examine the re-growth of a-Si films and identified the rate limiting
atomic scale mechanism.
156
Appendix A. Mathematica Fitting
In section 2.3 a brief discussion of the parameter fitting process was presented.
In this section we will examine the Mathematica codes employed in the fitting from the
perspective of fitting a SiGe binary. These Mathematica codes were designed and
written by Dewey Murdick during his graduate work on GaAs [102]. All modifications
pertaining to silicon and germanium have been made by me. The fitting code has been
split up into two primary programs: fitGSP and fitBO. These two programs call or load
additional files which are mandatory for their proper function, however only the CG-
NNL file will need to be altered for the fitting of the SiGe binary. Since modification of
the CG-NNL file is the first thing that needs to be done (it provides inputs for fitGSP and
fitBO) it will be discussed first, followed by discussions on fitGSP, fitBO, and adjusting for
promotion energy.
CG-NNL
Both the fitGSP and fitBO programs employ preconstructed crystal lattice data
(nearest neighbor lists, NNL) for each fitting structure to ease the computational cost of
the fitting algorithm. This crystal lattice data is constructed in the CG-NNL program.
157
This program serves two primary functions: first it generates the NNL’s that are
employed by fitGSP and fitBO, and second it outputs crystal files compatible with the
Fortran MD codes used in simulation. The uses of the second function are obvious and
will not be discussed further here. Each crystal NNL that you wish to construct has its
own subfolder in the CG-NNL program. A sample for the zinc-blende structure is shown
in Figure A.1. Similar subfolders are needed for each structure desired.
The only adjustments for the SiGe alloy system that need to be made are
changing the latparam entry and Btype information for the ZB and NaCl structures.
Additional structures can be created at this point such as the ordered SiGe alloy. The
entry for ordered SiGe can use the ZB subfolder as a template and change the atom type
positions in Btype. Structural data for the ZB and NaCl structures should be obtained
Figure A.1 Excerpt from the Mathematica program CG-NNL.nb that creates the
nearest neighbor list for the zinc blende structure of GaAs. The individual parts of
the code are labeled; additional information in the text.
cF8 / B3 (Zinc Blende); GaAs
latparam = {a -> 5.653}; Lattice Parameters
latlist = {a};
A = {{a, 0, 0}, {0, a, 0}, {0, 0, a}}; Primitive Vectors
B = {{0, 0, 0}, {1/4, 1/4, 1/4}, {0, 1/2, 1/2}, {1/4, 3/4, 3/4}, {1/2, 0, 1/2),
{3/4, 1/4, 3/4}, {1/2, 1/2, 0}, {3/4, 3/4, 3/4}}; Basis Vectors
Btype = {Ga, As, Ga, As, Ga, As, Ga, As}/. {Ga -> mat1, As -> mat2};
Nucsys = {3, 3, 3}; Crystal Size
CreateAll{“B3”, A, B, Btype, Nucsys, latparam} NNL Creation
CG-NNL.nb Mathematica Excerpt
158
from DFT calculations. After all necessary adjustments have been made to the crystal
structures of importance it is time to create a new input file for fitGSP and fitBO. This is
done by executing the “Write Orthogonal Form” subfolder (more specifically the
“wlist{…}”, “DeleteFile[…]”, and “Save[…]” lines).
fitGSP
As mentioned earlier in section 2.3, the BOP has been fitted using a three step
iterative process. The first fitting step is the parameterization of the GSP function. This
step obtains values for the 6 GSP parameters (r0, rc, m, n, n, nc) as well as a value for
the repulsive energy prefactor . It should be noted that the fitGSP program generalizes
the BOP format and assigns n and n the same value (this restriction can be relaxed in
the third fitting stage). The fitGSP program uses a discovery made by Albe et al. for the
Tersoff potential that the bond energy of cubic structures and dimers can be expressed
only as a function of bond length. The BOP has a similar feature if the promotion energy
term is neglected. For structures that have only the first nearest-neighbor shell within
the cutoff distance the bond orders and have a single value which remains
constant under hydrostatic strain. The fitGSP program derives parameters for the pair
terms and outputs bond order (BO) targets for the various structures considered to
fitBO. These BO targets reflect the environmental considerations of the BOP (i.e. the
angular dependencies).
159
A limitation on structures that can be employed in fitGSP is introduced by the
parameterization theory outlined above. Only those structures that can be fully
described by first nearest neighbors and where the angular environment of each atom
in the crystal is the same can be used. This limits the SiGe alloy structures that can be
fitted to the dimer, ZB, and NaCl structures. The theoretical ordered SiGe structure will
most likely not qualify because of the lattice anisotropy along (111) planes caused by
aligned Ge-Ge bonds. This limitation will not be detrimental in the long term.
The fitGSP program has a simple presentation. The “Setup & Functions”
subfolder should not need to be altered for fitting the SiGe binary. It is recommended
that a new “SiGe data” subfolder be created using previous examples as a template.
The data subfolder will need to adjusted significantly to reflect the properties of SiGe.
The DFT data obtained for the SiGe dimer, NaCl, and ZB structures is placed in the
“objdata” array along with weight priorities. Figure A.2 presents a sample objdata input
array for the Ge elemental system. Guidance on appropriate weight priorities is difficult
to provide. A trial and error approach to assigning weights to obtain insight into their
effects would be of benefit to a new user. Additional non 1st nearest neighbor
structures can be input as “auxiliary structures”. These structures are not fitted,
however, the fitGSP output will give estimates on the pair fitting and the corresponding
BO targets. Each auxiliary structure is placed in the “auxdata” array as a matrix
composed of structural information. An example for the Ge -Sn phase would be:
160
04944
122263.3"35"
028173.2"25"
0444
153314.2"15"
22
22
caA
cA
caA
The first column indicates the structure and which nearest neighbor the data refers to;
in this case “A5-1” refers to the first nearest neighbor in the -Sn structure. The second
column is the separation distance. The third column is an equation for the separation
distance as a function of the lattice constants. The fourth column is the atomic
Figure A.2 The primary data entry array found in the fitGSP.nb Mathematica fitting
program. The columns have the following definitions:
I. Crystal Structure
II. Nearest Neighbor Distance, r
III. Equation for r based on the lattice parameters
IV. Cohesive Energy per atom
V. Atomic Coordination
VI. Bulk Modulus in GPa
VII. Atomic Volume as a function of r
VIII. Fitting weight for cohesive energy
IX. Fitting weight for bulk modulus
objdata =
“dimer” 2.3 r -1.32 1 250 150
“A4” 2.4465
-3.82 4 62.17
500 200
“Ah” 2.6588 a -3.578 6 67.38 100 50
“A1” 2.9652
-3.484 12 63.49
600 100
I II III IV V VI VII VIII IX
fitGSP.nb Mathematica Excerpt
161
coordination of the 1st, 2nd or 3rd nearest neighbor shell. The fifth column defaults to
zero, corresponding to a feature in the code that was never finalized.
The final step before fitting the GSP parameters is choosing constraints on the
parameters. The preset variable ranges are generally sufficient for fitting, however
these can be changed if so desired (although a parameter value of zero or less would be
unphysical, this should be avoided for obvious reasons). The parameter k corresponds
to the m/n ratio (1/k = m/n) which determines the potential hardness. For group IV
semiconductors a m/n ratio target of ~2 is theoretically motivated (for example Si has a
m/n value of 1.8). To ensure a proper m/n ratio, the value of k can be constrained to 0.5
in fitGSP (this value may be adjusted later in the third stage; however it should not be
allowed to deviate significantly from the target of ~2).
Fitting the GSP parameters is an iterative process. The initial structure data for
SiGe may not provide a data trend with points that intersect a smooth GSP curve. This
makes the fitting process difficult, since all the point cannot be intersected. The initial
data must be iteratively adjusted to fit a smooth GSP curve (without significant sacrifice
of important considerations). A non-smooth GSP curve will provide unusual
abnormalities in simulation which are unintended and unrealistic, therefore it is
considered mandatory that the GSP function be smooth and continuous. The default
minimization technique uses Differential Evolution, a genetic algorithm that maintains a
population of specimens, x1… xn. At each iteration a random combination of specimens
is “mated” to form a new specimen which is compared to its progenitors. If the new
162
specimen represents an improvement it replaces one of the originals, if not it is
discarded. The differential evolution method is employed because it is considered quite
robust.
Once a suitably smooth GSP curve has been obtained that inTersoffects the
adjusted SiGe target values the fitGSP program will provide a set of three matrices that
are the data inputs for fitBO. These can be copy/pasted directly into the appropriate
fitBO file.
fitBO
The fitGSP program obtains parameters that determine the radially dependent
pair-wise interactions. The fitBO program determines the appropriate angular
dependencies and bond order strengths. the fitBO determines values for the following
parameters: (ds), f, k, c, p (p), p (pn), and (parameters in parenthesis are the
Mathematica equivalent). The f and k parameters are not appropriate for use with the
SiGe system; they are designed to accommodate semiconductor systems with non-half
filled valence shells (such as the III-V GaAs system). As such they are assigned values of f
= 0.5 and k = 1.0 (these values result in the valence skewing term being set to zero). For
both the Si and Ge elemental systems the angular parameter c has been set to zero; this
would be a good condition to start with for fitting SiGe, although additional angular
flexibility may be needed. If the parameter c is employed, Equation 2.23 in section 2
would become:
163
c
cpg
ijk
ijk
1
2cos)1(cos1
A.1
Additional parameters cprom and dprom (corresponding to the promotion energy terms
A and respectively) are listed in fitBO, however these parameters are not assigned any
significance in the Mathematica program (they are declared but never used). These two
parameters can be given any value desired, however they have been set to a default of
cprom = 0 and dprom = 1. Similarly the parameters b2n and b2o should also be
assigned values of one. These parameters were from an earlier formulation of the BOP
and are not employed in the Si and Ge potentials. After all these considerations only the
parameters , p, p, and should not be assigned any constant value.
Use the inputs from fitGSP and the preexisting samples in fitBO to construct the
input subfolder for the BO fitting. This subfolder should consist of initial variable
assumptions (detailed in previous paragraph), the three data matrices from fitGSP, and
assigned fitting weights for the energy, the BO and the pressure. The weight placed on
pressure indicates the priority on maintaining the provided structure data as a minimum
of energy. Typically the pressure weights are the highest to ensure that the correct
crystal structure is maintained. Choosing the appropriate values for the weights has
been described as an intuitive “art” rather than a science [102]. There is some truth to
this, as a significant amount of trial and error is poured into the fitting process. It is also
possible to include defect structures at this point of the fitting (i.e. a single germanium
atom in a silicon lattice or similar situations). This is done by reading in a
164
preconstructed data file containing the xyz lattice data for a crystal of interest. The
fitBO program can be set to calculate the energy of this system, from which you can
determine the energy of interest. There are four minimization schemes preset into
fitBO: automatic (Mathematica chooses a minimization scheme based on the nature of
the problem), Differential Evolution, Nelder Mead, and Simulated Annealing. Employing
a combination of these methods generally gives the best result.
The program fitBO outputs all the necessary parameters (including the
placeholder values for the promotion energy terms) in the “fitvalu” array. This array
lists the parameters in the following order (using the Mathematica nomenclature): b2n,
b2o, c, cprom, dprom, ds, f, k, nc, nf, ns, p, pn, r0, r1, rcut, 0, 0 and 0. This array is
the input for another Mathematica program file designed to generate a potential force
file for the Fortran MD code. Because we still need to input the promotion energy
functionality into the potential this 4th program cannot be employed. Also outputted by
the fitBO minimization subroutine is data regarding the quality of the fit. This data
includes the energy predictions for the structures provided as well as data about the BO.
A small graphic of the fitted angular dependence is also generated.
Fitting Uprom
In the third stage of the fitting process we introduce the promotion energy,
Uprom. The introduction of a non-zero promotion energy term will alter the potential
predictions in sometimes unexpected ways. It is therefore important to understand
165
how each parameter will influence the shape of the potential. A visual approach based
on the dimer energy curve is perhaps the simplest. Figures A.3 to A.6 compile these
effects. These images use the fitted Ge potential as their baseline. The parameters are
then adjusted by ±5% and ±10% to illustrate the nature of their influence on the shape
of the dimer energy. What follows are general comments on the nature of the changes
observed for varying each parameter individually. The effects of varying the parameters
in the many myriad combinations available are not discussed.
The bond energy coefficient terms (0, 0 and 0) behave as one would expect
when varied. Lowering 0, the repulsive energy coefficient, results in a lower energy for
Figure A.3 The effects of altering the
BOP parameters , and by small
increments. The direction of motion is
indicated by the + and – signs, where a
+ indicates the parameter was
increased and a – indicates the inverse.
+
+
+ -
-
-
166
all structures (i.e. stronger bonds). The reverse is also true; increasing 0 raises the
energy. A secondary effect of lowering 0 is that the separation distance, rij, at which
the potential predicts a minimum of energy also decreases. The parameters 0 and 0
influence the potential in a similar fashion, although because that are a measure of the
bond strength and not repulsion the trends are reversed. Increasing 0 and 0 results
in a lowering of the bond energy as well as shifting the energy minimum to the left
(lower NN separation). The bond energy coefficients can therefore be used to “tune”
the BOP to match the appropriate minimum energy configuration (in the case of silicon
and germanium, the dc structure). These energy terms also have a small influence on
the curvature of the potential near the minimum, although other terms will have a
greater influence (increasing 0 and decreasing 0 and 0 results in a lower curvature
and vice versa).
Figure A.4 The effects of altering the BOP promotion energy parameters A and by
small increments. The direction of motion is indicated by the + and – signs, where a
+ indicates the parameter was increased and a – indicates the inverse.
+ + - - A
167
The promotion energy terms A and (called cprom and dprom in Mathematica
respectively) have very similar effects on the shape of the potential to the repulsive
energy coefficient 0. This is because they are both positive terms in the potential. The
promotion energy represents the energy penalty incurred by hybridizing the atomic
orbitals; therefore as the magnitude of the promotion energy is increased the bond
energy is weakened. The promotion energy has the form
212
,11
ij
ij
prom AU
9.2
Therefore if A is very large Uprom = and if A is very small Uprom = 0. If is very large
Uprom = 0 and if is very small Uprom = 0. It is therefore important to keep a careful
balance for these parameters. For the parameterization of the SiGe binary initial values
Figure A.5 The effects of altering the BOP cutoff parameters r1 and rcut by small
increments. The direction of motion is indicated by the + and – signs, where a +
indicates the parameter was increased and a – indicates the inverse.
+ +
- - A
- -
+ +
r1 rcut
168
of A = 1.325 and = 8.342 would not be unreasonable (the average between the Si and
Ge elemental potential parameters).
The cutoff parameters r1 and rcut have little influence on the potential at or near
equilibrium separation distances. This is of course by design; the inclusion of a potential
cutoff is not physically motivated but required for computational purposes. By changing
these terms you can introduce non-physically motivated effects into the potential. This
is evidenced by the small separation distance around 3.4 Å in Figure A.5 for rcut in which
the dimer binding curve becomes positive. Increasing the cutoff distance also results in
Figure A.6 The effects of altering the BOP cutoff parameters r0, rc, rc and rc by
small increments. The direction of motion is indicated by the + and – signs, where a
+ indicates the parameter was increased and a – indicates the inverse.
+ +
+ +
-
-
- -
r0 rc
rc rc
169
the inclusion of additional atoms in the neighbor list for certain structures. This greatly
increases the computational cost of simulations for little benefit and must be avoided.
It is also important that the cutoff not be too sharp as this may result in aberrant
behavior in simulation. It is therefore important that r1 be sufficiently smaller than rcut
to provide a smooth cutoff. If r1 is too small however, structures with large NN
separation distances will be affected, most notably the fcc structure.
The parameters r0, rc, rc and rc have a profound effect on the shape of the
potential. The parameter r0 is intended to be the separation distance corresponding to
the minimum energy of the dimer; however the inclusion of the promotion energy term
prevents that. This is because the promotion energy term is not written in the form of a
GSP pair term. As a result the dimer minimum energy is located at a slightly higher
value (2.32 Å vs. 2.395 Å). Altering r0 allows one to control the position of the energy
minimum directly with little change to any other property. The parameter r0 can
therefore be used to “tune” the potential directly to the desired minimum energy
structure. The parameters rc and rc have the distinction that they are the only
parameters which were capable of altering the order of energy preference between
structures with only a 10% change in value. The dramatic changes observed in the
shape of the dimer curves in Figure A.6 alter the order of energy preference for atomic
structures. When rc is decreased, the energy minimum shifts to the right and is
lowered. As a result, the energy minimum is much closer to the fcc NN distance and the
fcc structure becomes the lowest energy structure. A small increase in rc does not
significantly shift the position of the dimer energy minimum, but does increase the
170
energy curve slightly at larger radius. This results in the fcc structure becoming
significantly less stable. The parameters rc and rc behave in a similar fashion. Great
care must be employed when altering these parameters.
The six GSP exponential parameters nx and ncx (where x is , or ) do not
significantly alter the shape of the potential when varied within 10% (as such they are
not shown in a separate figure). These parameters are therefore more suited for “fine
tuning” the potential than providing large blunt alterations such as can be achieved by
varying other terms such as r0. All six of these terms have a slightly larger influence on
the energy of structures with larger atomic spacing. Perhaps the most interesting
feature of these parameters is that there exists a value of r (the separation distance
between atom i and j) such that the predicted energy is invariant upon nx and ncx. This
occurs at r = r0 by design.
Using these parameter trends it is possible to incorporate a physically
meaningful promotion energy term into the pair-wise BOP obtained in the first two
fitting stages. A significant investment in time will be necessary to fit the SiGe binary
BOP. Using the suggestions laid out here should greatly assist a future researcher with
the project.
171
Appendix B. Density Functional Theory Calculations
A portion of the density functional theory (DFT) estimates that are presented in
this dissertation were calculated personally by the author using the Vienna Ab Initio
Simulation Package (VASP). VASP is a software package created, distributed, and
maintained by the Hafner Research Group at the University of Vienna that performs ab-
initio quantum mechanical calculations using pseudopotentials and a plane wave basis
set. The simulation approach implemented in VASP is founded on the finite
temperature local density approximation with the free energy as a variational quantity.
The VASP guide, currently available online at http://cms.mpi.univie.ac.at/vasp/, is an
excellent resource for any researcher attempting to use the software package and
served as the primary reference source for the calculations performed by the author.
The VASP guide provides significant assistance in understanding program specific
terminology, file names and control flags that control the program. References 146 to
149 provide extensive detail on the program itself and the theory behind it. This
dissertation makes use of VASP to obtain a consistent database of silicon and
germanium bulk properties for use in the fitting process. The input files employed by
172
VASP are INCAR, POTCAR, KPOINTS and POSCAR. These input files are described here
and the non-default values employed are shown.
B.1 INCAR
The file INCAR is the central input file of VASP. It provides the “what to do and
how to do it” for the program. There are a large number of adjustable parameters for
the researcher available within INCAR, however it is not necessary to change many of
them as the default values are typically convenient. The non-default values that were
specified in the VASP estimates performed are as follows:
ISMEAR = -5 (the recommended value for semiconductors)
ENCUT = 300 (good energy convergence was found using this value for the
cutoff energy)
B.2 POTCAR
This file contains the pseudopotential (PP) for each atomic species in the
calculation. The VASP package comes with a database of available PPs for the majority
of elements that one would consider. Variations in PPs are also available for many of
the elements, including silicon and germanium. The PP that was chosen for use here
was the projector augmented wave (PAW) potential. This was chosen because it comes
highly recommended by the developers of VASP.
173
B.3 KPOINTS
This input file, as its name suggests, specifies the specific k-point coordinates for
generating the k-point grid. Two methods exist for the input of the k-points, explicit and
automatic generation. The automatic k-point generation method has been selected
here because of the relatively simple nature of the calculations. In this method you are
given the option of choosing between either the Monkhorst or Gamma k mesh
generator methods. It is recommended in the VASP literature that the Monkhorst
method be chosen for cubic crystal cells and the Gamma method for hexagonal crystal
cells. A sample KPOINTS file to be used with a sc crystal would read as follows:
Automatic (first line is considered a comment line)
0 (number of k-points = 0 indicates automatic generation)
Monkhorst-Pack (indicates the use of the Monkhorst generator method)
4 4 4 (k-point mesh subdivisions)
0 0 0 (optional shift of the mesh)
B.4 POSCAR
This file contains the specific lattice information that generates the crystal.
Whereas each of the previous files were largely independent of the specific crystal being
examined, the POSCAR file must be specifically tailored to each crystal. A sample
POSCAR file to generate a fcc lattice is shown below.
fcc cubic crystal (first line is considered a comment)
3.819 (lattice constant)
174
1 0 0
0 1 0 (unit cell lattice vectors)
0 0 1
4 (number of atoms within the unit cell)
Direct (specifies the method atomic positions are provided)
0 0 0 (atomic positions)
0.5 0.5 0
0.5 0 0.5
0 0.5 0.5
175
Appendix C. Qualitative Analysis Determinants
Table C.1 This table describes the ranges used to determine the numerical value (1 to
5) assigned to each property in Table 8.1. The parameter x represents the percentage
difference between the DFT estimate and the MD prediction. The parameter
represents the numerical difference between the DFT estimate and the MD prediction.
Property 1 2 3 4 5
Va x > 10% 5% < x < 10% 2% < x < 5% 1% < x < 2% x < 1%
Ec > 0.5 0.3 < < 0.5 0.1 < < 0.3 0.02 < < 0.1 < 0.02
B x > 100% 50% < x < 100% 20% < x < 50% 6% < x < 20% x < 6%
Clusters* x > 40% 30% < x < 40% 20% < x < 30% 10% < x < 20% x < 10%
Defects** x < 30% 20% < x < 30% 10% < x < 20% 5% < x < 10% x < 5%
Tm ±400 K ±300 K ±200 K ±100 K ±50 K
Surfaces 0.1 < 0.04 < 0.02 <
* A -1 penalty was applied if the lowest energy cluster did not match the HF predicted
lowest energy structure
** A -2 penalty was applied if the lattice around the vacancy did not relax inward.
176
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