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Journal of Crystal Growth 286 (2006) 197–204
www.elsevier.com/locate/jcrysgro
Low-temperature atomic assembly of stoichiometric gallium arsenide
from equiatomic vapor
D.A. Murdick, X.W. Zhou, H.N.G. Wadley
Department of Materials Science and Engineering, School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22904, USA
Received 2 September 2005; received in revised form 30 September 2005; accepted 3 October 2005
Communicated by M. Uwaha
Abstract
The low-temperature atomic assembly of homoepitaxial GaAs thin films on the (0 0 1) surface has been investigated using molecular
dynamics with a Stillinger–Weber potential energy function. During equiatomic vapor deposition, crystalline growth was observed for
substrate temperatures above 35% of the melting temperature. Below this temperature, the critical epitaxial thickness began to rapidly
decrease as defects were increasingly incorporated and eventually nucleated an entirely amorphous structure. The atomic assembly
mechanisms of arsenic dimer incorporation, as well as gallium vacancy formation, were studied just above the amorphous/crystalline
growth transition temperature. The adsorption of arsenic dimers was found to show dependence upon the orientation of the deposited
molecule. Atomic processes responsible for the formation of the gallium vacancy defects were observed, and the influence of growth
temperature on defect formation was also identified.
r 2005 Elsevier B.V. All rights reserved.
PACS: 81.15.Kk; 82.20.Wt; 81.05.Ea
Keywords: A1. Molecular dynamics simulation; A1. Stillinger–Weber potential; A3. Molecular beam epitaxy; A3. Vapor deposition epitaxy; B1. Gallium
arsenide
1. Introduction
GaAs thin films can be grown by a variety of physical
vapor deposition methods [1,2], including molecular beam
epitaxy (MBE), digital alloy deposition with annealing [3],
and pulsed-laser melting [4]. All proceed under ultra-high
vacuum conditions. The major differences between the
methods are the chemical form in which the gallium and
arsenic species arrive at the growth surface and the growth
surface’s atomic structure. The surface atomic structure
depends in turn on the growth temperature and the relative
concentration of the vapor phase species [5,6].
The homoepitaxial vapor deposition of GaAs under
MBE conditions can be divided into high-temperature
Corresponding author. Tel.: +1 434 982 5673; fax: +1 434 982 5677.
E-mail address: [email protected] (D.A. Murdick).
0022-0248/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jcrysgro.2005.10.006
(HT) and low-temperature (LT) ranges. HT MBE is
performed at temperatures 0:45oT=T m o0:60 [1,7,8],
where T is the absolute growth temperature and T m is
the absolute melting temperature. LT MBE typically
occurs at growth temperatures in the range 0:34o
T=T m o0:38 [1,8].
Low-temperature growth conditions are often used when
a large concentration of dopants needs to be added to
GaAs [9]. Low temperatures [10] enable kinetic trapping of
dopants and impede the clustering and surface segregation
that normally follows when doped materials are processed
at a high temperature [9]. This is especially important for
some spintronic materials, such as manganese doped GaAs
alloys, where the clustering of manganese atoms in
Ga1x Mnx As alloys can destroy the ferromagnetic properties of interest [2,10–12]. Under LT MBE conditions,
stoichiometric GaAs can only be grown when the As:Ga
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ratio of the vapor flux is near unity [13]. If high As:Ga flux
ratios are used, the arsenic concentration in the sample
becomes elevated [14]. Therefore, the upper temperature
limit of doped GaAs films is set to restrict dopant
migration and the lower bound is set by the need to avoid
the onset of amorphous film growth under an equiatomic
flux [15].
This amorphous/crystalline transition between growth
modes is a function of the deposition rate, temperature,
and As:Ga flux ratio [8,7]. Experimentally, the transition is
assessed from estimates of the maximum thickness that an
epitaxial film can be grown—the critical epitaxial thickness
ðyepi Þ [15]. Using such an approach, epitaxial growth under
MBE growth rates has been shown to require a growth
temperature T=T m 40:32 [15]. Like crystallinity, the
concentration of defects is also strongly dependent on
the deposition rate, growth temperature, and flux composition [13,15,16]. Experimentally, defect concentrations
are below 1014 cm3 for high growth temperatures
ðT=T m ¼ 0:5420:61Þ, while defect concentrations are on
the order of 1018 21020 cm3 for low growth temperatures
ðT=T m 0:38Þ [13].
The atomic scale mechanisms responsible for crystalline/
amorphous growth mode transition and defect trapping are
not well understood in the GaAs system. Here we use a
molecular dynamics (MD) modeling approach to explore
some of the atomic-scale assembly mechanisms and their
linkages to the film growth conditions. Particular attention
is paid to the growth of GaAs thin films with a near
equiatomic As:Ga flux, which is relevant to the lowtemperature growth of highly doped crystalline films
[9,13,15].
2. Vapor deposition simulation
In MD simulations, the interatomic potential energy
function significantly affects the reliability of the results.
Here, a potential function proposed by Stillinger and
Weber (SW) [17] and parameterized by Angelo and Mills
[18] and by Grein et al. [19] has been utilized for
simulations of GaAs film growth. The suitability of this
potential for GaAs simulations has recently been assessed
and shown to be the best available potential for simulations
of LT GaAs (0 0 1) growth [20].
The MD simulations were conducted using the Nordsieck predictor corrector algorithm with a 2 fs time step
[21]. The initial GaAs substrate had a (0 0 1) surface that
measured 32 Å 32 Å. An arsenic-rich bulk-terminated
(0 0 1) surface was initially created. Upon annealing prior
to the start of deposition, it reconstructed to form a ð2 1Þ
dimer row surface [20]. The substrate consisted of six layers
and each layer contained 64 atoms per (0 0 1) plane. The
bottom layer was fixed and the five layers above were free
to vibrate. Simulations of vapor deposition were conducted
for 10 ns of real time, which is sufficient for the deposition
of a film with more than 12 monolayers. Many of the
simulations were repeated 7–10 times to generate more
statistically reliable results.
The time scales available to MD simulations require the
use of an accelerated deposition rate, as compared to those
used in experimental MBE. During an MD simulation,
either a gallium atom or an arsenic dimer was deposited
every 14 ps. The vapor made normal impact with the
surface. Each atom or molecule was given an initial
translational energy of 0.17 eV. No rotational energy was
initially given to the molecules. The average internal
(vibrational) energy of the As2 molecules was equivalent
to a vapor temperature of around 1100 K (0.25 eV/atom).
The total (translational and vibrational) energy was
comparable to the kinetic energies of gallium atoms and
arsenic dimers emitted from effusion cells [1,7].
During vapor condensation on a growth surface, the
latent heat release, and to a lesser extent the species kinetic
energy, are partitioned into the vibrational modes of the
lattice. This excess energy causes an increase in surface
temperature. To avoid this unrealistic overheating, the first
four substrate layers above the fixed layer were thermally
controlled via a Nosé-Hoover thermostat algorithm [22]. In
this approach, a dragging force is applied to each atom.
The force was proportional to ðT actual T goal Þ=q, where
T actual is the actual temperature of the atom, T goal is the
desired temperature of the system, and q is the drag
coefficient, which is a function of the time step, desired
temperature, and the mass of the atom. As the system
evolved over time, the thickness of the temperaturecontrolled region grew proportionally to the number of
atoms added to the thin film through vapor deposition.
The accelerated deposition rate used here effectively
limits long-range diffusion events that can occur between
impact events. One approach for estimating the equivalent
experimental temperatures is to equate the simulated and
experimental homologous temperatures. The homologous
temperature is calculated by scaling the substrate temperatures by the melting temperature. The melting temperature
of the SW potential has previously been determined to be
T m ¼ 2620 100 K [20], which is higher than the experimental GaAs melting temperature T exp
m ¼ 1513 K [23], so
that T exp
m =T m ¼ 0:577 0:066. For this particular potential, with an overestimated melting temperature, this
approach is likely a reasonable approximation. Therefore,
unless otherwise indicated, the temperatures will be
reported using these homologous temperatures, T=T m .
3. Results
3.1. Crystalline order and temperature
Two representative low-temperature simulation results
are shown in Fig. 1 using equal As and Ga fluxes. Over the
range of temperatures subsequently investigated, the
deposited films had an equiatomic (stoichiometric) composition, consistent with experimental observations of growth
under these low-temperature conditions [13].
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199
defects. In addition to direct measurement of crystalline
order, the critical epitaxial thickness ðyepi Þ was measured
for the simulated samples. We found that yepi for the
simulated films changed from 12:7 Å at T=T m ¼ 0:34 to the
full film thickness at T=T m ¼ 0:38. Experimentally, crystalline growth is observed for T=T m 40:32 [15], which is
below the simulated transition temperature estimate.
3.2. Arsenic adsorption and desorption mechanisms
Fig. 1. Simulated GaAs thin films are shown for (a) T=T m ¼ 0:27 (mostly
amorphous growth) and (b) T=T m ¼ 0:38 (crystalline growth), where T is
simulated growth temperature and T m is the melting temperature
predicted by the SW potential. The As:Ga flux ratio is close to 1. The
initial substrate is marked with a reflective plate placed behind the lattice.
Amorphous
Crystalline
Fig. 2. The crystalline order parameter, S, for simulated GaAs thin films
as a function of growth temperature, T=T m . The As:Ga flux ratio was held
close to unity, and the order parameter was calculated after cooling the
lattice to 0 K.
To quantify the crystallinity of the as-grown films, a
dimensionless crystalline order parameter, S, was defined
such that a non-vibrating bulk crystal, with all lattice sites
occupied has a value of unity, while a system with no atoms
on a crystal lattice has a value of 0 (see the Appendix for its
definition). The crystallinity parameter is shown as a
function of temperature in Fig. 2. Atomic position resolved
images of these thin films indicate that crystalline films
were grown when the order parameter, S, was greater than
0.5, while increasingly disordered films were produced
when S was below this value.
The simulated amorphous/crystalline structure transition temperature occurred near T=T m ¼ 0:35 0:04 for
S 0:5. The large error bars shown in Fig. 2 around this
transition temperature illustrate the significant randomness
of amorphous growth nucleation near the transition
temperature. In this region, the degree of crystalline
growth is sensitive to the random creation of atomic scale
The mechanisms of molecular arsenic adsorption and
desorption can be investigated using time resolved imaging
of atom locations. Two mechanisms were identified for As2
incorporation under LT MBE conditions. One mechanism
incorporated both atoms of the dimer into the film and
occurred the majority of the time (96 3% of dimer
impacts) during LT MBE. The time resolved atom position
sequence shown in Fig. 3 summarizes this mechanism. The
arsenic dimer (atoms marked 1 and 2) vertically approached the (0 0 1) surface with the dimer at an angle of
about 45 to the surface normal, Fig. 3(1). This dimer was
introduced with little internal vibrational energy and no
angular momentum, and it retained these characteristics as
it approached the surface. Upon impact, both arsenic
dimer atoms bonded rapidly to the surface (atoms 3 and 4),
Fig. 3(2). They then slowly reoriented to form a second
pair of strong bonds to atoms 5 and 6, Fig. 3(2–6).
The binding energy and dimer position above the surface
for this adsorption and reorientation sequence are shown
in Fig. 4. The ‘‘surface/dimer binding energy,’’ Fig. 4(a),
was determined at each time step by calculating the
difference in the total energy between two identical systems
with the dimer removed to a position far above the surface
and with the dimer at the position indicated in Fig. 4(b).
After the initial contact between the arsenic dimer molecule
and two surface atoms, Figs. 3(2) and 4(2), the arsenic
atoms reoriented to form bonds with additional surface
atoms. During this reorientation, the dimer/surface binding
energy data exhibited a temporary transitional bonding
plateau that persisted for 1.5–2.0 ps. The dimer/surface
binding energy of this transitional bonding state was roughly
70% less than the stable fully adsorbed state, Fig. 3(6). The
existence of such a transitional state during GaAs vapor
deposition is consistent with DFT calculations [24].
A second mechanism of partial arsenic dimer incorporation was also observed, but it occurred infrequently
(4 3% of dimer impacts), Fig. 5. This mechanism is very
sensitive to the strength of the dimer bond, which is
underestimated by the potential used here. We therefore
suspect that the observed probability of this mechanism
may be substantially overestimated [20]. Further work with
enhanced potentials will be needed to ascertain its true
significance. For this mechanism, the arsenic dimer (atoms
1 and 2) approached the surface vertically with a dimer
bond angle that was almost perpendicular to the (0 0 1)
surface, Fig. 5(1). Atom 1 initially bonded to the surface
atom 3, Fig. 5(2). Atom 2 remained vertical and unbonded
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(a)
(b)
Fig. 4. Incorporation of As2 into a gallium-rich (0 0 1) surface at
T=T m ¼ 0:34: (a) dimer binding energy and (b) distance above the
surface. The vertical dashed lines are numbered to match the snap-shots
shown in Fig. 3.
with atoms 4 and 5, the potential energy between atoms 1
and 2 increased as evidenced by the decreasing distance
between the atoms in Fig. 5(b). The first dimer atom then
received a thermal kick, which when combined with the
increased potential energy was sufficient to break the dimer
bond and enable desorption of atom 2. The initial
orientation of the incoming arsenic dimer appeared to
determine the likelihood that this mechanism occurred.
Fig. 3. Incorporation of As2 into a gallium-rich surface at T=T m ¼ 0:34
over 5 ps with 1 ps time-series snapshots of surface bonding.
to the surface, while the kinetic energy of the dimer pushed
atom 3 into the surface, Fig. 5(3). Atom 1 then began to
reorient on the surface and eventually bonded to atom 4,
Figs. 5(4)–(5). Although, the separation between atoms 1
and 2 reduced during the initial impact, it was eventually
kicked off and desorbed, Fig. 5(6). Atom 2 desorbed with
an extra 0:3 eV of energy more than it had initially.
The dimer binding energy and distance above the surface
are plotted as a function of time in Fig. 6. A transitional
bonding state was again seen, Fig. 6(a), but this time it
persisted for only about 0.6 ps, which was sufficient for the
dimer atoms to find a low-energy binding site on the
surface (as opposed to two atoms in the previous
mechanism). During the period of time that atom 1 bonded
3.3. Point defect formation
The assembly of atoms and dimers on the GaAs (0 0 1)
surface is a random process that can introduce defects
under low-temperature growth conditions. The defect
formation energy directly impacts the likelihood that a
point defect will form and strongly contributes to the
concentration of each point defect type. The defect
formation energy, O, can be calculated following a method
proposed by Zhang and Northrup [25,26]. They showed
that the defect formation energy in a binary system is not
only determined by the calculated change in system energy,
but is also strongly impacted by the environmental
conditions above the grown surface. The defect formation
energy is calculated by
O ¼ E 0D 12 DnDm,
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(a)
(b)
Fig. 6. Incorporation of one of the atoms from an arsenic dimer into a
gallium-rich surface at T=T m ¼ 0:34: (a) dimer binding energy and
(b) distance above GaAs (0 0 1) surface. The light gray dashed lines are
numbered to match the snap-shots shown in Fig. 5.
As i
As Ga
Ga i
VAs
Fig. 5. Incorporation of one of the atoms from an arsenic dimer into a
gallium-rich surface at T=T m ¼ 0:34 over 1 ps with 0.2 ps time-series
snapshots of surface bonding.
VGa
where E 0D is the defect formation energy calculated for
the computational cell, Dn is the difference between the
number of gallium and arsenic atoms ðnGa nAs Þ in the
computational cell, and Dm is the relative chemical
potential that accounts for the effect of relative partial
pressures of arsenic and gallium in the vapor environment
above the surface. A neutral charge state for each defect
was assumed.
The defect formation energy, E 0D , was previously
reported for vacancies (V Ga and V As ), antisites (GaAs
and AsGa ), and simple tetrahedral interstitials (Gai and
Asi ) [20,27]. It was calculated at 0 K using a conjugate
-2.53 -1.90
-1.27
Ga As
-0.63
0
0.63
µ
1.27
1.90
2.53
Fig. 7. The defect formation energies for gallium and arsenic vacancies
(V Ga and V As ), antisites (AsGa and GaAs ), and interstitials (Gai and Asi )
are plotted as a function of chemical potential, Dm.
gradient energy minimization algorithm [28]. The chemical
potential, Dm, was varied between DH f , where DH f ¼
2:53 eV per formula unit is the heat of formation for the
GaAs zinc blende structure [20]. The defect formation
energies are shown over a range of arsenic- to gallium-rich
equilibrium environments in Fig. 7. The equiatomic vapor
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condition used in this study was neither significantly
arsenic- or gallium-rich. The expected environment therefore lies somewhere between these two extremes.
The defect concentration is also dependent upon kinetic
factors during film growth, such as deposition rate and
growth temperature as well as vapor flux composition
[13,15,16]. For films grown at constant As:Ga flux ratio
and deposition rate (as is the case for this computational
study), the sensitivity of defect concentration to temperature can be explored. The defect concentration is plotted as
a function of temperature for gallium vacancies ðV Ga Þ,
arsenic vacancies ðV As Þ, gallium antisites sitting on arsenic
sites ðGaAs Þ, and arsenic antisites sitting on gallium sites,
Fig. 8. For crystalline samples with S40:5, the occurrence
of gallium and arsenic interstitials was not observed
sufficiently frequently to plot as a function of temperature.
In fact only a single gallium interstitial was observed in
these many simulations. Defect complexes were also
observed including a V Ga 2Asi defect complex.
Lattice defect concentrations were measured by counting
the gallium and arsenic atoms for eight of the grown layers
above the substrate. These layers were sufficiently below
the surface to avoid the disorder inherent in free surfaces.
Lattice vibration was removed from the system by first
minimizing the system energy to help the atoms find their
local low-energy positions. The method was quite effective
for films grown with S40:5 ðT=T m X0:34). The defect
concentration detection limit for each run was 8:6 1019 cm3 due to the limited film thickness sampled. By
increasing the number of simulations at a given temperature, this was reduced to 1:3 1019 cm3 .
The defect concentration for all four defect types
dropped as the growth temperature was increased. The
defect concentrations for both antisite defects dropped to
levels at or below the defect detection limit for
T=T m 40:40. Gallium antisites were slightly more prevalent at the lower temperatures. The arsenic vacancies
Fig. 8. Defect concentrations for vacancy and antisite defects are plotted
as a function of temperature, T=T m . As:Ga flux ratio was held near unity
during the simulated GaAs thin film growth.
showed a slightly higher concentration than either of the
antisite defects. The fact that the concentration of arsenic
vacancies was higher than the gallium antisites is interesting because the defect formation energy of GaAs is less than
that of V As for these two gallium-rich point defects.
Clearly, kinetic processing slightly favors the formation of
vacancies for simulations with the Stillinger–Weber potential. Gallium vacancies were commonly observed for the
growth conditions simulated. Concentrations reached
3:4 1021 cm3 at T=T m ¼ 0:35. This is higher than the
concentration of gallium vacancies ð125 1018 cm3 Þ
reported experimentally at a similar temperature [13]. The
defect concentrations were the highest ð5 1021 Þ for films
grown very close to the transition temperature. This is
higher than the defect concentrations of 1018 to 1020 cm3
for low growth temperatures ðT=T m p0:38Þ [13]. The onset
of amorphous growth, Fig. 2, is directly related to the
concentration of defects in the as-grown film, Fig. 8.
Therefore, the elevated amorphous/crystalline transition
temperature, which was noted earlier, can now be
explained in terms of the overprediction of defect
concentration that drives the amorphous/crystalline transition in simulated thin films.
Several factors may have influenced the high defect
concentrations seen in the LT simulations. Firstly, the
accelerated deposition rates used in our simulation would
likely trap a higher concentration of defects than experimental MBE growth rates. Secondly, the vacancy defect
formation energies predicted by the potential were reduced
by 17% for gallium and 13% for arsenic when compared to
formation energies predicted by DFT [20,27]. This
discrepancy will, therefore, cause vacancy formation to
be overpredicted. Thirdly, in regard to the high arsenic
antisite concentration, arsenic atoms around a gallium
vacancy are often displaced. This increases the probability
of finding the neighboring arsenic atoms in defect
positions. In this way, arsenic atoms can form defect
complexes with gallium vacancies. Such a formation
mechanism would help account for the observation of
arsenic antisites despite their prohibitively large formation
energy. Lastly, the 9% variation of the As:Ga ratio for
the 64 runs analyzed represents a deviation from stoichiometric growth conditions and could increase the defect
concentration.
Gallium vacancies were the most common defects
observed in the simulation. The formation of such a defect
at discrete time steps during film growth is shown in Fig. 9.
A single cross-sectioned plane of the thin film is shown over
4000 ps of simulation time. In all the snapshots, atoms
oscillated in and out of the cross-sectioned plane. This
oscillation (and the related surface diffusion) was particularly evident at surfaces and around defects as is clearly
seen in Fig. 9(a) and (d). Initially (Fig. 9(a)), As2 was
deposited and one of the arsenic atoms (atom 1) entered
the shown plane. Atom 1 was strongly bonded to a gallium
atom to the left (not shown) and very weakly bonded to the
arsenic atom to the right. (Note that in these figures the
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vacancies. This is largely a kinetic effect and could be
annealed out at higher temperatures. Defect concentration
clearly increased when the growth temperature was lowered
and is likely to be the dominant factor in crystalline to
amorphous growth [13,16,29].
4. Conclusions
Fig. 9. The formation of gallium vacancies is observed in a time-series
sequence. The images were prepared by cross-sectioning the simulation
computational cell. The film was grown at T=T m ¼ 0:41 with an As:Ga
flux ratio of 1.
strength of the bond is proportional to the width of the
connecting bar.) In frame (b) an additional arsenic atom
(atom 2) trapped a gallium vacancy and the system of two
atoms and a vacancy was capped by atoms 3 and 4. In
frames (c) and (d) of Fig. 9, additional atoms (5–10) were
added that trapped another gallium vacancy. The double
vacancy induced some further disorder in the crosssectioned plane, see atoms 6 and 7 in Fig. 9(d). In
Fig. 9(e), atom 3 and one of the gallium vacancies switched
places, as indicated by the arrow in the previous panel (d).
This migration of vacancies toward the surface was
temperature dependent and becomes more common as
temperature increases. The final snapshot shows two
gallium vacancies. Atoms 6 and 7, which surrounded the
upper gallium vacancy, were displaced more than atoms 2
and 8, which were more embedded in the bulk.
It is clear from Fig. 9 that the formation of defects in
GaAs semiconductors takes thousands of picoseconds.
Essentially the formation is driven by an initially misplaced
atom, which is capped sufficiently rapidly that the defect
does not have time to move over the surface to a more
highly coordinated site during growth. The concentration
of defects would therefore be related to deposition rate and
temperature, as was observed to be the case for gallium
1. The Angelo, Mills, Grein et al. [18,19] parameterization
of the Stillinger–Weber (SW) [17] potential energy
function was utilized to simulate stoichiometric growth
from a vapor phase.
2. The potential predicted crystalline thin films for
temperatures above T=T m 0:35 and below the evaporation temperature. Increasingly amorphous growth
was observed for T=T m o0:34.
3. Incorporation of arsenic dimers was the dominant
mechanism observed. A secondary mechanism (and
much less frequent) of a single arsenic atom incorporation was also identified, but was likely overpredicted by
the SW potential. Initial dimer orientation was observed
to play a significant role in this assembly mechanism.
4. Gallium vacancy defects have the lowest formation
energies and dominate during low-temperature stoichiometric vapor deposition. The formation of gallium
vacancies during film growth was accomplished by
capping vacancies before surface diffusion filled them.
5. The SW potential has limitations with regard to the
modeling of elemental binding energies [20]. These
properties, among others, should be improved to
increase the confidence in the veracity of the observed
arsenic molecular bonding and desorption mechanisms.
Acknowledgements
We gratefully acknowledge the support of DARPA/
ONR under contract No. N00014-03-C-0288, Carey
Schwartz and Julie Christodoulou program managers.
Appendix A. Crystalline order parameter
The assessment of crystalline order of grown samples can
be quantified by the definition of a crystalline order
parameter, S. The calculation of S used in our analysis
was motivated by the long-range order parameter for
binary solids [30]. It has been generalized to quantify the
order of a system that allows for atom displacements as
well as substitutions. The general form of the crystalline
order parameter is then S ¼ N % =N, where N is the total
number of deposited atoms and N % is the number of
deposited atoms that have a crystalline environment.
The atoms in the as-grown region, i ¼ 1 . . . N, are
determined to be on a lattice site by comparing the vectors
of j neighboring atoms to that of a perfect crystal. For a
tetrahedral zinc blende lattice, a minimum of four
vectors must be considered ðj ¼ 4Þ. For each one of the
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neighboring atoms, j, the vector from the central atom, i, is
subtracted from the perfect lattice vector and the magnitude of the resultant, Drij , is computed. A normal
distribution function is then computed for each resultant
magnitude as cij ¼ expðaDr2j Þ, where the parameter a is
set equal to 45 to strongly penalize deviation from the
crystalline vectors. Therefore, the on-site count can be
determined by
N% ¼
N X
4
X
cij =4.
i¼1 j¼1
The region used to compute S must be determined to best
approximate the crystallinity of the as-grown thin film by
avoiding the counting of relaxed or reconstructed surface
atoms. This is done by counting only the N deposited
atoms four monolayers ð5:65 ÅÞ below the top plane. (The
top plane is defined as the highest plane that has a planer
atomic density of at least 25%.) Hence, neither the
substrate nor surface atoms are counted in the determination of S. For the simulations discussed in this paper, there
are 64 atoms per layer and a total of 600–800 atoms were
deposited. Excluding the initial substrate and surface
atoms, all calculations were performed on grown thin films
with around 500 atoms in a bulk-like environment.
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