Computational Materials Science 39 (2007) 541–551
www.elsevier.com/locate/commatsci
A potential for simulating the atomic assembly of cubic AB compounds
X.W. Zhou
b
a,*
, H.N.G. Wadley
b
a
Department of Materials Mechanics, 7011 East Avenue, Sandia National Laboratories, Livermore, California 94551-0969, United States
Department of Materials Science and Engineering, 116 Engineer’s Way, University of Virginia, Charlottesville, VA 22904-4745, United States
Received 23 June 2006; received in revised form 1 August 2006; accepted 2 August 2006
Abstract
Stillinger–Weber interatomic potentials can be used to study the crystal growth of AB compounds with the zinc-blende (B3) structure,
but have been unable to be used for other cubic structured compounds. Here we extend a recently modified Stillinger–Weber potential for
cubic elements so that it is suited for studying the growth of cubic compounds with the B1 and B2 structures. We also parameterize the
potential for the Mg–O system. The potential is shown to accurately model the lattice constants and the cohesive energies of the fcc Mg,
fcc O, and the B1 MgO structures. It also correctly ensures that the equilibrium phase of each of these materials possesses the lowest
cohesive energy with respect to all their other phases. The potential correctly predicts crystalline growth during molecular dynamics simulations of the vapor deposition of fcc Mg and B1 MgO thin films. The results also reveal the formation of various defects in the films,
including islands and twins in Mg, interstitials and disrupted regions on the MgO growth surface, and local un-oxidized regions at the
MgO/Mg interface during the oxidation of Mg. The simulation approach enables the study of atomic assembly processes controlling the
formation of these defects.
2006 Elsevier B.V. All rights reserved.
Keywords: Molecular dynamics simulations; Interatomic potential; Vapor deposition; Thin films
1. Introduction
Large tunneling magnetoresistance ratios at ambient
temperature [1,2] have been measured for some magnetic
tunnel junction (MTJ) multilayers that utilize thin dielectric tunnel barrier layers to separate pairs of thin ferromagnetic metal layers [3,4]. Various metal oxides have been
proposed for the tunnel barrier layer [3–6]. Aluminum
oxide has been widely used [3,4], in part because thin, continuous, amorphous aluminum oxide films can be relatively
easily synthesized by first depositing a thin Al layer on a
ferromagnetic metal substrate and then oxidizing the surface. The ambient temperature tunneling magnetoresistance ratio achieved using these amorphous barriers has
been reported to be as high as 70% [1]. Recent experiments
indicate that crystalline, B1 structure MgO barriers can be
*
Corresponding author. Tel.: +1 925 294 2851; fax: +1 925 294 3410.
E-mail address: [email protected] (X.W. Zhou).
0927-0256/$ - see front matter 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.commatsci.2006.08.005
grown on ferromagnetic metals, resulting in tunneling magnetoresistance ratios as high as 230% [7].
The tunneling magnetoresistance is believed to be sensitive to the atomic scale structure of the tunneling barrier
at the barrier/metal interface [8]. This is affected by the
growth conditions used to fabricate the tunneling barriers
[9]. Atomistic simulations of growth using molecular
dynamics are beginning to allow visualization of the
atomic assembly mechanisms and the evolution of atomic
scale structures as atoms are added to a surface [9,10].
These simulations require an interatomic potential that
satisfactorily defines the interatomic forces between the
atoms in the system of interest. For systems such as aluminum oxide, ionic (Coulomb) interactions between the
metal and oxygen ions can become important. Modified
charge transfer, embedded atom method potentials have
been used to simultaneously address metallic bonding in
metal regions and the variable ionic bonding associated
with a continuous variation of ion charge across the
542
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
metal/metal oxide interfaces [11,12]. However, these
potentials are unsuitable for studying the crystalline
growth of MgO barrier layers because they are based on
the embedded atom method approach which lacks the
angular dependence needed to simulate the lowest cohesive
energy, (equilibrium) B1 structure of MgO crystals. Ideally, an integrated charge transfer, angular dependent
potential needs to be used for these material systems. As
a first step, we develop a suitable angular dependent
potential while ignoring the charge transfer effects.
Because the variable charge across interfaces cannot be
addressed, such a potential can be best utilized for uniform MgO compound.
Numerous angular dependent potentials have been proposed for the study of atomic systems with strong angular
dependence. They include Stillinger–Weber (SW) [13],
Tersoff [14–17], and bond-order potentials [18–22]. To correctly simulate the assembly of MgO crystals whose equilibrium phase has a B1 structure, it is necessary that the
interatomic potential correctly predict the lowest (most
negative) cohesive energy for the B1 crystal [23,24]. Tersoff
and other related bond order potentials are sufficiently flexible that they can be well fitted to the lattice constants and
cohesive energies of many phases including those that are
relatively unstable under ambient conditions [23,24]. However, this flexibility also introduces difficulties to the fitting,
and an extensive set of trials are usually required to identify
a set of parameters that correctly predict the lowest cohesive energy for the equilibrium phase and its structural
and mechanical properties [24].
Any potential describing interactions among an atom
ensemble can be generally written as a sum of two-body,
three-body, . . . , N-body contributions. Two-body (pair)
interactions are usually sufficient to describe the binding
of atoms with only radial (non-directional) bonding symmetry. However, crystals consisting of atoms with highly
directional bonds require at least three-body interactions
to be included. If properly chosen, the two-body and
three-body interactions can reasonably well describe the
bonding of many of these systems and other many-body
interactions can be ignored [13]. SW potentials link the
cohesive energy, Ec, to a sum of a two- (U) and a three(H) body interactions: Ec = U + H.
In SW potentials, the three-body term is set equal to
zero when the bond angle equals that encountered in diamond-cubic (dc) or zinc-blende (B3) crystal structures.
This allows the two-body term alone to fully describe
the cohesive energy, the lattice constant, and the bulk
modulus of equilibrium dc or B3 phase. Other phases
can have lower energy two-body (U) terms than the equilibrium dc or B3 structure. However, these phases are
always associated with different bond angles. The lowest
energy can always be retained for the dc or B3 phase as
long as the three-body H term is designed to increase sufficiently rapidly as the bond angle deviates from that of dc
or B3 structure. As a result, SW potentials can be easily
parameterized to ensure that no other phases have a lower
cohesive energy than the dc or B3 structure while simultaneously predicting accurate lattice constant and cohesive
energy for the dc or B3 structured crystal [13,25]. Like
most other potentials, the main drawback of the SW
potential is that it can only capture well the most stable
solid crystal of a material with little or no ability to predict
the cohesive energies of metastable solid phases and stable
molecular gases (e.g., O2 dimer). This precludes studies of
some problems such as O2 molecule formation and evaporation from any O-rich surfaces. When the simulated
structure is not far away from the most stable solid crystalline phase, the SW potential can be successfully applied
[25–28].
Unfortunately, the usual form of a SW potential can
only be applied for dc elements or B3 compounds. One
approach to modify the SW potential for wurtzite GaN
compound has been proposed [29]. A similar modification to the SW potential has recently been proposed for
elemental cubic crystals including those with either the
dc, simple-cubic (sc), body-centered-cubic (bcc), or facecentered-cubic (fcc) structure [30]. Here we investigate the
further extension of this potential to binary compound
crystals with any of the cubic (B1, B2, or B3) structures.
To illustrate its application, we fit the potential for the
Mg–O system to the exact ab initio values of the cohesive
energies and lattice constants of the equilibrium Mg, O,
and MgO solid phases. We then use this Mg–O potential
to simulate the growth of Mg and MgO films from atomic
Mg and Mg + O vapor fluxes.
2. Structural parameters of compound cubic crystals
The B1, B2, and B3 unit cells of AB compound crystals
and representative crystal lattices are shown in Fig. 1.
These structures can be distinguished by various structural
parameters. For example, the ratio of the radius of the ith
nearest neighbor shell, ri, to that of the nearest neighbor
shell, r1, ni = ri/r1, depends on the structure. An atom of
type l can have an ith neighbor shell composed of either
identical atoms (ll neighbors) or dissimilar atoms (lm
neighbors, l 5 m). The number of neighbors in the ith
shell, Zi, and a volume conversion factor F relating the
atomic volume V to the nearest neighbor distance r1
through V ¼ F r31 are also dependent on the structure.
Values of these parameters up to the first four nearest
neighbor shells are listed in Table 1 for the three binary
AB compound crystal structures. For all ordered AB compound structures, the nearest neighbors are of a dissimilar
atomic type while the next nearest neighbors are of an identical type, Table 1.
We denote the bond angle formed between atom i and
its two nearest neighbors, j and k, by hjik as shown in
Fig. 2. The angle, hjik, can be conveniently represented by
its cosine value. All possible cosine values of the nearest
neighbor bond angle, hjik, their degeneracies, Njik, and the
species of atoms i, j, k, are listed in Table 2 for the three
compound cubic crystal structures.
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
(a)
1.0
B1 structure
Representative film crystals
Unit cells
0.9
y [010]
0.8
y [111]
k
gνμκ(cosθ)
0.7
A
B
x [100]
z [110]
(b)
543
0.6
θijk
j
B3
0.5
B1
0.4
0.3
z [001]
x [112]
i
0.2
B2
0.1
B2 structure
0.0
-1.0 -0.8 -0.6 -0.4 -0.2
y [110]
y [010]
0.0
0.2
0.4
0.6
0.8
1.0
cosθ
A
Fig. 2. gmlj functions for mlj = BAB or ABA in B1, B2, and B3 crystals.
B
x [100]
x [001]
z [110]
Table 2
Cosine values of the bond angle, hjik, their degeneracy, Njik, and the species
of i, j, and k for B1, B2, and B3 crystal structures
z [001]
Compound
structures
(c)
B3 structure
y [100]
y [010]
cos hjik
Number of bond
angles Njik
Species jik
(l, m = A, B, l 5 m)
B1
1
0
3
12
mlm
mlm
B2
1
1/3
1/3
4
12
12
mlm
mlm
mlm
B3
1/3
6
mlm
A
B
x [100]
z [001]
x [011]
z [011]
Fig. 1. (a) B1 (such as NaCl and MgO); (b) B2 (such as CsCl); and (c) B3
(such as GaAs) structures.
energy between atoms i and j separated by a distance rij,
ulm(rij) is a positive pair function used in the angular interaction, and gmlj(cos hjik) is a function of the cosine of the
bond angle cos hjik. All the functions /lm(rij), ulm(r), and
gmlj(cos hjik) depend on the types of interacting atoms, with
the subscripts l, m, and j corresponding to the species of
atoms i, j, and k, respectively.
We seek an interatomic potential for ordered binary
compound cubic crystals with an equilibrium B1, B2, or
B3 crystal structure. We assume that the elemental potentials are already known or can be determined using the
procedures described previously [30]. Since atomic interactions decay quickly as the distance between atoms
increases, only closely separated atoms need to be considered during calculations. This can be efficiently achieved
by designing a potential that smoothly decays to zero at
3. Modified Stillinger–Weber potential
The cohesive energy of an ensemble of atoms predicted
by a SW potential [30] can be expressed in the form
Ec ¼
iN
iN X
iN
N X
N X
1 X
1 X
/lm ðrij Þ þ
ulm ðrij Þ
2N i¼1 j¼i1
2N i¼1 j¼i1 k¼i
1
k6¼j
ulj ðrik Þ gmlj ðcos hjik Þ
ð1Þ
where Ec is the cohesive energy (eV/atom) for the computational system, N denotes the total number of atoms in
the system, i1, i2, . . . , iN are all the neighbors of atom i
(including image atoms), /lm(rij) is the pair interaction
Table 1
Structural parameters for B1, B2, and B3 crystal structures
Compound structures
Relative atom spacing
ni = ri/r1
i=1
B1
1
B2
1
B3
1
Neighbor species
(l, m = A, B, l 5 m)
i=2
pffiffiffi
q2ffiffi
i=3
pffiffiffi
q3ffiffi
2 ffiffiffiffi
q
qffiffi
qffiffiffiffi
qffiffiffiffi
4
3
8
3
8
3
11
3
i=4
i=1
i=2
i=3
Coordination Zi
i=4
i=1
i=2
F-factor
i=3
i=4
lm
ll
lm
ll
6
12
8
6
11
3
lm
ll
ll
lm
8
6
12
24
p4ffiffiffiffi
27
1
16
3
lm
ll
lm
ll
4
12
12
6
p8ffiffiffiffi
27
544
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
a designated cutoff distance between atoms. The pairwise
(two-body) and the angular (three-body) terms may be
assigned different cutoff distances. Because the structural
stability is mostly determined by the angular dependent
short range interactions, and the calculation cost associated with angular dependent terms in a potential is significantly increased when the number of neighbors increases,
we focus attention upon a short-range angular term
between the dissimilar species in an AB compound that
involves only the nearest neighbor interactions. For the
pairwise term between dissimilar species, the cutoff distance is allowed to go beyond the second nearest neighbor
distance.
The interaction ranges between identical species are fully
defined by the elemental potentials. The cutoff distances of
elemental potentials have been chosen to include only the
nearest neighbors in the dc, sc, and fcc structures but have
been extended to include the second nearest neighbors for
the bcc structure [30]. The distance between neighbors
scales with the lattice constant. When the elemental structure is bcc, or when the lattice constant of the elements is
significantly larger than that of the compound, the interaction between identical species AA and BB may occur
beyond the nearest neighbors in the equilibrium AB
compound structure.
To account for the considerations described in the
above, the interaction range (and hence the neighbor list
i1, i2, . . . , iN) used in Eq. (1) is viewed to extend to the nth
nearest neighbor shell of the AB compound, where n could
be larger than unity and is dependent on the pre-determined elemental potentials.
The radially dependent functions between a pair of
atoms with species l and m, /lm(r) and ulm(r), are given by
r 4
rlm
rlm
lm
/lm ðrÞ ¼ Alm S lm
exp
Alm exp
r
r rc;lm
r rc;lm
ð2Þ
and
clm
ulm ðrÞ ¼ C lm exp
r ruc;lm
ð3Þ
where Alm, Slm, Clm, rlm, clm, rc,lm and ruc,lm are seven fitting
parameters for each of the three types of atom pairs
lm = AA, BB, and AB (or BA). Efficient simulations can
be achieved by truncating the potentials at selected cutoff
distances and the parameters rc,lm and ruc,lm represent these
cutoff distances for the two functions respectively. Note
that because the elemental potentials are assumed to be
known, only seven free parameters between dissimilar species, AAB, SAB, CAB, rAB, cAB, rc,AB and ruc,AB, need to be
determined.
The angularly dependent function, gmlj(h), can be
expressed using quadratic splines
gmlj ðcos hÞ ¼ go;n;mlj þ vn;mlj ðcos h cos hn;mlj Þ2 ;
xmin;n;mlj 6 cos h < xmax;n;mlj ;
n ¼ 1; 2; . . . ; M mlj
ð4Þ
where go,n,mlj, vn,mlj, cos hn,mlj, xmin,n,mlj, xmax,n,mlj and
Mmlj are six sets of parameters corresponding to six types
of three-bodies, mlj = AAA, BBB, BAB, ABA, AAB
(equivalent to BAA), and ABB (equivalent to BBA). The
parameters for mlj = AAA or BBB are known. There are
no AAB/BAA or ABB/BBA three-bodies in the perfect
compounds, and their angular functions do not therefore
affect the bulk properties of these compounds. As a result,
these angular interactions can be ignored and it is possible
to then simply set Mmlj = 1, go,1,mlj = 0 and v1,mlj = 0 for
mlj = AAB/BAA or ABB/BBA. The remaining parameters that need to be defined for the binary system are then
go,n,mlj, vn,mlj, cos hn,mlj, xmin,n,mlj, xmax,n,mlj and Mmlj for
mlj = BAB and ABA. Further reductions are achieved
from symmetry since the parameters for mlj = BAB are
identical to those for mlj = ABA.
The first and the second terms on the right-hand side of
Eq. (1) represent the two-body (U) and the three-body (H)
contributions respectively. The modified SW potential
takes the normalized form so that the first term is set equal
to the cohesive energy of the equilibrium phase regardless
of its (B3, B2 or B1) structure. This normalized potential
can predict accurate cohesive energy for the equilibrium
phase if the second term vanishes when the bond angles
approach those of the equilibrium phase. To ensure the
lowest energy for the equilibrium phase, the second term
is designed to increase rapidly as the bond angles deviate
from those of the equilibrium phase.
Eq. (1) shows that the second angularly dependent term
in a compound structure is composed of contributions
from identical and dissimilar three-body interactions. Since
the nearest neighbors in the compound are dissimilar, the
angular contribution from the identical three-bodies are
zero unless the cutoff distance of the elemental potentials
exceeds the second nearest neighbor distance in the compound. This only occurs when the lattice constant of the
elements is significantly larger than that of the compound.
It should be noted that the pre-determined elemental
potentials are also normalized. This means that the identical three-body angular contribution is zero even for large
elemental cutoff distance as long as the two sublattices of
the equilibrium AB compound structure coincide with the
equilibrium lattices of the two elements occupying the
two respective sublattices.
When the predetermined identical three-body contribution to the angular term is zero, a normalized compound
potential can be easily obtained by setting the dissimilar
three-body contribution to be zero. This can be achieved
when the quadratic spline function, Eq. (4), is used for
the dissimilar three-body interactions. The parameters
needed for the quadratic spline functions for the three
(B1, B2, and B3) crystal structures are shown in Table 3.
The gmlj functions defined in Table 3 are plotted in
Fig. 2 for the three crystal structures. It can be seen that
these functions approach zero when the bond angles
become equal to those of the corresponding crystal structures, Table 2.
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
Table 3
Angular function parameters for the three-body mlj = BAB or ABA in
B1, B2, and B3 crystal structures
Structures
Mmlj n xmin,n,mlj
B1
3
1 1.00000 0.75000 1.00000
1.0
2 0.75000 0.25000 0.50000 1.0
3 0.25000
1.00000
0.00000
1.0
0.00000
0.12500
0.00000
B2
5
1
2
3
4
5
1.00000 0.83333 1.00000
1.0
0.83333 0.50000 0.66667 1.0
0.50000 0.16667 0.33333
1.0
0.16667
0.16667
0.00000 1.0
0.16667
1.00000
0.33333
1.0
0.00000
0.05556
0.00000
0.05556
0.00000
B3
1
1 1.00000
xmax,n,mlj
cos hn,mlj
vn,mlj go,n,mlj
1.00000 0.33333
1.0
0.00000
4. Potential parameterization for Mg–O system
At room temperature, MgO and Mg have B1 and hexagonal-close-packed (hcp) crystal structures [31] and O exists
as a diatomic gas. Local density function approximation
(LDA) calculations using the Vienna Ab-initio Simulation
Package (VASP) [32] were first performed to determine the
cohesive energies of diamond-cubic (dc), simple-cubic (sc),
body-centered-cubic (bcc), face-centered-cubic (fcc) and
hcp structures for elemental oxygen and magnesium and
B1, B2 and B3 structures for the stoichiometric compound
MgO. We found that the fcc O had the lowest cohesive
energy of all the solid phases studied. Because fcc and
hcp structures have identical nearest neighbor bonding,
the fcc and hcp Mg structures were found to have similar
cohesive energies that were both lower than those of any
other phases. The B1 structure of MgO was found to have
the lowest cohesive energy. The predictions that the hcp/fcc
phase of Mg and the B1 phase of MgO have the lowest
cohesive energies are in good agreement with the observations of these phases under ambient conditions [31].
According to this analysis, we selected fcc O, fcc Mg, and
545
B1 MgO as the model equilibrium phase structures for subsequent potential parameterization.
The elemental potentials for O and Mg were determined
using the approaches described in the elemental analysis
[30]. Once the elemental potentials are determined, the only
unknown functions in Eq. (1) are /AB and uAB. Some analytical equations relating these two functions to material
properties have been derived and are given as Eqs. (A.1)–
(A.3) in Appendix A and as Eq. (B.1) in Appendix B.
Eqs. (A.1)–(A.3) enable the parameterization of the /AB
term so that the potential predicts exactly ‘‘target values’’
for the lattice constant, a, the cohesive energy, Ec, and
the bulk modulus, B, of the equilibrium compound crystal
structure. Eq. (B.1) enables the parameterization of the uAB
term in such a way that the potential well predicts the shear
moduli, C11 and C44, of the equilibrium compound phase.
However, these equations do not enable the fitting of other
metastable phases and they cannot ensure that the equilibrium phase has the lowest cohesive energy.
To correctly simulate the crystalline growth of the equilibrium crystal phase, it is essential that the potential predicts the lowest cohesive energy for the equilibrium phase
and yields physical cohesive energy vs. atomic volume curves
for all the phases. This was achieved by adjusting the target
values of the bulk modulus (B) and elastic moduli (C11 and
C44) during parameterization using Eqs. (A.1)–(A.3) and
(B.1). Such a fitting procedure does not yield a highly accurate prediction of these elastic constants, but results in a
potential that adequately describes these and other properties. A complete set of potential parameters determined
using this procedure are listed in Table 4. The cohesive
energy, lattice constant, and elastic moduli predicted by
the potentials for the equilibrium crystal phases are compared with the results of the LDA calculations in Table 5.
Table 5 shows that the potentials predict exactly the
target values of cohesive energy and lattice constant for
Table 4
Parameters for the Stillinger–Weber potentials for Mg–O system
Pair lm
Alm (eV)
Slm
Clm (eV1/2)
rlm (Å)
clm (Å)
rc,lm (Å)
ruc,lm (Å)
MgMg
OO
MgO
2.936977
2.264679
2.842769
1.836933
1.030683
123.5643000
0.000000
0.000000
1.583780
2.322254
2.140179
0.418141
0.000000
0.000000
0.350756
4.511678
3.530965
2.979658
0.000000
0.000000
2.979658
Table 5
Cohesive energy, Ec, lattice constant, a, bulk modulus, B, elastic constants, C11, C12, and C44, for Mg (fcc), O (fcc), and MgO (B1)
Elastic constants (eV/Å3)
Materials
Data type
Cohesive energy
Ec (eV/atom)
Lattice constant
a (Å)
B
C11
C12
C44
fcc Mg
Potential
LDA
1.472
1.472
4.513
4.513
0.426
0.213
0.639
0.292
0.320
0.174
0.320
0.154
fcc O
Potential
LDA
0.761
0.761
3.532
3.532
0.570
0.380
0.855
0.581
0.428
0.280
0.428
0.194
B1 MgO
Potential
LDA
4.997
4.997
4.234
4.234
0.930
0.930
2.034
1.666
0.378
0.562
1.316
0.916
eV/Å3 = 160 GPa.
546
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
Table 6
Cohesive energy Ec (eV/atom) of various phases for Mg, O, and MgO
Material
Mg
O
MgO
(a)
(b)
Mg, T = 300 K
y [111]
Structure
dc/B3
sc/B1
bcc/B2
fcc
0.491
0.254
2.853
0.736
0.380
4.997
1.470
0.734
4.458
1.472
0.761
–
MgO, T = 650 K
y [111]
Mg
disrupted region
O Mg
interstitial
Table 7
Lattice constant a (Å) of various phases for Mg, O, and MgO
Material
Structure
dc/B3
sc/B1
bcc/B2
fcc
Mg
O
MgO
7.378
5.768
4.695
3.190
2.498
4.233
3.501
2.752
2.677
4.512
3.532
–
the three equilibrium crystal phases. The prediction of the
elastic constants for the B1 MgO structure is also very
good. However, the elastic constants for the closely packed
fcc Mg and fcc O cannot be well captured by the pair function format used in the current version of the SW potential.
This is consistent with earlier findings [30].
Calculations were carried out to determine the cohesive
energies for other Mg, O and MgO phases. The results for
the cohesive energy and the lattice constant are summarized in Tables 6 and 7 respectively.
Table 6 confirms that the model equilibrium (fcc) Mg,
(fcc) O, and (B1) MgO phases all have the lowest cohesive
energies compared with their other structures.
5. Molecular dynamics simulations
Both Tersoff and bond order potentials have superior
formats to SW potentials that can be justified from firstprinciples [22]. However, few analytical bond order potentials are currently available and we find that most published
Tersoff potentials predict anomalous amorphous structures
when used for molecular dynamics simulations of crystal
growth [23]. To correctly capture the crystalline growth, a
potential must predict the lowest energy for the equilibrium
phase against any of the other configurations, reproduce
well the defect energies, satisfy the mechanical stability conditions, and have reasonably long cutoff distance to sample
the atomic interactions on a surface. We find that most published Tersoff potentials fail at least one of these requirements. In general, molecular dynamics simulation of
vapor deposition is a sensitive test of the performance of a
potential. They are therefore used to test the modified SW
potential for AB compounds.
5.1. Thin film deposition
The Mg–O potential parameterized above was tested by
using it to simulate the growth of Mg and MgO films from
the vapor and the oxidation of an Mg film to create an
MgO layer on Mg metal. A molecular dynamics approach
twin
z [110]
5Å
x [112]
z [110]
5 Å x [112]
Fig. 3. Atomic configurations for the films deposited using an adatom
incident energy of 1.0 eV, an adatom incident direction normal to the
surface, and a growth rate of 0.5 nm/ns. (a) Mg, T = 300 K; and (b) MgO,
T = 650 K. The substrate regions prior to deposition are shaded.
[9,10] was first used to simulate the growth of the fcc Mg
and B1 MgO crystals on their close-packed (1 1 1) crystal
surfaces. The simulations utilized an adatom energy of
1.0 eV, an adatom incident direction normal to the surface,
and a growth rate of 0.5 nm/ns. The Mg film was grown at
a temperature of 300 K while the MgO film was grown at a
temperature of 650 K. An equiatomic Mg–O vapor flux
was used for the MgO growth.
The atomic configurations of the films obtained after
2000 ps of deposition are displayed in Fig. 3(a) and (b),
where the shaded region represents the substrate prior to
deposition. It can be seen from Fig. 3(a) that the simulation
of Mg deposition on an fcc Mg substrate correctly predicted
the epitaxial growth of an fcc crystalline Mg layer. An island
of several atomic layer thick is seen to have developed at the
relatively low growth temperature of 300 K and the extremely high simulated deposition rate. The formation of such
islands is commonly seen in simulations of the accelerated
growth of fcc materials in the [1 1 1] direction [10,33]. This
arises because adatoms condensed on the surface of an island
have insufficient time to overcome the Erlich–Schwobel barriers and migrate to the lower energy ledge sites responsible
for gradual expansion of the island and reduction in its
height. Fig. 3(a) also shows a thin twinned region between
the initial substrate and the deposited layer. This arises
because fcc Mg is stacked with ABCABC. . . (1 1 1) planes
in the [1 1 1] (growth) direction. On a given A plane, adatoms
falling on twinned (C) sites are likely to be retained at these
sites since their binding energy is almost identical to that of
the lattice (B) sites. As a result, twins are nucleated and merge
to form structures like that seen in the simulations [34].
Fig. 3(b) shows that MgO deposition resulted in the epitaxial growth of a B1 crystalline MgO film. This is consistent with the low energy electron diffraction (LEED)
experiments, which indicate that a fully oxidized MgO
layer has good registry with an Mg substrate [35].
Fig. 3(b) also reveals regions of defective structure. These
defects include an interstitial near the original substrate
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
surface and a disrupted region on the surface. Experimentally, the MgO film in the MTJ multilayers with the highest
magnetoresistance ratio was grown using a low pressure
sputtering process [7]. These low pressure sputtering
processes result in a high adatom energy which facilitates
flattening of the surface [9], improved crystallinity [26],
and reduced defect concentrations [36].
5.2. Mg surface oxidation
The molecular dynamics approach can also be used to
simulate the oxidation of a (1 1 1) Mg surface. The initial
Mg crystal was generated using the potential predicted
lattice constant. Its oxidation was simulated by continuously injecting oxygen atoms perpendicularly to the top
(1 1 1) Mg surface using an oxygen flux of 2.24 ·
104 atom ps1 Å2, a substrate temperature of 800 K,
and an oxygen adatom energy of 1.0 eV. The atomic scale
configuration of the initial Mg crystal is shown in Fig. 4(a),
and the structure of the surface after 1200 ps of oxidation is
shown in Fig. 4(b). To more clearly examine the Mg–O
bonds in the MgO oxide layer, the bars used in Fig. 4(a)
to connect the Mg atoms are dropped in Fig. 4(b). It can
be seen that a crystalline B1 MgO film formed on the
(1 1 1) Mg surface. No misfit dislocations were observed
in the relatively small system simulated. This is consistent
with the relatively small lattice mismatch between Mg
and MgO (Table 5). Fig. 4 indicates that the MgO oxide
formed under the simulated conditions was non-uniform
as the bottom MgO/Mg interface is not flat. The transformation towards a flat MgO/Mg interface requires additional oxygen diffusion to the incompletely oxidized
interface regions. In experiments, MgO tunnel barrier layers are normally synthesized by co-depositing Mg and O
rather than by the oxidation of a pre-deposited Mg layer.
(a)
before oxidation
y [111]
(b)
1200 ps after oxidation
The modified MgO SW potential may provide a useful
means to explore improved ways to grow more uniform
MgO tunnel barrier layers.
6. Conclusions
A modified SW potential is proposed for ordered AB
compounds with cubic B1, B2, and B3 structures.
This modified SW potential is well suited for molecular
dynamics simulations of vapor deposition and has been
parameterized for the Mg–O system. The parameterized
potential reproduces exactly the cohesive energy and
lattice constant for the lowest energy crystal phases: fcc
Mg, fcc O, and B1 MgO. For the B1 structure of MgO,
the modified potentials also predict well the elastic constants obtained from ab initio calculations. As previously
identified, the elastic constants for the closely packed fcc
Mg and fcc O crystals are not satisfactorily predicted [30].
The potentials do predict crystalline growth of the model
equilibrium phases and the formation of the B1 MgO oxide
phase during oxidation of a (1 1 1) Mg surface. Preliminary
simulations indicated the formation of islands and twins
during Mg growth, the formation of interstitials and
disrupted regions during MgO growth, and the formation
of a non-uniform MgO/Mg interface due to locally incomplete oxidation of Mg. The MgO potential and the simulation approach will enable the role of synthesis conditions
upon the formation of these defects to be explored.
Acknowledgements
We are grateful to the Defense Advanced Research
Projects Agency and Office of Naval Research (C. Schwartz
and J. Christodoulou, Program managers) for support of
this work through Grant N00014-03-C-0288.
Appendix A. Lattice constant, cohesive energy, and bulk
modulus
y [111]
Analytical relations defining lattice constant, cohesive
energy (relative energy per atom in a solid with that of
an isolated atom), and bulk modulus are derived for the
potential. From the considerations discussed above, the
angular term for the equilibrium compound phase remains
at zero during hydrostatic straining (i.e., bond angles
remain constant). The cohesive energy defined by Eq. (1)
is then radial dependent only. If the pair potentials include
an interaction up to the nth nearest neighbor shell, the
cohesive energy can be rewritten in a normalized form as
O Mg
Mg
z [110]
547
5Å
x [112]
z [110]
5Å
x [112]
Fig. 4. Atomic configurations of the Mg surface before and after
oxidation using the conditions: a substrate temperature of 800 K, an
oxygen adatom incident energy of 1.0 eV, an oxygen adatom incident direction normal to the surface, and an oxygen flux of
2.24 · 104 atom ps1 Å2. (a) Before oxidation and (b) 1200 ps after
oxidation.
Ec ¼
¼
n
n
1X
1X
Z j /AlA;j ðrj Þ þ
Z j /BmB;j ðrj Þ
4 j¼1
4 j¼1
n
n
1 X
1 X
Z j /AlA;j ðrj Þ þ
Z j /BmB;j ðrj Þ
4 j¼1
4 j¼1
lA;j ¼A
mB;j ¼B
548
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
þ
n
n
1 X
1 X
Z j /AlA;j ðrj Þ þ
Z j /BmB;j ðrj Þ
4 j¼1
4 j¼1
lA;j 6¼A
mB;j 6¼B
n
n
1 X
1 X
Z j /AA ðrj Þ þ
Z j /BB ðrj Þ
¼
4 j¼1
4 j¼1
lA;j ¼A
AAB
þ
2
n
X
mB;j ¼B
"
Z j S AB
j¼1
lA;j 6¼A
rAB
r1
!
#
4 4
rAB
1
r1
1 exp
r
nj
nj c;AB
r1
ðA:1Þ
In Eq. (A.1), j loops from the nearest neighbor (j = 1) to
the nth nearest neighbor (j = n), and lm,j represents the species in the jth neighbor shell from an atom of species m. Two
subscripts (m and j) are needed in lm,j because the species in
the jth nearest neighbor of an atom type m depends on both
m and j, Table 1. Note that /AA and /BB are known. Eq.
(A.1) therefore defines a relationship between a material
property (the lattice cohesive energy) and the free parameter of the potential, AAB, SAB, rAB, and rc,AB.
At the equilibrium condition, the radial derivative of the
cohesive energy equals zero: E0c ¼ dEc =dr1 ¼ 0. From Eq.
(A.1), we can therefore write
n
1 X
E0c ¼ 0 ¼
Z j nj /0AA ðrj Þ
4 j¼1
lA;j ¼A
n
X
þ
1
4
2
j¼1
lB;j ¼B
Z j nj /0BB ðrj Þ þ
4S AB rrAB
1
6
4
exp
n
AAB rAB X
Z j nj
2
2 r1
j¼1
l 6¼A
3
3 2
A;j
4
rc;AB
rAB
nj r 1
S AB r1
nj þ n5j
7
5
2
r
n5j nj c;AB
r1
!
rAB
r1
r
nj c;AB
r1
ðA:2Þ
Eq. (A.2) gives a second relationship between a material
property (the equilibrium nearest atomic spacing r1 that corresponds to lattice constant a) and potential free parameters.
The elastic bulk modulus B ¼ V d2 Ec =dV 2 ¼ r21 00
Ec =ð9V Þ is related to the second derivative of the cohesive
energy with respect to r1, E00c (V is the atomic volume). Differentiation of Eq. (A.2) yields
E00c ¼
Eq. (A.3) provides a third relationship between material
property (the bulk modulus) and the free parameters of
the potential. Eqs. (A.1)–(A.3) involve three equations with
four unknown parameters AAB, SAB, rAB and rc,AB. Once a
value for rc,AB is selected, the values of AAB, SAB, and rAB
can be solved. If the solutions exist, they then allow an
exact prediction of target values for the lattice constant,
cohesive energy, and bulk modulus of an equilibrium
compound crystal structure.
Appendix B. Shear moduli
In addition to bulk modulus B, the shear moduli C11
and C44 can be selected as the other two independent elastic
constants of a cubic crystal. Analytical equations relating
the potential function uAB to C11 and C44 are derived.
Using an approach described previously[30], the shear
moduli Caa (a = 1, 2, . . . , 6) for a equilibrium compound
crystal structure can be written as
"
2
iN
N X
1 X
orij
o2 rij
00
C aa ¼
/lm ðrij Þ
þ /0lm ðrij Þ 2
2N V i¼1 j¼i1
oea
oea
2 #
iN
X
o
cos
h
jik
þ
ulm ðrij Þulj ðrik Þg00mlj ðcos hjik Þ
oea
k¼i
1
k6¼j
"
2
iN
N X
1 X
orij
o2 rij
00
/AA ðrij Þ
þ /0AA ðrij Þ 2
¼
2N V i¼1 j¼i1
oea
oea
l¼A m¼A
þ2
iN
X
k¼i1
k6¼j
j¼A
2 #
o
cos
h
jik
u2AA ðrij Þ
oea
"
2
iN
N X
1 X
orij
00
þ
/BB ðrij Þ
2N V i¼1 j¼i1
oea
l¼B m¼B
þ
o2 rij
/0BB ðrij Þ 2
oea
þ2
iN
X
k¼i1
k6¼j
j¼B
9V B
r21
rAB
n
n
n
1 X
1 X
AAB rAB X
r1
2
¼
Z j n2j /00AA ðrj Þ þ
Z j n2j /00BB ðrj Þ þ
Z
n
exp
j
rc;AB
j
4 j¼1
4 j¼1
2 r31
n
j
r1
j¼1
lA;j ¼A
2
lB;j ¼B
u2BB ðrij Þ
o cos hjik
oea
2 #
!
lA;j 6¼A
4 3
4
5 3
r
rc;AB
4rc;AB
rc;AB
6 rAB
2 rAB
rAB
rAB
2n6j nj c;AB
n
þ
2S
n
5n
þ
S
n
n
þ
20S
n
AB
AB
AB
j
j
j
j
j
j
r1
r1
r1
r1
r1
r1
r1
r1
6
7
4
5
4
r
n6j nj c;AB
r1
ðA:3Þ
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
"
2
iN
N X
1 X
orij
o2 rij
00
þ
/AB ðrij Þ
þ /0AB ðrij Þ 2
2N V i¼1 j¼i1
oea
oea
m6¼l
þ2
iN
X
k¼i1
k6¼j
j6¼l
¼
u2AB ðrij Þ
o cos hjik
oea
where
2
iX
N
N ðmÞ
1 X
orij
;
F m;AA ðaÞ ¼
2N r21 i¼1 j¼i ðmÞ oea
2 #
1
l¼A
2
iX
n 6
N
N ðmÞ
X
1 X
orij
6 00
6/AA ðrm Þ
2N V m¼1 4
oea
i¼1 j¼i ðmÞ
1
l¼B
1
l¼A
m¼A
F m;AB ðaÞ ¼
iX
N
N ðmÞ
X
o2 rij
oe2a
i¼1 j¼i ðmÞ
þ 2u2AA ðrm Þ
3
2
o cos hjik 7
7
7
oea
5
ðmÞ
þ
Gm;AB ðaÞ ¼
þ 2u2BB ðrm Þ
2
N
X
3
2
o cos hjik 7
7
7
oea
5
ðmÞ
iX
N ðmÞ
i¼1 j¼i1 ðmÞ k¼i1
l¼B m¼B
j¼B
k6¼j
2
iX
n 6
N
N ðmÞ
X
1 X
orij
6 00
þ
6/AB ðrm Þ
2N V m¼1 4
oea
i¼1 j¼i ðmÞ
i¼1 j¼i1 ðmÞ
m6¼l
þ 2u2AB ðrm Þ
N
X
iX
N ðmÞ
2
o rij
oe2a
3
2
o cos hjik 7
7
7
oea
5
ðmÞ
iX
N ðmÞ
i¼1 j¼i1 ðmÞ k¼i1
m6¼l
j6¼l
k6¼j
¼
n
1 X
½r2 /00 ðrm ÞF m;AA ðaÞ þ r1 /0AA ðrm ÞGm;AA ðaÞ
V m¼1 1 AA
n
1 X
½r2 /00 ðrm ÞF m;BB ðaÞ
þ u2AA ðrm ÞH mm;AA ðaÞ þ
V m¼1 1 BB
þ r1 /0BB ðrm ÞGm;BB ðaÞ þ u2BB ðrm ÞH mm;BB ðaÞ
n
1 X
þ
½r2 /00 ðrm ÞF m;AB ðaÞ þ r1 /0AB ðrm ÞGm;AB ðaÞ
V m¼1 1 AB
þ u2AB ðrm ÞH mm;AB ðaÞ
2
iX
iX
N
N ðmÞ
N ðmÞ
1 X
o cos hjik
H mm;AA ðaÞ ¼
;
N i¼1 j¼i ðmÞ k¼i ðmÞ
oea
l¼A
1
1
m¼A
j¼A
k6¼j
2
iX
iX
N
N ðmÞ
N ðmÞ
1 X
o cos hjik
H mm;BB ðaÞ ¼
;
N i¼1 j¼i ðmÞ k¼i ðmÞ
oea
1
1
m¼B
j¼B
k6¼j
2
iX
iX
N
N ðmÞ
N ðmÞ
1 X
o cos hjik
H mm;AB ðaÞ ¼
N i¼1 j¼i ðmÞ k¼i ðmÞ
oea
1
þ /0AB ðrm Þ
iX
N
N ðmÞ
1 X
o2 rij
;
2N r1 i¼1 j¼i ðmÞ oe2a
l¼B
m6¼l
iX
N
N ðmÞ
X
m¼B
m6¼l
1
iX
N ðmÞ
iX
N
N ðmÞ
1 X
o2 rij
;
2N r1 i¼1 j¼i ðmÞ oe2a
1
iX
N
N ðmÞ
X
o2 rij
oe2a
i¼1 j¼i ðmÞ
m¼B
m¼A
1
m¼B
l¼B
1
l¼B
1
/0BB ðrm Þ
iX
N
N ðmÞ
1 X
o2 rij
;
2N r1 i¼1 j¼i ðmÞ oe2a
l¼A
Gm;BB ðaÞ ¼
2
iX
n 6
N
N ðmÞ
X
1 X
orij
6 00
þ
6/BB ðrm Þ
2N V m¼1 4
oea
i¼1 j¼i ðmÞ
l¼B
Gm;AA ðaÞ ¼
iX
N ðmÞ
i¼1 j¼i1 ðmÞ k¼i1
l¼A m¼A
j¼A
k6¼j
2
2
iX
N
N ðmÞ
1 X
orij
;
2N r21 i¼1 j¼i ðmÞ oea
1
m¼A
iX
N
N ðmÞ
X
m¼B
m6¼l
1
l¼A
m¼A
2
iX
N
N ðmÞ
1 X
orij
;
F m;BB ðaÞ ¼
2
2N r1 i¼1 j¼i ðmÞ oea
2
þ /0AA ðrm Þ
549
ðB:1Þ
1
1
m6¼l
j6¼l
k6¼j
ðB:2Þ
In Eqs. (B.1) and (B.2), i sums over atoms in the system,
j and k sum over i’s neighbors, m sums over neighbor
shells, and the notation i1(m), i2(m), . . . , iN(m) indicates a
sub-list of i’s neighbors that are in the mth nearest neighbor shell. Fm,lm, Gm,lm, and Hmm,lm are independent of the
lattice constant, and can be viewed as constants for each
crystal structure. Selected values of Fm,lm, Gm,lm, and
Hmm,lm needed for the C11 and C44 calculations are
shown in Tables B1–B3 for the three binary compound
crystals.
Because only the nearest angular interaction is considered for the dissimilar species AB, the only unknown in
Eq. (B.1) is uAB(r1). One way to optimize uAB(r1) is to
minimize the square deviation of the predicted C11 and
C44 from their target values. This yields
550
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
Table B1
Selected values of Fm,lm for B1, B2, and B3 crystals
Structures
m
Fm,AA(1)
Fm,AA(4)
Fm,BB(1)
Fm,BB(4)
Fm,AB(1)
Fm,AB(4)
B1
1
2
3
4
0.00000
1.00000
0.00000
2.00000
0.00000
0.50000
0.00000
0.00000
0.00000
1.00000
0.00000
2.00000
0.00000
0.50000
0.00000
0.00000
1.00000
0.00000
1.33333
0.00000
0.00000
0.00000
1.33333
0.00000
B2
1
2
3
4
0.00000
0.66667
1.33333
0.00000
0.00000
0.00000
0.66667
0.00000
0.00000
0.66667
1.33333
0.00000
0.00000
0.00000
0.66667
0.00000
0.44444
0.00000
0.00000
10.06061
0.44444
0.00000
0.00000
2.30303
B3
1
2
3
4
0.00000
1.33333
0.00000
2.66667
0.00000
0.66667
0.00000
0.00000
0.00000
1.33333
0.00000
2.66667
0.00000
0.66667
0.00000
0.00000
0.22222
0.00000
5.03030
0.00000
0.22222
0.00000
1.15152
0.00000
Table B2
Selected values of Gm,lm for B1, B2, and B3 crystals
Structures
m
Gm,AA(1)
Gm,AA(4)
Gm,BB(1)
Gm,BB(4)
Gm,AB(1)
Gm,AB(4)
B1
1
2
3
4
0.00000
0.70711
0.00000
0.00000
0.00000
0.35355
0.00000
0.50000
0.00000
0.70711
0.00000
0.00000
0.00000
0.35355
0.00000
0.50000
0.00000
0.00000
1.53960
0.00000
0.50000
0.00000
0.38490
0.00000
B2
1
2
3
4
0.00000
0.00000
0.81650
0.00000
0.00000
0.28868
0.40825
0.00000
0.00000
0.00000
0.81650
0.00000
0.00000
0.28868
0.40825
0.00000
0.88889
0.00000
0.00000
2.40544
0.22222
0.00000
0.00000
2.62699
B3
1
2
3
4
0.00000
0.81650
0.00000
0.00000
0.00000
0.40825
0.00000
0.57735
0.00000
0.81650
0.00000
0.00000
0.00000
0.40825
0.00000
0.57735
0.44444
0.00000
1.20272
0.00000
0.11111
0.00000
1.31350
0.00000
Hmm,AB(1)
Hmm,AB(4)
Table B3
Selected values of Hmm,lm for B1, B2, and B3 crystals
Structures
m
Hmm,AA(1)
Hmm,AA(4)
Hmm,BB(1)
Hmm,BB(4)
B1
1
2
0.00000
14.00000
0.00000
6.00000
0.00000
14.00000
0.00000
6.00000
0.00000
0.00000
8.00000
0.00000
B2
1
2
0.00000
0.00000
0.00000
4.00000
0.00000
0.00000
0.00000
4.00000
18.96296
0.00000
3.16049
0.00000
B3
1
2
0.00000
14.00000
0.00000
6.00000
0.00000
14.00000
0.00000
6.00000
4.74074
0.00000
0.79012
0.00000
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uH
u 11;AB ð1Þ V C 11 þ H 11;AB ð4Þ V C 44
(
)
u
u
Pn
r21 ½/00AA ðrm ÞF m;AA ð1Þ þ /00BB ðrm ÞF m;BB ð1Þ þ /00AB ðrm ÞF m;AB ð1Þ
u
u H 11;AB ð1Þ m¼1
u
þr1 ½/0AA ðrm ÞGm;AA ð1Þ þ /0BB ðrm ÞGm;BB ð1Þ þ /0AB ðrm ÞGm;AB ð1Þ
u
(
)
u
u
Pn
r21 ½/00AA ðrm ÞF m;AA ð4Þ þ /00BB ðrm ÞF m;BB ð4Þ þ /00AB ðrm ÞF m;AB ð4Þ
u H
11;AB ð4Þ
u
m¼1
þr1 ½/0AA ðrm ÞGm;AA ð4Þ þ /0BB ðrm ÞGm;BB ð4Þ þ /0AB ðrm ÞGm;AB ð4Þ
u
u
u H
2
2
2
2
11;AB ð1Þ½uAA ðr2 ÞH 22;AA ð1Þ þ uBB ðr 2 ÞH 22;BB ð1Þ H 11;AB ð4Þ½uAA ðr2 ÞH 22;AA ð4Þ þ uBB ðr2 ÞH 22;BB ð4Þ
t
uAB ðr1 Þ ¼
H 211;AB ð1Þ þ H 211;AB ð4Þ
ðB:3Þ
X.W. Zhou, H.N.G. Wadley / Computational Materials Science 39 (2007) 541–551
Eq. (B.3) gives a fourth relationship between material
properties (elastic shear moduli) and free parameters of
the potential.
References
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