Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
ATOMISTIC SIMULATION METHODS FOR STUDYING
SELF HEALING MECHANISMS IN Al/Al2O3
Ivan Lazic, Marius Ernst and Barend Thijsse
Department of Materials Science and Engineering,
Delft university of Technology
+31 (015) 2789518
+31 (015) 2786730
I.Lazic@tudelft.nl
www.structureandchange.3me.tudelft.nl
New methods for modeling initial oxide film growth using MD (Molecular Dynamics) simulations are explored,
in order to study the atomic mechanism of self-healing oxidation in metal/oxide systems, in particular in
Al/Al2O3. Computationally, this is a challenging undertaking. A new version of the MEAM (Modified
Embedded Atom Method) potential is proposed and extended by a variable charge ionic potential model from
the literature. This makes it possible to include angular forces, which are needed for relatively open crystal
structures such as metal oxides, as well as localized and dynamic charge transfer between Al and O, which is
necessary to handle time-dependent local composition variations. As Coulomb solver we use the PPPM method
(particle-particle-particle-mesh). In this work the first results are reported. Tests of the ionic potential model in
combination with the PPPM solver yield excellent results. Simulations of oxygen atom arrivals at an Al surface
will be presented.
Keywords: molecular dynamics, oxides, self-healing
1
Introduction
The surface oxides on aluminum and aluminum alloys can be called “self-healing” in that
they quickly re-form after scratching. The initial stage of oxidation is very rapid, much faster
than for a more noble metal such as e.g. Ru (Fig. 1) and is therefore difficult to follow
experimentally. Here simulations would be able to extend time scales and resolution. Such
self-healing phenomena of metal oxides using Molecular Dynamics (MD) simulations require
sophisticated interatomic potentials of the underlying metal-oxygen systems, which is a
challenging undertaking. The main target of our investigation is Al2O3. For oxides and
metal/metal-oxide systems in general, extra difficulties appear because of the presence of
ionic bonds, giving rise to long-range Coulomb interactions. Moreover, at interfaces, surfaces
and defects, the atomic charges cannot be considered fixed, and models that allow charge
transfer should be applied. Finally, aluminum oxide has a relatively open structure, which
suggests that angular terms in the interatomic potentials could play an important role. For
these reasons we are developing a potential for Al/Al2O3 that combines a charge-transfer
potential for the electrostatic interactions with the Modified Embedded Atom Method
(MEAM) [1] for the non-electrostatic interactions. Such a potential would also be applicable
to the relatively open oxides of e.g. Si and to the mildly ionic III-V and II-VI semiconductors.
This paper reports on the potential under development and on the implementation of the
Coulomb solver.
1
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
Previous MD work on Al/Al2O3 has been done using MEAM but without considering charges
explicitly [2], and using charge transfer ionic potentials but in combination with EAM, which
does not includes angular dependencies [3][4]. Here we propose to use the Charge Transfer Ionic
Potential (CTIP) published in ref. [4] in combination with novel version of the MEAM. Unlike
previous work, we use the iterative Particle-Particle-Particle-Mesh (PPPM) method as Coulomb solver
[5]. PPPM easily adapts to periodic and nonperiodic boundary conditions, and, because of its localized
computability, it is convenient for code parallelization. An additional reason is that the computational
efficiency of PPPM is ϑ ( N log N ) while, e.g., for Ewald Summation it is ϑ ( N 3 / 2 ) [6]. A
disadvantage of PPPM is that the maximal error can not be predicted analytically.
Figure 1: Oxide growth measured by in-situ ellipsometry on oxidizing single crystals of aluminum and
ruthenium. Formation of the initial, chemisorbed oxide layer is easily noticeable for ruthenium but not for
aluminum. Very likely it does take place also in aluminum but very fast
2
Theory and methods
In our MEAM+CTIP model, the energy U of a system of N atoms is the sum of a nonelectrostatic and an electrostatic part, U = U nes +U es , where the non-electrostatic MEAM part
is given by
N
N N
U nes = ∑ ∑ 12Φ IJ (rij ) + ∑ FI (xi )
(1)
i=1
i=1 j=1
j≠i
in which upper case I, J denote the chemical types of the atoms i, j. The pair potential and
embedding functions for each of the chemical types or type combinations are given by
Φ(r) = −E p (1+ η + c pη 2 + d pη 2 )e−η ,
2
η = α p (r / rp −1),
(2)
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
F(x) = Ae x ln x + Be x + Ce x ,
(3)
where the p and e subsripts denote adjustable parameters. The quantity xi is proportional to the
square of what in (M)EAM terminology is called the “background electron density at atom i”
and contains all dependencies on the i-angle in the ijk atom triplets,
3
xi = ∑ t I(l ) ∑ ∑ p(lJ )e
l=0
−qJ(l ) rij
pK(l )e−qK
(l )
rik
P (l ) (cosθ jik ) . (4)
j≠i k≠i
Here, t(l), p(l), q(l) are fitparameters for each chemical type, and P(l) is the Legendre
polynomial. For simplicity, Eqs. (1-4) are given without angular and radial cutoff functions.
New in this MEAM format is that the embedding function has been extended by two terms,
that the concept of “reference structure” has been abandoned, and that the pair potential is a
parametrized (Morse-like) function rather than a function determined by a prescribed equation
of state of a particular crystal phase. Eqs. (1-4) are more flexible than the classical format,
without changing much in the underlying physical picture. With these expressions it proved
possible to represent the well-known Stillinger-Weber and Tersoff-III silicon potentials to a
very high degree of accuracy [7], which suggests that they will be appropriate to the current
Al/Al2O3 case as well.
In our implementation of the electrostatic CTIP part, the long-range Coulomb interactions are
handled by the PPPM method which is briefly explained in Figs. 2 and 3. The grids on the
figures represent the mesh, which holds the charge distribution in the middle panel of Fig. 3.
Assuming pointlike charges, the interaction between a charged atom i and all other charged
atom j separated by rij, including all periodic images, is divided into two parts by adding and
subtracting a charge distribution ρ j (r) around every atom j. The volume integral of this
distribution must be equal to qj. If it is chosen to be Gaussian, ρ j (r) = q jσ −3π −3/2 exp(−r 2 / σ 2 ) ,
it can be proven that the interaction energy between point charge i and point-charged atom j
minus its own charge distribution is qi q j erfc(rij / σ ) / rij . The complementary error function
decreases rapidly with distance, which means that this part of the interaction has become
short-ranged and a cutoff radius can be used (Fig. 2), normally on the order of 5 Å. The rest of
the interaction (Fig. 3) is calculated by placing point charge i in the electrostatic potential on
the mesh, solved by the Poisson equation ∇ 2ϕ mesh = −ρ mesh / ε 0 . Here, ρ mesh is the total charge
of all individual charge distributions ρ j (r ) , plus i’s own charge distribution, allocated to the
mesh points through a diffusion-like algorithm. In this way ρ mesh is the same for all atoms i
and the Poisson equation needs to be solved only once. For each atom, the spurious
interaction between its point charge and its charge distribution can then be subtracted
analytically (Fig. 2, right). In summary, one obtains
N
qi q j
j =1
j ≠i
rij
∑ kc
N
qi q j
j =1
j ≠i
rij
= ∑ kc
⎛ rij
erfc⎜⎜
⎝σ
⎞
⎟ + q i (ϕ mesh ( q ) − ϕ self ( q i ))
⎟
⎠
3
(5)
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
Figure 2: Short-range part (“PP”) of the interaction between atom i and all other atoms, which is taken into
account directly, showing the cutoff radius. The mesh is not used here
Figure 3: The rest of the interaction (“PM”). The middle frame shows the interaction of i with the charges
attributed to the mesh points. The Poisson equation is used here. The frame on the right shows the correction for
self-interaction
where kc = 1 / 4 πε 0 , and ϕ self (qi ) = 2kc qi / σ π is the self-electrostatic potential. The vector
notation indicates the set of all charges. The mesh size should be no more than one quarter of
the cutoff radius.
According to the CTIP model the charges in the system are not constant. Every atom is
allowed to become ionized by charge transfer to and from other atoms. The ionization
energy Eiionization (qi ) = χ i qi + J i qi2 / 2 , with χ i the electronegativity and Ji the electrostatic
hardness of atom i, is compensated by the appearance of Coulomb electrostatic energy
1
EiCoulomb = k c ∑ qi q j / rij . The charges that atoms obtain for a certain atomic arrangement are
2
determined by searching for the minimum of total electrostatic energy Ues (ionization plus
Coulomb) with the condition of total system charge neutrality.
This energy is given by
N
1
⎛
⎞ 1 N N
U es = ∑ ⎜ χ i q i + J i q i2 ⎟ + ∑ ∑ (k c q i Z j ([ j | f i ] − [ f i | f j ]) + k c q j Z i ([i | f j ] − [ f i | f j ]) ) +
2
⎠ 2 i =1 j =1
i =1 ⎝
j ≠i
⎛⎛
N
q i − q min,i
1 N N
k c q i q j [ f i | f j ] + ∑ ω ⎜ ⎜1 −
∑
∑
⎜⎜
2 i =1 j =1
q i − q min,i
i =1
⎝⎝
j ≠i
⎞
⎛
⎟(q − q ) 2 + ⎜1 − q max,i − q i
i
min,i
⎟
⎜
q i − q max,i
⎠
⎝
4
(6)
⎞
⎞
2 ⎟
⎟( q − q
i
max,i )
⎟
⎟
⎠
⎠
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
where notations [a | f b ] and [ f a | f b ] (a=i,j; b=i,j; a≠b) denote the Coulomb interaction
f (r , q ) f (r , q )
f (r , q )
integrals [a | fb ] = kc ∫ b b b dVb and [ f a | f b ] = k c ∫ ∫ a a a b b b dVa dVb . This is
rvv
rav
V V
V
a
b
b
according to the used charge model, in which each atom has a pointlike charge part and a
distributed part: ρ iatom (r ) = Z i δ (r ) + (qi − Z i ) f i (r ) . For mathematical simplicity the
distributed part of the atomic charge is chosen to be f i ( r ) = ξ i3 exp( −2ξ i r ) / π . The last term
in Eq. (6) is an extra term that softly bounds the charges between the values q min and q max
( ω > 0 ) to account for chemical valence. Parameters that need to be specified for each type of
atom in the system are then: χ i , J i , Z i , ξ i , q min , and qmax .
After applying PPPM (Eq. (1)) to Eq. (6) we achieve
U es =
⎛⎛
q − qmin,i
1 N
2ω ⎜ ⎜1 − i
∑
2 i =1 ⎜ ⎜⎝
qi − qmin,i
⎝
⎞
⎛
⎟(q − q ) 2 + ⎜1 − qmax,i − qi
min, i
⎟ i
⎜
qi − qmax,i
⎝
⎠
(
)
N
⎞
⎞
⎟(q − q ) 2 ⎟ + 1 ∑ 2 χ q + J q 2 + q ⋅ (ϕ (q ) − ϕ (q )) +
i
i i
i i
i
mesh
self
i
max, i
⎟
⎟ 2 i =1
⎠
⎠
N N
⎛
⎞
1 N N
(kc qi Z j ([ j | fi ] − [ fi | f j ]) + kc q j Zi ([i | f j ] − [ fi | f j ])) + 1 ∑∑ kc qi q j ⎜⎜ 1 erfc⎛⎜⎜ rij ⎞⎟⎟ + ([ fi | f j ] − 1 ) ⎟⎟
∑∑
2 i =1 j =1
2 i =1 j =1
rij ⎠
⎝σ ⎠
⎝ rij
j ≠i
j ≠i
(7)
The minimization of Ues together with the condition
∑
N
i =1
q i = 0 is done using the Conjugate-
Gradient method [8]. For the line minimization in the charge space the Brent algorithm was
used [8].
In order to test the accuracy of the CTIP/PPPM implementation, we use the fact that the
electrostatic energy for the simple case of, e.g., a NaCl crystal type can also be determined
and minimized analytically, in terms of the Madelung constant,
N
1
⎛
⎞ 1 N N
U es (q ) = ∑ ⎜ χ i qi + J i qi2 ⎟ + ∑∑ (kc qi Z j ([ j | f i ] − [ f i | f j ]) + kc q j Z i ([i | f j ] − [ f i | f j ]) ) +
2
⎠ 2 i =1 j =1
i =1 ⎝
j ≠i
N
N
(8)
2
1
1
N NaCl q
kc qi q j ([ f i | f j ] − ) − α madelung
∑∑
2 i =1 j =1
rij
2
r0
j ≠i
NaCl
where qi = q if atom i is Na and q i = − q if atom i is Cl, α madelung
is the Madelung constant
for the NaCl crystal type, and r0 nearest neighbor distance. The double summation terms
include only short-range functions and converge very fast. In this formula the charge
bounding energy term is not included for simplicity.
3
Results
(1) Results of how well PPPM works are given in Table I. The basic tests of Madelung
constant calculations were done for several artificially made but convenient structures.
5
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
If a unit point charge is placed inside the heavily drawn box (at fractional coordinates X, Y, Z)
in Fig. 4 and a mirroring operation is applied, infinite three-dimensional structures as shown
in Fig. 4 can be generated. If the charge is placed in the middle of the box we achieve exactly
the NaCl crystal type, but by varying the coordinates of its position inside the box we obtain
other types. For all of these we can calculate the Madelung constant exactly. A comparison of
the exact and the PPPM results for the Madelung constant is shown in Table I.
Figure 4: Generating the crystal structure samples for the Madelung constant test
Table I: Madelung constants for different structures calculated exactly and using PPPM
X
0.2
0.6
0.4
0.1
0.5
0.8
0.7
0.9
0.6
Y
0.5
0.6
0.9
0.1
0.5
0.2
0.9
0.7
0.5
Z
0.3
0.1
0.8
0.7
0.5
0.5
0.8
0.1
0.4
Madelung constant
Exact
PPPM
2.83447
2.83447
5.10754
5.10755
5.27626
5.27628
6.47665
6.47664
1.74755
1.74756
3.26069
3.26071
5.29997
5.29998
6.47665
6.47664
1.85929
1.85934
(2) Values of charges and potential energies per atom for the NaCl structure in which one
allows charge transfer can be calculated exactly (Eq. 8) and compared with the minimization
carried out by PPPM (Eq. (7)). Results are given in Fig. 5. The curves “exact” and “PPPM
( ω = 0 )” overlap almost perfectly. As expected, the PPPM calculations with bound charges
(in our case ω = 20 eV/e2 ) yield different curves in the region where charge bounding is
actually taking place. The parameters used for these calculations are shown in Table II.
These parameters are just test parameters, not literature values or values achieved by fitting.
cut −off
The cut-off radius for all short-range terms in PPPM method was chosen to be rpppm
= 5.4 Å.
6
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
Figure 5: Charges q (left) and electrostatic potential energy per atom Ues/N (right) for the NaCl crystal structure
type after energy minimization calculated analytically and using PPPM for different nearest neighbor distances.
The inserts show the deviation between PPPM (ω = 0) and the exact method
Table II.: Test parameters for Na and Cl atomic types
Element q
q max χ[eV] J [eV ] ξ [Å-1] Z
min
Na
Cl
0
-1
1
0
-3.4
2.0
10
14
1.0
2.1
0.6
0.0
(3) The effect of angular forces is illustrated by a simplified calculation of the III-V
compound AlP. Using a realistic MEAM potential, the cubic ZnS phase is the stable crystal
phase. By changing only one parameter value in the MEAM potential, namely t(3) in Eq. 4 ,
which controls the energy associated with the third spherical harmonic describing an atom’s
geometrical environment, the cubic CuAu phase can be made the stable crystal phase (Fig. 6).
Figure 6: Energies of two phases of a simplified model of AlP as a function of nearest-neighbor distance. By
simply changing a single parameter in the MEAM potential, the relative stability of the two phases can be tuned
(left and right). Charge transfer varies from approximately 0.25 e at small R to 0.13 e at large R
7
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
(4) The first simulation results of an oxygen atom approaching an Al surface kept at 375 K are
shown in Fig. 7. The oxygen atom was directed at the Al sample at normal incidence, with an
initial kinetic energy of 0.0125 eV, which corresponds to the average thermal speed at 300 K.
For this simulation, the EAM potential of ref. 4 were used, i.e. a potential without the angular
terms l = 1, 2, 3 in Eq. (4). After impact, the oxygen atom reaches the second Al atomic
monolayer from the surface and stays there as an interstitial. As expected, no permanent
change was produced in the Al lattice structure. The amount of charge that the oxygen atom
has acquired as interstitial is –0.54 e, which is far away from the minimum value of –2 e, and
this charge is almost totally transferred to the six closest neighboring Al atoms (around +0.08
e each).
Figure 7: Oxygen atom approaching Al surface held at 375 K. a) Oxygen atom reaching the first Al plane. b)
Final oxygen position as interstitial. Upper two figures: view along the Al <110> direction. Lower two figures:
top view along the Al <001> direction. The squares indicate the same Al unit cell. Colors represent different
charge ranges: light blue (–0.02 e, +0.02 e), dark blue (+0.07 e, 0.25 e), magenta (–0.55 e, –0.45 e)
4
Discussion
Results of calculating Madelung constant using PPPM, given in Table I, are in extremely
good agreement with analytically determined values for all structures generated as described
above (Fig. 4). In agreement with this, the results of using PPPM as a long-range Coulomb
solver in minimization process are also very satisfactory (Fig. 5); the deviations using PPPM
in comparison with the exact calculations do not exceed 8 meV/atom for the electrostatic
potential energy and are less than 0.009 e for the charges in the NaCl crystal type.
8
© Springer 2007
Proceedings of the First International Conference on Self Healing Materials
18-20 April 2007, Noordwijk aan Zee, The Netherlands
Ivan lazic et al.
In the range of nearest neighbor distances where the equilibrium positions are expected,
between 2 and 3.5 Å, these errors are even less then 1 meV/atom and 0.001 e. Using an
MEAM instead of EAM potential, the possibility of fine tuning of nonelectrostatic angular
dependent part of the potential interaction should allow us better fitting to experimental and
ab initio data for open structures as Al2O3. Finally, the approach of a single oxygen atom
towards the surface of a pure Al crystal gives us some insight in very beginning of process of
aluminum oxidation. Further work is needed to understand why the interstitial oxygen atom in
Fig. 7 is only partially charged, while one could expect a full charge of –2 e. This may be due
to a flaw in the charge transfer model, although it correctly leads to fully charged atoms in
Al2O3.
5
Conclusions
In this work the first steps towards building a sophisticated enough simulation system for
metal-oxide/metal-alloy systems and some first results are reported. The combination of a
charge transfer ionic potential (CTIP) [4] together with PPPM as long-range Coulomb
interaction solver was tested against analytical calculations for simple system such as the
NaCl crystal type. The results for charges and energies are excellent. Also, a modified version
of the MEAM potential is proposed, one that is designed for more flexibility then the original
one. Control of the phase stability by adjusting parameters of the angular dependent terms in
the model has been demonstrated to work well, even together with atomic charges and charge
transfer in the system. A first simulation on oxygen atom approaching an Al surface yields
realistic results on initial oxidation. These first tests will be followed by further simulations
that combine MEAM and CTIP in the studying of self-healing effect in Al/Al2O3.
ACKNOWLEDGMENTS
The authors would like to thank Prof. Dr. Marcel Sluiter, Darko Simonovic, MEng, and Dr. Wim Sloof from the
Department of Material Science and Engineering, Delft University of Technology, for discussions and critical
review of the manuscript. This research is supported by the Netherlands Foundation for Fundamental Research
on Matter (FOM) and the Netherlands Institute for Metal Research (NIMR) within the project 02EMM31.
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© Springer 2007