Pd diffusion on MgO(100): The role of defects and small cluster mobility
Density functional theory is used to explore the energy landscape of Pd atoms adsorbed on the terrace of MgO(1 0 0) and at oxygen vacancy sites. Saddle point finding methods reveal that small Pd clusters diffuse on the terrace in interesting ways. The monomer and dimer diffuse via single atom hops between oxygen sites with barriers of 0.34 eV and 0.43 eV respectively. The trimer and tetramer, however, form 3D clusters by overcoming a 2D–3D transition barrier of less than 60 meV. The trimer diffuses along the surface either by a walking or flipping motion, with comparable barriers of ca. 0.5 eV. The tetramer rolls along the terrace with a lower barrier of 0.42 eV. Soft rotational modes at the saddle point lead to an anomalously high prefactor of 1.3 × 1014 s−1 for tetramer diffusion. This prefactor is two order of magnitude higher than for monomer diffusion, making the tetramer the fastest diffusing species on the terrace at all temperatures for which diffusion is active (above 200 K). Neutral oxygen vacancy sites are found to bind Pd monomers with a 2.63 eV stronger binding energy than the terrace. A second Pd atom, however, binds to this trapped monomer with a smaller energy of 0.56 eV, so that dimers at defects dissociate on a time scale of milliseconds at room temperature. Larger clusters bind more strongly at defects. Trimers and tetramers dissociate from monomer-bound-defects at elevated temperatures of ca. 600 K. These species are also mobile on the terrace, suggesting they are important for the ripening observed at ⩾600 K during Pd vapor deposition on MgO(1 0 0) by Haas et al. [G. Haas, A. Menck, H. Brune, J.V. Barth, J.A. Venables, K. Kern, Phys. Rev. B 61 (2000) 11105].
Pd diﬀusion on MgO(100): The role of defects
and small cluster mobility
, Graeme Henkelman
, Charles T. Campbell
, Hannes Jo
Department of Chemistry 351700, University of Washington, Seattle, WA 98195-1700, United States
Department of Chemistry and Biochemistry, University of Texas at Austin, 1 University Station Stop A5300, Austin, TX 78712-0165, United States
Faculty of Science, VR-II, University of Iceland, 107 Reykjavı
Received 20 December 2005; accepted for publication 20 January 2006
Available online 9 February 2006
Density functional theory is used to explore the energy landscape of Pd atoms adsorbed on the terrace of MgO(1 0 0) and at oxygen
vacancy sites. Saddle point ﬁnding methods reveal that small Pd clusters diﬀuse on the terrace in interesting ways. The monomer and
dimer diﬀuse via single atom hops between oxygen sites with barriers of 0.34 eV and 0.43 eV respectively. The trimer and tetramer, how-
ever, form 3D clusters by overcoming a 2D–3D transition barrier of less than 60 meV. The trimer diﬀuses along the surface either by a
walking or ﬂipping motion, with comparable barriers of ca. 0.5 eV. The tetramer rolls along the terrace with a lower barrier of 0.42 eV.
Soft rotational modes at the saddle point lead to an anomalously high prefactor of 1.3 · 10
for tetramer diﬀusion. This prefactor is
two order of magnitude higher than for monomer diﬀusion, making the tetramer the fastest diﬀusing species on the terrace at all tem-
peratures for which diﬀusion is active (above 200 K). Neutral oxygen vacancy sites are found to bind Pd monomers with a 2.63 eV stron-
ger binding energy than the terrace. A second Pd atom, however, binds to this trapped monomer with a smaller energy of 0.56 eV, so that
dimers at defects dissociate on a time scale of milliseconds at room temperature. Larger clusters bind more strongly at defects. Trimers
and tetramers dissociate from monomer-bound-defects at elevated temperatures of ca. 600 K. These species are also mobile on the
terrace, suggesting they are important for the ripening observed at P600 K during Pd vapor deposition on MgO(100) by Haas et al.
[G. Haas, A. Menck, H. Brune, J.V. Barth, J.A. Venables, K. Kern, Phys. Rev. B 61 (2000) 11105].
! 2006 Elsevier B.V. All rights reserved.
Keywords: Density functional calculation; Models of surface kinetics; Palladium; Magnesium oxide
The growth, migration and agglomeration of small me-
tal clusters on oxide surfaces are of importance in many
technological applications, including catalysis and chemi-
cal sensing by oxide-supported metal nanoparticles, the
production of metal thin ﬁlms and coatings, and the
fabrication of microelectronic, magnetic, photonic and
photovoltaic devices [1–7]. The dynamics of small metal
clusters on oxides is also of importance in fundamental
scientiﬁc research concerning such systems. Key issues in-
clude the mechanisms and kinetics of diﬀusion of the metal
adatoms and metal clusters, their nucleation dynamics and
sintering mechanisms, the role of surface defects in these
processes, and how the size and distribution of the metal
clusters inﬂuence their energetics, thermal stability, elec-
tronic properties and functionality in applications.
A widely held view of the growth of late transition metal
ﬁlms on single crystal oxide surfaces by vapor deposition is
that metal atoms land primarily on ﬂat terr aces and diﬀuse
over the surface by hopping from one site to another. Step
edges or point defects, however, bind the metal adatoms so
strongly that they get trapped there. A second diﬀusing
adatom which encounters such a metal-defect complex will
0039-6028/$ - see front matter ! 2006 Elsevier B.V. All rights reserved.
Corresponding author. Tel.: +1 512 471 4179; fax: +1 512 232 4655.
E-mail address: email@example.com (G. Henkelman).
Surface Science 600 (2006) 1351–1362
also get trapped there, and by the further addition of other
diﬀusing a datoms, the metal cluster builds up at the defect
site [3,4,8]. This is referred to as heterogeneous nucleation.
In homogeneous nucleation, the clusters nucleate instead
on the majority (terrace) sites. However, the detai ls of the
elementary steps and intermediate structures involved are
still poorly understood, and indeed it is not even clear
whether this is the proper mechan ism. Certainly, no atom-
istic model that reproduces the growth kinetics for any me-
tal-on-oxide system has ever been developed which is based
on independently determined kinetic parameters.
Since oxide-supported metal clust ers with diameters less
than 4 nm are very unstable relative to large metal parti-
cles, they also tend to sinter into fewer, larger clusters,
especially at high temperatures . This is a major mecha-
nism for deactivation of oxide-supporte d transition metal
catalysts. Sinter ing can occur either by ‘‘Ostwald ripening’’,
in which individual metal atoms (possibly in the form of a
complex with another species, hence ‘‘monomers’’) leave a
metal cluster, diﬀuse over the support, and join another
metal cluster, or by ‘‘cluster diﬀusion/agglomeration’’
wherein the metal clusters diﬀuse across the oxide surface,
encounter other clusters and merge into one (as do two
water droplets when they touch) [4,9–16]. Again, the details
of the elementary steps and intermediate structures in-
volved are still poorly understood, and no atomistic model
that fully reproduces sintering kinetics has been developed
yet. Nevertheless, a complete understanding of sintering
kinetics would be very valuable in the development of
In this paper, we combine density functional theory
(DFT) with the nudged elastic band method (NEB)
[17,18], the dimer method [19,20], and harmonic transition
state theory [21,22] to estimate the structures, energies,
entropies, diﬀusion mechanisms, transition states and diﬀu-
sion kinetic parameters of small metal clusters for a proto-
typical metal-on-oxide system. The results reveal several
important phenomena involving clusters larger than dimers
which have previously been ignored but which clearly need
to be considered when analyzing the growth and sintering
of metal islands on oxide surfaces and the role of point de-
fects. The energies of the many cluster structures reported
here also will help any eﬀort to develop a simple parameter-
ized Pd–MgO( 1 0 0) potential for molecular dynamics sim-
ulations of growth and sintering kinetics.
To date, the most thoroughly studied metal/oxide sys-
tem in terms of structural elucidation during growth and
sintering is Pd on MgO(1 0 0) [3,23–29]. We have, therefore,
chosen to carry out theoretical calculations on this system,
but its properties are qualitatively typical of many other
interesting metal/oxide systems . Experimental measure-
ments of Pd islands grown by metal atom deposition on
MgO(1 00) at low temperature have shown that the num-
ber density of the islands is remarkably insensitive to the
surface temperature during growth . This is a clear indi-
cation that defects play an important role in the growth
process. It is well known that oxide surfaces tend to have
a high density of point defects (typical estimates being
) in addition to steps and grain bound-
aries. HREELS spectra have been assigned and interpreted
in terms of neutral oxygen vacancies, so called F-centers,
where an oxygen atom has been removed from the surface
. It has also been suggested that di-vacancies are present
where both a Mg- and O-atom are missing  and, in
some cases, charged oxygen vacancies [32,33]. We have as-
sumed that the defects nucleating Pd clusters are neutral
oxygen vacancies. This is co nsistent with experiments using
CO as a chemical marker to determine the Pd bonding at
surfaces. The vibrational frequency of CO bound to Pd
atoms at point defects on MgO are observed to be red-
shifted relative to Pd–CO . Theoretical calculations,
however, show a blueshift for Pd atoms at charged F-cen-
ters , suggest ing that the primary point defects are (neu-
Measurements of Pd island densities from vapor deposi-
tion experiments on MgO(1 0 0) have been previously ana-
lyzed by applying the common growth model mentioned
above. Various energy parameters, such as monomer diﬀu-
sion activation energy, Pd/defect trapping energy, and di-
mer binding energy, were adjusted to ﬁt the measured
island density over a wide range of substrate temperatures.
There are two problems, however, with this simple picture
of Pd island formation. Diﬀerent types of experiments lead
to diﬀerent sets of best-ﬁt parameters [3,30], and the de-
rived energy landscape is inconsistent with ﬁrst principles
calculations. Speciﬁcally, a strong dimer binding energy
of 1.2 eV is required to match the ripening observed at
600 K . Giordano et al. ﬁnd the binding energy of a sec-
ond Pd atom to an adatom/F-center complex to be very
weak (0.39 eV), insigniﬁcantly larger than the dimer bind-
ing energy on the defect-free terrace (0.35 eV) . The
weak binding has also been seen for Pt on MgO(1 0 0), in
which a Pt adatom prefers the terrace over binding to
another adatom trapped at an F-center . This sug-
gests that some assumptions in the model must be
We present here theoretical results that indicate a diﬀer-
ent role for F-centers in the Pd-island growth process,
which presumably will also be important in understanding
sintering kinetics. A brief communication of some of these
kinetic results has been published previously  (see also
Ref. ). Here we describ e a systematic atomic-scale study
of the formation and diﬀusion mechanisms of small Pd
) on MgO(1 0 0), with and wi thout oxy-
gen vacancy defects.
2. Calculation methodology
Calculations are performed with the Vienna Ab initio
Simulation Package, VASP, using the PW91 functional
 and ultra-soft pseudo-potentials of the Vanderbilt
form . A plane wave cutoﬀ of 270 eV, appropriate for
the pseudo-potentials, and a C point sampling of the Brill-
ouin zone, are found to be suﬃciently converged, as shown
1352 L. Xu et al. / Surface Science 600 (2006) 1351–1362
in Appendix A. Spin polarization was tested for each clus-
ter size. The only cluster found to have a magnetic moment
is the tetramer, for which spin polarization reduces the
adsorption energy by 0.12 eV. For this reason, all calcula-
tions of the tetramer are done with spin polarization.
The MgO(1 0 0) surface is modeled as a slab with three
layers, using either 24 or 36 atoms per layer in each unit
cell. The simulation box is chosen with a height of
, perpendicular to the (1 00) surface. The lower two
layers are held frozen at the optimal DFT lattice constant
of 4.23 A
, which compares well with the 4.21 A
tal lattice constant. Atomic positions in the top layer are
fully optimized in each calculation. When Pd is adsorbed
on the surface, the substrate atoms are found to relax by
no more than 0.1 A
, and adsorption energies are lowered
by 0.1–0.2 eV. The relat ive energies of diﬀerent cluster sizes
and geometries, however, are largely unaﬀected by the
relaxation of the top layer. Furthermore, the addition of
more MgO layers and the relaxation of more surface layers
do not change our results (see Appendix A).
Adsorption energies, E
, for a cluster of n Pd atoms on
the MgO surface are calculated as
is the energy of n Pd atoms adsorbed on the
MgO surface, E
is the energy of an isolated gas phase
Pd atom, and E
is the energy of the MgO substrate.
Diﬀusion barriers are calculated by ﬁnding saddle
points on the potential surface. When both the initial
and ﬁnal state is known for a diﬀusion process, the
nudged elastic band (NEB) method [17,18] is used to
determine the reaction mechanism and diﬀusion barrier.
The dimer saddle point ﬁnding method [19,20] was also
used both to investigate the possibility of unexpected
mechanisms leading from known minima, and to eﬃ-
ciently converge upon nearby saddle points. All geome-
tries are optimized so that the maximum force in every
degree of freedom is less than 0.001 eV/A
. Normal mode
frequencies are calculated with a ﬁnite diﬀerence method
for all relaxed atoms at stationary points to evaluate the
entropy in the harmonic approximation, and the prefactor
of reaction rates. Tests show that ﬁnite diﬀerence displace-
ments ranging from 0.005 to 0.001 A
resulted in prefac-
tors diﬀering by less than 10% from converged values.
Reaction rates are estimated using the harmonic form of
transition state theory [21,22],
In this expression E
is the energy of the saddle point, E
is the energy of the local minimum corresponding to the
initial state, k
is Boltzman’s constant, T is the tempera-
ture, N is the number of moving atoms, and m
are the real
normal mode frequencies at the saddle point (!) and initial
state (init) respectively.
Three binding sites were considered for the adsorption
of Pd atoms. The Mg and O sites, directly above Mg and
O ions respectively, and the hollow site between tw o Mg
and two O ions. We ﬁnd that the Pd monomer only binds
at the O-site, in agreement with previous studies [42,43].
The monomer adsorption energy is found to be 1.37 eV
with a Pd–O distance of 2.09 A
. The oxygen on which
the Pd sits relaxes inward by 0.05 A
relative to the topmost
plane surface ions. The monomer binding energy is rela-
tively insensitive to the details of the DFT calculation.
Cluster and supercell models using the BP and PW91
density functio nals give a binding energy of ca. 1.4 eV
A Bader atoms-in-molecules (AIM) analysis of the
charge density [47,48] shows a small but noticeable and
well localized charge transfer of 0.15 e from the substrate
oxygen ion to the on-top adsorbed Pd adatom and almost
no charge transfer from Mg atoms in the substrate. The
direction an d amount of charge transferred between O
and Pd is close to the value of 0.19 e given by a recent
AIM calculation from a very diﬀerent kind of DFT calcu-
lation, using a cluster model with the B3LYP functional
. An earlier cluster model calculation using the BLYP
functional with and without a relativistic c orrection pre-
dicted a charge transfer from O to Pd of 0.2 and 0.3 e
respectively using a Mulliken population analysis .
The agreement between these diﬀerent calculations suggests
that the charge transfer from O to Pd is a robust pred iction
of DFT calculations.
The diﬀusion mechanism for a Pd monomer (see Fig. 1)
is found to be a direct hopping process between oxygen
sites. In the saddle geometry, the Pd atom sits above a hol-
low site. The energy barrier for monomer diﬀusion is
0.34 eV; consistent with what has been found previously
. An Arrhenius prefactor for monomer hopping is
0 1 2 3
Reaction Coordinate (Å)
Fig. 1. Pd monomer binds to O-sites with adsorption energy 1.37 eV, and
diﬀuses via a hop mechanism. White circles represent Pd atoms, small gray
circles Mg atoms and large dark gray circles are O atoms.
L. Xu et al. / Surface Science 600 (2006) 1351–1362 1353
calculated from the harmonic approximation (Eq. (2) ) as
7.4 · 10
. This is approximately an order of magnitude
less than a typical prefactor of 5 · 10
The monomer is found to bind very strongly to a neutral
oxygen vacancy (F-center) with an adsorption energy of
4.0 eV, in agreement with a recent study  using both
cluster and supercell models.
The most stable dimer on the MgO(100) terrace, (la-
beled as D1, as shown in Fig. 2), has two Pd adatoms sit-
ting at neighboring oxygen sites. The Pd–Pd distance of
is smaller than the distance of 3.00 A
gen atoms in the surface. The Pd dimer sits 2.12 A
the MgO substrate surface plane and has an adsorption en-
ergy of 3.28 eV, relative to two isolated atoms in the gas
phase. A Bader analysis shows a charge transfer, similar
to the monomer, of 0.15 e to each Pd adatom from the oxy-
gen atom ben eath. The dimer binding energy, deﬁned as
the energy released when two Pd monomers on the
MgO(1 00) terrace form a dimer, is found to be 0.54 eV.
The second most stable dimer, labeled as D2, has two Pd
adatoms adsorbed on next nearest neighboring oxygen
sites, with a Pd–Pd distance of 3.03 A
and a Pd-surface
plane distance of 2.10 A
. The total adsorption energy for
D2 is 2.96 eV, which is 0.22 eV larger than the adsorption
energy of the monomer.
The next most stable conﬁgurations, D3 and D4 (see
Fig. 2), both have adsorpt ion energies of 2.73 eV, within
0.01 eV of that for two separate monomers. The Pd–Pd
interactions are weak enough that they can be considered
as two non-interacting monomers. This is consistent with
the fact that the monomer adsorption energy can be found
accurately with a periodic substrate containing only four
Mg and four O atoms per layer in each unit cell [42,43].
Finally, we considered a vertical dimer in which the two
Pd atoms are arrange d perpendicular to the surface above
an oxygen site. Although this geometry is found to be a
(shallow) local minimum, it is almost an eV less stable than
D1, and is not relevant to ripening dynamics.
The dimer binds at an oxygen vacancy with an adsorp-
tion energy of 5.94 eV and a geometry similar to D1 with a
missing oxygen ion beneath one of the Pd atoms. The di-
mer binding energy at the defect, relative to two mono-
mers, one on an F-center, is 0.57 eV. This is somewhat
higher than the 0.37 eV binding energy found using the
B3LYP functional (which is expected to be more accurate
than the PW91 functional used in this study) .
3.3. Dimer diﬀusion
To determine the mechanisms of dimer diﬀusion, we did
a systematic study of saddle points between the Pd dimer
conﬁgurations described above. The most relevant three
such trans itions are shown in Fig. 3. The lowest barrier
of 0.43 eV corresponds to the partial dissociation of D1
to D2 (D1-2), by a single Pd hopping mechanism. This pro-
cess has a prefactor of 2.5 · 10
. The reverse barrier is
0.1 eV and has a prefactor that is 60 times larger. Fig. 4
shows how two such processes lead to net diﬀusion through
a partially dissociated intermediate structure.
D1 can also diﬀuse by a concerted slide mechanism (D1-
1) along the dimer axis with a barrier of 0.60 eV, and it can
dissociate to D3 (two isolated monomers) with a barrier
(D1-3) of 0.84 eV.
Trimer conﬁgurations are constructed by adding one Pd
atom to the D1 and D2 dimer conﬁgurations. The lowest
energy trimer, labeled as T1 in Fig. 5, is constructed by
adding a Pd atom on the bridge between the atoms in
D1. Remarkably, this vertical quasi-equilateral triangle
structure with the Pd–Pd distances of 2.52 A
has the largest
adsorption energy of 5.69 eV (1.89 eV/atom) relative to 3
Pd atoms in gas phase . The second most stable trimer,
T2, with an adsorption energy of 5.42 eV (1.81 eV/atom)
can be constructed in a similar manner as T1, using
D2 as the base. The Pd bonds in T2 are stretched to
between the base atoms, and 2.50 A
top atom and the base atoms. T2 is less stable than T1 in
the same way that D2 is less stable than D1. The binding
D1: 3.28 D2: 2.96 D3: 2.73 D4: 2.73
Fig. 2. The four most stable dimer conﬁgurations, and adsorption
energies in eV. The adsorption energy of D3 and D4 is the same,
indicating that the monomers in both conﬁgurations are separated beyond
their interaction distance.
0 1 2 3 4
Reaction Coordinate (Å)
Fig. 3. Pd dimer diﬀusion pathways.
1354 L. Xu et al. / Surface Science 600 (2006) 1351–1362
energy of T1 (relative to D1 plus a monomer) on the ﬂat
MgO(1 00) terrace is 1.06 eV.
Trimer conﬁgurations lying ﬂat on the MgO(1 0 0) sur-
face are constructed by placing a Pd atom next to the D1
and D2 dimers. The most stable ﬂat timer, T3, has a Pd
atom next to D1 forming a triangle on oxygen sites with
two Pd–Pd bond lengths of 2.72 A
and one stretched bond
of 3.02 A
. T3 has an adsorption energy of 5.25 eV (1.75 eV/
atom), which is 0.44 eV less stable than the lowest energy
vertical trimer, T1; similarly, vertical Pd trimers have been
also predicted to be 0.4 eV more stable than ﬂa t ones in a
recent study of Pd
The fourth trimer conﬁguration , T4, is simply a linear
structure, which can be constructed by adding a Pd atom
to the end of the D1 dimer conﬁguration. The Pd–Pd bond
lengths are 2.74 A
and the energy is 5.18 eV (1.73 eV/
atom). This T4 trimer can partially dissociate to the bent
trimer (T5) or fully dissociate to a monomer plus a dimer
(i.e., structure T6) in a similar mechanism to the D1-2
and D1-3 processes shown in Fig. 3.
A Bader analysis of T1 shows that 0.1 e charge transfers
from the MgO substrate to each of the two Pd atoms in
contact with the substrate, while the extra charge on the
top Pd atom is smaller (0.06 e).
Adsorption of trimers on F-centers has been studied by
Giordano et al. with a very similar model. The two most
favorable trimer conﬁgurations were found to be almost
degenerate in energy. One structure is a tilted vertical tri-
mer similar to T3 with one of two oxygen atoms being re-
moved to form the F-center, and the other is a bent linear
trimer similar to T4 with the oxygen atom beneath the cen-
tral Pd atom missing to form the F-center . Our calcu-
lations conﬁrm this result and predict an adsorption energy
of 8.08 eV. The trimer binding energy (i.e., the energy of
the T1/F-center complex minus that of the D1/F-center
complex plus the monomer on a terrace) is found to be
0.76 eV; very similar to the value of 0.75 eV found previ-
3.5. Trimer diﬀusion
The lowest energy migration pathways for the Pd trimer
are shown in Fig. 6. The most stable trimer (T1) diﬀuses via
a walking mechanism similar to the D1-2 dimer diﬀusion
shown in Fig. 3. The topmost Pd atom remains above
0 1 2 3 4 5 6
Reaction Coordinate (Å)
Min.SaddleInitial Saddle Final
Fig. 4. Dissociation and complete diﬀusion mechanism of the Pd dimer.
T4: 5.18 T5: 4.84
T3: 5.25T2: 5.42T1: 5.69
Fig. 5. Most stable Pd trimer conﬁgurations: E
0 1 2 3 4 5
Reaction Coordinate (Å)
Fig. 6. Pd trimer diﬀusion pathways.
L. Xu et al. / Surface Science 600 (2006) 1351–1362 1355
and between the base dimer during its lowest energy migra-
tion process (T1-2). The barrier for this step is 0.48 eV,
which is similar to that found for dimer diﬀusion
(0.43 eV); the barrier for the reverse process (T2-1) is only
0.21 eV. Fig. 7 (lower pathway) shows how the walking
mechanism can lead to net diﬀusion of the trimer.
A second, ﬂipping mechanism, has a very similar barrier
to the walking mechanism: T1 can diﬀuse via the ﬂat con-
ﬁguration T3 (see Fig. 6). By crossing a slightly higher bar-
rier of 0.50 eV, the top atom from T1 can ﬂip down to the
surface, forming T3, from which a diﬀerent atom can lift
onto the other two forming T1 in a diﬀerent location (see
Fig. 7, process T1-3-1). The reverse step has a small barrier
of only 0.055 eV, so that the overall barrier for diﬀusion by
this ﬂipping mechanism is only 0.50 eV. In a similar way,
T3 can diﬀuse via the T2 vertical conﬁguration (process
T3-2), crossing the barrier of 0.18 eV or the overall barrier
of 0.63 eV with respect to the lowest energy T1 structure.
The T1-3 (ﬂipping) and T1-2 (walking) mechanisms
have similar barriers (0.50 eV and 0.48 eV respectively). A
higher prefactor of 1.1 · 10
for the ﬂipping mecha-
nism as compared to 5.4 · 10
for the walking mecha-
nism, means that the ﬂipping mech anism will dominate
trimer diﬀusion at high temperature; in fact, it is even
slightly favored at liquid nitrogen temperature (77 K).
The ﬂat T3 trimer can change conﬁguration via a dou-
ble-slide mechanism in which two Pd atoms slide across
hollow sites along the short edge of the ﬂat triangle (pro-
cess T3-3). The barrier for this mechanism is 0.25 eV with
respect to the ﬂat trimer T3, and 0.69 eV with respect to
the lowest energy trimer T1. Not only is the overall barrier
very high compared to T1-2 and T1-3 but also this mecha-
nism alone does not lead to trimer diﬀusion: the trimer re-
mains trapped in a square of ﬁve oxygen binding sites.
Finally, the higher energy ﬂat trimer structures (T4, T5
and T6 in Fig. 5) can diﬀuse on the MgO(100) surface
without forming the low-energy vertical structure T1 and
T2. These processes (including T3-5, T4-5, T5- 5 and T4-6
shown in Fig. 6) involve single Pd atom hops between oxy-
gen sites with barriers in the range 1.0–1.4 eV with respect
to T1. At temperatures less than 1000 K these high energy
processes will not play signiﬁcant roles due to the over-
whelming stability of the vertical structures.
Similar to the Pd trimer, both 2D (ﬂat) and 3D (tetrahe-
dral) structures are stable. Two tetramer conﬁgurations
were studi ed on a large r MgO unit cell, with 36 atoms
per layer. The tetramer is stable in a ﬂat square planar con-
ﬁguration [53,54] , in which each Pd atom is bound to an
oxygen site. This structure has an adsorption energy of
7.71 eV (1.93 eV/atom), with Pd–Pd bond distances of
and Pd–O distances of 2.28 A
. As with the ﬂat tri-
mer, the ﬂat tetramer can easily convert to a more stable
3D structure. Fig. 8 shows the minimum energy path, in
which a Pd atom in the 2D square planar structure rises
on top of the other three atoms to form a 3D tetrahedron,
while crossing a barrier of only 0.04 eV.
The 3D tetrahedral structure, with an adsorption energy
of 8.80 eV (2.20 eV/atom) is more stable than the 2D pla-
nar structure because it has more Pd–Pd bonds, each of
which is stronger than the Pd–O bonds to the surface.
The Pd–O bond length is 2.23 A
, and the Pd–Pd bond
length is 2.59 A
. The base triangle of Pd atoms is close to
equilateral (angles of 56.5", 56.5" and 67.0") despite the
rectangular MgO(1 0 0) surface. This structure has been
conﬁrmed as the most stable tetramer conﬁguration previ-
ously . The binding energy of the tetrahedron tetramer
is 1.7 eV, i.e., lower by 1.7 eV than the energy of T1 and a
A Bader analysis of the tetrahedron tetramer on
MgO(100) shows some combined features of the ﬂat and
vertical trimers. The two equivalent Pd atoms on the base
each have 0.17 e extra charge, while the third has a smaller
charge of 0.05 e. The top Pd atom also shows a small extra
charge of 0.09 e.
Removing an oxygen atom from below one of the Pd
atoms in the tetrahedron base leads to a tetrahedron/F-
0.0 0.2 0.4 0.6 0.8 1.0
Reaction Coordinate (Å)
Fig. 7. Diﬀusion mechanism of the vertical trimer (T1). Upper insets show
the ﬂipping mechanism (T1 ! T3 ! T1) and lower insets show the
walking mechanism (T1 ! T2 ! T1).
0 1 2 3 4
Reaction Coordinate (Å)
Fig. 8. Low energy barrier for the 2D–3D tetramer transition.
1356 L. Xu et al. / Surface Science 600 (2006) 1351–1362
center complex. The binding energy of the tetramer, as
compared to the T1/F-center complex plus a monomer
on the ﬂat terrace is 1.4 eV.
3.7. Tetramer diﬀusion
Due to the greater stability of the tetrahedron on the
MgO(1 00), and the low 2D–3D transition barrier, the tet-
rahedron conﬁguration will dom inate tetramer dynamics
on the MgO(1 0 0) surface. The lowest energy tetrahedron
diﬀusion paths are shown in Fig. 9.
A qualitatively new type of mechanism for cluster diﬀu-
sion was found with this tetramer, in which the tetramer
rolls to a new binding site on the surface. We call this the
‘‘rollover’’ mechanism. There are two such processes, la-
beled as roll-I and roll-II in Fig. 9, in which a base atom
and the top atom roll over (i.e., rotate around) the axis
formed by the other two base atoms. The higher energy
process, roll-II does not lead to net diﬀusion, but the lower
energy, roll-I process, with a barrier of only 0.42 eV, does.
This rollover mechanism was found to be the low est barrier
pathway for tetramer diﬀusion. Once the tetrahedral tetra-
mer is formed, it will rapidly diﬀuse as a unit by this mech-
anism at room temperature, as shown in Fig. 10. The
1.14 eV barrier (labeled as 2D–3D in Fig. 9) required to
form the ﬂat tetramer from this tetrahedron is highly
Two processes were found which do not, by themselves,
lead to tetramer diﬀusion, but do allow diﬀusion when
combined sequentially. The ﬁrst ‘‘rotation’’ process (rot-
I) with a low barrier of 0.15 eV involves the hop of a single
base Pd atom, and the second (rot-II) with a barrier of
0.51 eV involves the concerted slide of two base Pd atoms.
This second mechanism is analogous to the T3-3 trimer
process, shown in Fig. 6. There is no trimer mechanism
corresponding to the ﬁrst (rot-I) mechanism; the diﬀusing
atom moves on top of the other into the intermediate min-
imum T1. In both cases, the tetrahedron remains trapped
in a square of oxygen binding sites. The combination of
the two mechanisms leads to net diﬀusion, with an overall
barrier of 0.51 eV (see Fig. 11).
Prefactor calculations show that the roll-I mechanism
has a very large prefactor: 1.3 · 10
and the roll-II
mechanism has a prefactor of 1.1 · 10
, while the
rot-I and rot-II mechanisms have prefactors of 2.3 ·
and 1.0 · 10
respectively. Hence, the rollover
(roll-I) mechanism is favored both by energy and entropy.
3.8. Binding at oxygen vacancies
The binding energies of small Pd clusters to a neutral
oxygen vacancy are tabulated in Table 1. Here, the binding
energy for a cluster (Pd
) at an F-center is deﬁned as the
energy released when a smaller cluster (Pd
) from the ter-
race combines with a Pd monomer trapped at the F-center.
The calculations show that adso rbed monomers are
0 2 4 6
Reaction Coordinate (Å)
Fig. 9. Pd tetramer diﬀusion pathways.
0 3 6 9 12 15
Reaction Coordinate (Å)
Fig. 10. The Pd tetrahedron rolls over the MgO(1 0 0) surface with a low
barrier of 0.42 eV.
0 3 6 9
Reaction Coordinate (Å)
Fig. 11. Pd tetramer diﬀusion via two rotation mechanisms.
L. Xu et al. / Surface Science 600 (2006) 1351–1362 1357
strongly trapped at F-centers (by 2.63 eV). A second mono-
mer, however, binds less strongly (0.56 eV). Larger ad-
sorbed clusters (n = 2–4) bind somewhat more strongly to
the monomer/F-center complex, with energies in the range
4.1. Structure of Pd clusters
DFT calculations have been used to understand the
bonding between Pd clusters and the MgO(100) surface.
Our calculations are consistent with previous theoretical
predictions that oxygen sites are favorable for monomer
adsorption [55,50,56,46,43]. Recent in situ Grazing Inci-
dent X-ray Scattering (GIXS) studies also conﬁrmed that
Pd adatoms adsorb at ox ygen anions [25,26], in contrast
to an earlier surface (extended) electron energy loss ﬁne
structure (SEE LFS) study which suggested that the Pd ada-
toms bind to magnesium sites . The Pd-surface distance
for monomers is calculated here to be 2.09 A
with previous calculations [43–46]. Experimental mea-
surements of 2.22 ± 0.03 A
for 1 ML Pd deposite d on
MgO(1 00) [25,26] are somewhat larger, perhaps due to
the weaker Pd–O bonds expected in a Pd monolayer. Sup-
port for this can be seen in calculations of the ﬂat tetramer
and a Pd monolayer, which show a Pd–O bond length of
ca. 2.28 A
, showing that the Pd–O bond length does in-
crease with Pd island size.
The competition between Pd–O and Pd–Pd bonding is
also shown in the evolution of cluster shape from dimers
to tetramers: Pd clusters tend to assume vertical or three-
dimensional structures. Only the dimer is most stable as a
ﬂat (i.e., 2D) island. Experimental measurement s have
found three-dimensional truncated octahedral shape for
larger Pd particles on MgO(1 0 0) [3,29].
As many properties of materials and chemical processes
can be explained directly and indirectly by the (re-)distribu-
tion of electron density within the material or upon to the
change, it is reasonable to look into the charge density in-
volved in the adsorption energetics. Our Bader charge anal-
ysis shows a clear charge transfer between MgO and Pd
adatoms as electron density transfer from oxygen to palla-
dium. This is diﬀerent than most experimental measure-
ments for similar metals on semiconducting oxides such
and ZnO, where the direction of charge transfer
can be assessed . (Such measurements are not possible
on insulators like MgO). This calcul ated charge transfer
direction is also veriﬁed in Mulliken charge analysis of
DFT results for the same system . Repulsion between
the negative charges on the Pd atoms in ﬂat-lying clusters
may contribute to the driving force for trimers to stand ver-
tically on the surface and larger clusters to be 3D.
Our charge analysis also shows that charge transfer is
localized to the Pd–O bond. Pd atoms separated from the
MgO surface, such as the topmost atom in 3D clusters,
are charged by less than 0.1 e. This leads to a possible
explanation as to why 3D structures form so easily; Pd
atoms can maintain their strong Pd–Pd bonds in a 3D
structure, while reducing their electrost atic repulsion from
the O-ions in the surface layer. Alternatively, the frontier
orbitals on these Pd clusters may favor bonding in certain
4.2. Diﬀusion of Pd clusters
Three-dimensional conﬁgurations of Pd clusters larger
than the dimers, on the MgO(100) surface, are both stable
energetically and also easily formed from less stable two-
dimensional conﬁgurations. For example, the transitions
from 2D trimer and tetramer conﬁgurations to the corre-
sponding most stable 3D structures are shown to have bar-
riers of only ca. 0.05 eV. This indicates that 3D structures
cannot be neglected in a model of Pd cluster diﬀusion
We ﬁnd here that Pd clust ers, at least up to tetramer, are
quite mobile on the MgO(1 0 0) terrace at room tempera-
ture. In a mean ﬁeld diﬀusion model , monomer diﬀu-
sion is assumed to be the fastest and most important for
cluster growth. In order to ﬁt this model to AFM studies
of island density, the monomer diﬀusion barrier needs to
be as low as 0.2 eV , so that monomer hopping is a ra-
pid process during the initial stage of island growth above
200 K. Our calculations show that the monomer hopping
barrier is somewhat higher, 0.34 eV, but that larger clusters
are also mobile. The dimer can diﬀuse with a barrier of
0.43 eV through a partial dissoc iation mechanism (see
Fig. 3) and the trimer can diﬀuse with an overall barrier
of only 0.50 eV through either a ﬂipping or walking mech-
anism (see Fig. 7). The tetramer diﬀuses with a barrier of
only 0.41 eV—even lower than the trimer diﬀusion barrier.
This decrease of the diﬀusion barrier from trimer to tetra-
mer adsorbate was also predicted for Al cluster self-diﬀu-
sion on Al(1 1 1) through ﬁrst principle calculations, and
is expected to hold for other metal/metal(1 1 1) systems
. Studies of Au on Al
also indicate that a higher
mobility of small clusters than adatoms must be considered
to match experiment [59,60]. In this work, it is found that
The binding energy (E
) of a Pd
(n = 2–5) cluster at an F-center (neutral
oxygen vacancy site)
Monomer 4.01 1.37 –
Dimer 5.94 3.28 0.56
Trimer 8.08 5.72 0.79
Tetramer 10.85 8.79 1.12
Pentamer 13.63 11.55 0.83
is calculated as E
,F) = E
,F) ! E
,F) ! E
,F) is the adsorption energy of the Pd
cluster on an F-
center relative to the gas-phase Pd atoms and the relaxed MgO(1 00) with
an F-center while E
) refers to the adsorption energy on the ter-
race. The most stable structures for Pd
(n = 1–4) clusters bound to F-
centers were assumed to be those found by Giordano et al. .
1358 L. Xu et al. / Surface Science 600 (2006) 1351–1362
each cluster size, from the monomer to the tetramer, have
similar diﬀusion barriers in the range 0.34–0.50 eV. This
means that entropic eﬀects could be important to distin-
guish the impor tant pathways in a kinetic analysis of
growth and sintering, even at moderate temperatur es.
In order to better understand and evaluate the entropic
eﬀects on the kinetics of Pd cluster diﬀusion, we calculated
harmonic Arrhenius prefac tors (m) for the low energy diﬀu-
sion processes of each small Pd cluster. Prefactors were
evaluated as the product of the stable harmonic mode fre-
quencies of the system at the saddle point divided by those
at the minimum (see Eq. 2). Frequencies were obtained
using ﬁnite diﬀerence displacements of all relaxed atoms
in the system, including the top MgO surface layer. The
importance of surface modes on the reaction prefactors
was tested by ﬁnding prefactors using a reduced Hessian
matrix constructed from ﬁnite diﬀerence displacements of
only the Pd atom(s). The adatom diﬀusion prefactor was
found to be diﬀerent by a factor of 2, and the tetramer
by a factor of 8, indicating that it is important to include
motions in the top MgO layer as we have done here. The
prefactors and barriers are used to calculate rates with
the harmonic form of transitio n state theory, Eq. (2), which
are presented in Table 2. Surprisingly, the prefactors of dif-
ferent processes vary by more than a factor of 2000. This is
quite diﬀerent than recent molecular dynamics simulations
using an analytic potential, which found that the prefactors
for these processes ‘‘all fall in the range of 2 · 10
Due to a high prefactor, the trimer ﬂipping mechanism
(T1-3) has a higher rate than the walking mechani sm
(T1-2) for all relevant temperatures (above 50 K), despite
the fact that the walking mechanism has a slightly lower
barrier. It is remarkable that the tetramer has a lower over-
all diﬀusion barrier than the trimer. This low barrier, com-
bined with a high prefactor, means that the tetramer
diﬀuses faster than the other clusters above 180 K. The
high mobility of clusters as large as tetramers woul d give
an unusual picture of mobile Pd island on the MgO(1 0 0)
surface. In fact, at room temperatur e, our hTST rate con-
stants suggest the tetramers should diﬀuse even faster than
To determine why these prefactors vary by more than
2000-fold, we analyzed the vibrational modes associated
with the minimum and saddle geometries for the pr ocesses
listed in Table 2. We found many soft vibrational modes in
both the minima and saddle geometries, associated with
collective motions of the adsorbed Pd atoms.
For the trimer walking mechanism (T1-2), the initial
state has a very low mode of 11 cm
associated with the
frustrated rotation of the Pd cluster on the MgO surface.
This mode (and other modes) is stiﬀer at the transition
state (40 cm
), resulting in a low prefactor of 5 ·
. (A typical prefactor for diﬀusion in solid systems
is 5 · 10
). In the trimer ﬂipping process (T1-3) the
modes are comparable in the saddle geometry and the pre-
factor is fairly typical, 1 · 10
The Pd tetramer cluster has one more Pd–O bond than
the trimer, and the collective rotational modes are stiﬀer. In
the saddle geometry for the tetramer rollover process, one
Pd atom loses its bond to the surface and two low rota-
tional modes appear with frequencies of 20 cm
low modes at the saddle, which are not present in the initial
minimum, lead to the very high 1 · 10
4.3. Kinetics of island formation
According to the energetics in Tables 1 and 2, once the
oxygen vacancies saturate with monomers, mobile Pd
monomers accumulate during deposition at 300 K and
above. At high enough concentrations, mobile monomers
will form small clusters of dimers, trimers and tetramers,
instead of getting irreversibly trapped on the monomers
at defects (F-centers).
Given the high mobility of the small clusters at room
temperature in Table 2, one must ask whether it is the dif-
fusion of small clusters instead of mono mer hopping that
dominates mass transport during Pd particle growth, by
moving around and combining with monomers or clusters
already trapped in F-centers (illustrated in Fig. 12). This
would give a very diﬀerent picture of island growth for
Pd/MgO syste m than usually assumed.
When Pd adatoms are ﬁrst deposited on the MgO(100)
surface, they will hop over the surface and get trapped irre-
versibly as monomers in oxygen vacancies. After the vacan-
cies become saturated, additional monomers will bind to
the monomers at vacancies. The dimer binding energy on
an F-center (0.56 eV) is not strong enough to hold the di-
mer together at 300 K and above, so that the dimers will
rapidly dissociate and reform as monomers, diﬀusing from
defect to defect. The binding to another monomer on a ter-
The lowest diﬀusion barriers DE for Pd
(n = 1–4) clusters on MgO(10 0), the corresponding Arrhenius prefactors, m, and harmonic transition state theory
rate k(T) at two temperatures, T , of 200 K and 300 K
Pd cluster (process) DE (eV) m (s
) k (200 K) (s
) k (300 K) (s
Monomer 0.34 7.4 · 10
2000 2 · 10
Dimer (D1-2) 0.43 2.5 · 10
4 2 · 10
Trimer (T1-2) 0.48 5.4 · 10
0.04 5 · 10
Trimer (T1-3) 0.50 1.1 · 10
3 4 · 10
Tetrahedron 0.41 1.3 · 10
3000 1 · 10
L. Xu et al. / Surface Science 600 (2006) 1351–1362 1359
race site is also weak (0.54 eV), so these dimers on terraces
will also dissociate rapidly. Thus, an equilibrium will
quickly establish between mobile species on the terrace
and at defects.
Over time, the islands at F-centers will grow larger.
Above 600 K, trimers and tetrameters are able to break
away from F-centers, leaving behind tightly bound mono-
mers. At these high temperatures, ripening will occur rap-
idly. In this picture, the ﬁnal conﬁgurations will still have
Pd clusters trapped on those F-centers, and will show an is-
land densit y that can be dominated by the defect density.
This helps ration alize the experimental study of the kinetics
of Pd cluster growth on MgO(1 0 0) with defects  , where
the measured island density versus ﬂux and temperature
was ﬁtted to a model involving a single moving species, that
was assumed to be monomer. The binding energy of that
moving species above 600 K was found to be 1.2 eV .
This is clearly too large compared to the 0.56 eV calculated
here for a dimer at a vacancy dissociating into a monomer
at the vacancy and a diﬀusing monomer on a terrace, which
was originally assumed to be the process associated with
this 1.2 eV diﬀerence. Instead, this 1.2 eV binding energy
compares much more favorably to the binding energy of
bigger clusters binding energy on neutral oxygen vacancy
sites which is found here by DFT to be ca. 1.0 eV (see
Further study needs to be carried on to show how well
the energetics and preex ponential factors found here agree
with the experimental observations. This will require kinet-
ics simulations, which are ongoing in this lab.
The higher mobility of clusters larger than dimers, and
the low adsorption energies of all clusters (less than
2.3 eV/atom) relative to the bulk cohesive energy of solid
Pd (3.9 eV/atom) suggest a Volmer-Weber (or 3D) growth
mode for Pd atoms on a perfect MgO(1 0 0) surface. This
can be ascribed to the stronger Pd–Pd interactions as com-
pared with Pd–MgO interactions. A 3D growth mode is
consistent with experiments at temperatures between 400
and 600 K [61,62,25].
This prediction of 3D growth at room temperature
should be taken with caution since there may be more com-
plicated defects on real MgO(1 0 0) surfaces. Strong trap-
ping eﬀect from those defective sites may reduce the
small clusters’ mobility, as suggested by the studies of Pd
atoms bonded on defective MgO(100) [56,43,36,51]. There-
fore, island growth on real MgO(1 0 0) surfaces with defects
could be very complicated and 2D isl ands at low coverage
and low temperatures cannot be completely ruled out. Fur-
ther study is needed, of the type presented in the molecular
dynamics simulation of Pd island growth on MgO(100)
, but a long time scale simulation seems to be more
appropriate to disclose the predicted formation of large is-
lands and even the growth of overlayers  .
Formation and diﬀusion of small Pd clusters up to tetra-
mers on MgO(100) has been studied using DFT combined
with the NEB and dimer saddle point ﬁnding methods. The
oxygen sites are conﬁrmed to be the lowest energy adsorp-
tion sites for Pd adatoms and larger Pd clusters have a
strong tendency to associate their ﬁrst-layer Pd atoms clo-
sely with these sites but also to coalesce into 3D structures.
The diﬀusion of Pd clusters involves stretching transitions
for small clusters and 2D–3D or rollover type transitions
for larger clusters. The diﬀusion barriers coupled with har-
monic prefactors suggest a high mobility for large clusters,
reﬂected by the faster diﬀusion for tetramers than trimers,
dimers or monomers at room temperature. These results
challenge the traditional view of monomer-dominated is-
land growth. Vibrational mode analysis rationalizes the
large variation in harmonic Arrhenius prefactors, which
arise due to the strong er Pd–Pd bonding than Pd–O bind-
ing and low energy modes of cluster rotation with respect
Traditional model in which
only monomers are mobile
Ab initio based model
with mobile small clusters
T < 600 K
T ≈ 600 K
T > 600 K
Fig. 12. Cartoon of the expected growth in the traditional model as
compared to a model based upon the present DFT energy landscape.
Arrows indicate typical events that are active in the diﬀerent models. (A)
At low coverage and at temperatures between 200 and 600 K, monomers
diﬀuse rapidly on the surface. In the traditional model, monomers bind to
other atoms or defects irreversibly at this temperature. The DFT based
model has a weak dimer binding energy so that monomer will saturate the
defect sites before larger clusters form in high concentration. Monomers
and dimers which form on the terrace or at defect sites continue to be
mobile and diﬀuse or dissociate until larger clusters form. (B) At ca. 600 K
ripening occurs. In the traditional model, ripening is associated with
activation of dimer dissociation. In the DFT-based model, monomers and
dimers are free to move or dissociate and thus quickly form larger clusters
at defects. At ca. 600 K, trimers and tetramers have enough energy to
dissociate from monomer-bound defects, thus rapidly forming even larger
and more stable clusters at defects. (C) Above 600 K the traditional model
predicts that monomers can leave defects Pd-free, and that large 2D
clusters form, until at elevated temperatures, the more stable 3D clusters
can ﬁnally form. DFT, however, predicts that monomers will not leave
defects at temperatures below 1000 K. Small clusters, however, leave
monomers behind at defects, and diﬀuse rapidly to coalesce into large 3D
clusters at defects.
1360 L. Xu et al. / Surface Science 600 (2006) 1351–1362
to the surface. The calculations support a Volmer-Weber
or 3D island growth mode for Pd on the perfect
MgO(1 00) surface, given the greater stability of 3D than
2D clusters, even for the trimer and tetramer. The stability
of 3D clusters can be rationalized by charge density analy-
sis of the adsorbed Pd clusters.
The authors a ppreciate useful discussions with Nu
pez, Gianfranco Pacchioni and Arthur Voter. This work
was supported by the National Science Foundation, award
No. CHE-0111468, by the Department of Energy OBES-
Chemical Sciences, and by the Robert A. Welch Foun-
dation, grant No. F-1601. LX thanks the Center of
Nanotechnology in the University of Washington for the
UIF Fellowship support on this project . We are grateful
to the Texas Advanced Computing Center and the MSCF
computer facility at the Paciﬁc Northwest National Labo-
ratory for use of their computational resources.
Appendix A. Convergence
The binding of a Pd atom on the MgO surface has been
calculated using a range of computational parameters. Ta-
ble A.1 shows the sensitivity of our calculations to increas-
ing the accuracy of the calculation. The typical errors, on
the order of 10 meV, indicate that the calculation is con-
verged with respect to each parameter.
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Sensitivity of the Pd monomer binding energy to calculation parameters
Pd adsorption energy (eV) System/parameter change
1.375 Reference system
1.356 2 · 2 · 1 Monkhorst-Pack k-point mesh
1.376 396 eV plane wave cutoﬀ
1.380 36 atoms per layer in the substrate
1.380 4 MgO layers with two relaxed
1.375 20.5 A
vacuum gap between images
1.380 Spin polarization
1.370 Dipole correction
Each parameter was tested by setting it more accurate than in a suﬃciently
converged reference system. The converged reference system for monomer
adsorption has a unit cell with 24 substrate atoms per layer, 3 layers with a
relaxed top layer, a 12.5 A
vacuum gap between periodic images of the
MgO slab, a soft oxygen pseudo-potential with an appropriate 270 eV
plane wave cutoﬀ, C point sampling of the Brillouin zone, and no spin-
polarization (except for tetramer calculations) or dipole correction.
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