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Proc. R. Soc. A (2011) 467, 2020–2042
doi:10.1098/rspa.2011.0009
Published online 18 March 2011
Atomistic and electronic structure of (X2O3)n
nanoclusters; n = 1–5, X = B, Al, Ga, In and Tl
BY SCOTT M. WOODLEY*
Department of Chemistry, Kathleen Lonsdale Materials Chemistry, University
College London, 20 Gordon Street, London WC1H 0AJ, UK
The stable and metastable, as measured using an all-electron density functional theory
approach, stoichiometric clusters of boron, aluminium, gallium, indium and thallium
oxide are reported. Initial candidate structures were found using an evolutionary
algorithm to search the energy landscape, defined using classical interatomic potentials,
for alumina and india followed by data mining or rescaling. Characterization of the
refined structures was performed by electronic structure techniques at the hybrid density
functional and many-body GW levels of theory. We make accurate predictions of the
spectroscopic properties represented by mean ionization potentials of 11.4, 9.9, 9.8, 8.8
and 8.4 eV and electron affinities of 0.05, 1.1, 1.6, 1.9 and 2.5 eV for boria, alumina,
gallia, india and thallia, respectively. The changes in the global minima, atomistic and
electronic properties with respect to the cluster and cation size are discussed.
Keywords: nanoclusters; density functional theory; atomistic structure; electronic structure;
sesquioxides; global optimization
1. Introduction
Sesquioxides span a wide range of physical and chemical properties and are
used as catalysts, lasing and more generally light-emitting materials, elements
of complex photoelectric devices, etc. (Catlow et al. 2010). In particular, india is
widely used as a transparent conducting oxide in modern optoelectronic devices,
ranging from solar cells to flat-panel displays (Edwards et al. 2004), whereas
alumina is not only important in many industrial applications (Hart 1990), in
particular as a support for catalysts, but also as particles in rocket exhaust where
it is possibly responsible for the depletion of stratospheric ozone (Robinson et al.
1997; Fernandez et al. 2003). Questions arise as to the nature of their physicochemical properties below the bulk regime. Moreover, is it possible to exploit the
size–property relationship to further enhance useful properties? The difficulty in
structure and property characterization below 5 nm presents an ideal opportunity
for theoretical guidance and prediction. In this regime, strong deviations are
expected from bulk-like local coordination, and quantum confinement effects will
*[email protected]
One contribution of 16 to a Special feature ‘High-performance computing in the chemistry and
physics of materials’.
Received 6 January 2011
Accepted 9 February 2011
2020
This journal is © 2011 The Royal Society
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Structure of sesquioxide nanoclusters
2021
be greatly enhanced. Low-energy atomic structures of (X2 O3 )n nanoclusters are
therefore investigated for n = 1–5 and X = B, Al, Ga, In and Tl. There is already
a number of published papers on clusters of alumina (Ying et al. 1993; Fernandez
et al. 2003; Van Heijnsbergen et al. 2003; Demyk et al. 2004; Karasev et al. 2004;
Song et al. 2004; Fernandez et al. 2005; Patzer et al. 2005; Neukermans et al. 2007;
Sierka et al. 2007; Feyel et al. 2008; Santambrogio et al. 2008; Sun et al. 2008;
Sierka 2010), gallia (Deshpande et al. 2006) and for india (Walsh & Woodley
2010). The atomic structure of small india clusters was predicted by refining
plausible low-energy structures that were generated by an evolutionary algorithm
(EA) search over the energy landscape. Rather than a direct search of the
density functional theory (DFT) energy landscape (Santambrogio et al. 2008; Doll
et al. 2010), classical interatomic potentials were used during the initial global
searches and DFT during the local refinements. The same efficient approach of
switching energy functions is applied here. The initial cluster structures for all
five compounds were generated by scaling the low-energy local minimum (LM)
structures, for both alumina and india, data mined from our initial global searches
and from the literature. Importantly, the relaxation of scaled clusters can lead to
new LM configurations not yet found, even when the configurations of the new
compounds are expected to be similar to those of the original compound (Sokol
et al. 2010; Woodley et al. 2010). The spectra of the electronic levels are then
generated for the all-electron DFT refined, lowest energy structures.
2. Methodology
(a) Energy of formation
It is assumed that the configuration adopted by the cluster (X2 O3 )n is that which
minimizes its energy of formation. As in previous work (Walsh & Woodley 2010;
Woodley et al. 2010), four different energy functions are employed, each assumed
to be better than the previous and, importantly, more expensive to compute.
The first energy function is based on the rigid-ion model (RM), where each
cation and anion is represented as a point charge of +3|e| and −2|e|, respectively,
and the analytical interatomic potential (IP),
qi qj
−rij
A
C
+ 12 + B exp
(2.1)
− 6,
V (rij ) =
rij
r
rij
rij
is used to model the interaction between ions i and j, which are distance rij apart.
The first term is simply the Coulomb contribution (in electronvolts) between
point charges qi and qj , and the latter terms are the standard Lennard-Jones
and Born–Mayer terms. For the RM, the energy of formation is simply the sum
of all two-body interactions defined by equation (2.1). The values for B and C
are taken from earlier work where the bulk phases are modelled (Ooms et al.
1988; Walsh et al. 2009). We have included the r −12 term to penalize interaction
distances that are less than a typical bond length (A = 1.0 eV Å12 for like-charged
species and 10.0 eV Å12 otherwise). In order to represent electronic polarization of
oxygen, the Shell model (SM; Dick & Overhauser 1958) is employed (our second
energy function). In the SM, each point charge from the RM is replaced by two:
Proc. R. Soc. A (2011)
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S. M. Woodley
one represents the nucleus and core electrons, and the other, which is connected
via a harmonic spring to the first, the centre of charge for the shell of valence
electrons. Typically, the last three terms in equation (2.1) act between the shells,
and the first term does not act between two point charges of the same ion. Both
models are implemented within the GULP code (Gale & Rohl 2003).
For the third and fourth energy functions, we use an all-electron DFT method
with a local numerical orbital basis set, as implemented in the FHI-AIMS code
(Blum et al. 2009; Havu et al. 2009). The FHI-AIMS two-tiered basis set is
employed for the oxygen and boron atoms, and only the one-tier basis set for the
larger cations, as they have a stronger cation nature, which provides an accuracy
of −5.39, −2.20, −10.63, −26.19, −24.61 and −15.24 meV per atom for O, B, Al,
Ga, In and Tl, respectively. Within the current approach, the eigenstates obtained
from the solution of the Kohn–Sham equations are Gaussian smeared with a
0.05 eV width. For our third definition, the PBEsol functional (Perdew et al. 2008)
is chosen as it is unbiased and is not too computationally expensive, while scalarrelativistic effects are treated at the scaled zero-order relativistic approximation
(ZORA) level (Vanlenthe et al. 1994). In contrast, for our fourth definition,
the hybrid PBEsol0 functional and the full ZORA level of theory (Vanlenthe
et al. 1994) are employed. The hybrid PBEsol0 functional replaces 25 per cent of
the PBEsol electron exchange with exact Hartree–Fock-like exchange (Ernzerhof
& Scuseria 1999; Perdew et al. 2008), where the effective exchange–correlation
functional becomes
PBEsol0
= 34 ExPBEsol + 41 ExHF + EcPBEsol .
Exc
(2.2)
Furthermore, quasi-particle corrections to the PBEsol0 Kohn–Sham eigenvalues
(3nPBEsol0 ) are included through the GW approximation (Hedin 1965; Hybertsen &
Louie 1985), which combines the single-particle Green function (G) with the
dynamically screened Coulomb interaction (W). The resulting quasi-particle
energies (EnQP ) emerge as a first-order perturbation, calculated using Kohn–Sham
molecular orbitals (jn ),
EnQP ≈ 3nPBEsol0 + jn |GW(3nPBEsol0 ) − VxcPBEsol0 |jn ,
(2.3)
which have a direct correspondence with photoemission (N − 1 states) and inverse
photoemission (N + 1 states) or, in the case of the highest occupied and lowest
unoccupied molecular orbital (HOMO and LUMO), levels of the nanoclusters of
interest, measures of the respective ionization potentials and electron affinities.
The GW approximation has had recent successes in describing both the band
structure and defect properties of metal-oxide systems (Jiang et al. 2009; Rinke
et al. 2009).
Throughout the paper, we will use ‘RM’, ‘SM’, ‘PBEsol’ and ‘PBEsol0’ to
indicate which of the four definitions, given above, is employed.
(b) Generating the stable and metastable configurations
The initial aim is to obtain a set of low PBEsol0 energy clusters, including the
global minimum (GM), formed from stoichiometric (X2 O3 )n units.
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Structure of sesquioxide nanoclusters
2023
First, we employ an EA technique (Al-Sunaidi et al. 2008; Woodley et al.
2010), implemented in the GULP package (Woodley et al. 1999), to search
for low-energy local minima (LM) on the RM energy landscapes of (Al2 O3 )n
and (In2 O3 )n . Here, we assume that the set of lowest PBEsol0 energy LM
structures can be readily obtained using standard local optimization techniques
to refine the low-RM energy geometries obtained from the EA. For each of
the two cations and each n, at least 10 runs of the EA is performed, each
starting from a different population of 30 cluster structures. The initial cluster
structures are composed of 2n cations and 3n oxygen anions placed randomly
within a sphere of radius 8.0 Å. Competition between structures within the
current population is simulated in order to determine which structures procreate
(phenotype crossover and mutation operators used to generate 30 children
structures from 30 parents in each EA cycle) and also which survive into
the next EA cycle (where the population consists of only the 30 best unique
cluster structures). A Lamarckian version of the EA is used, i.e. structures
generated in each of the 1000 EA cycles are subject to an immediate local
optimization, in which both LM and saddle points on the RM energy surface
are generated. The probability of a structure winning (either for procreation
or survival) is based purely on a comparison of its relaxed RM energy and
that of the other clusters in the current population. More details of our
EA implementation to generating stable and metastable low-energy cluster
structures can be found elsewhere (Woodley 2004; Woodley et al. 2004, 2010).
We also note that the india LM cluster configurations found from searching
over the (In2 O3 )n landscapes have been previously published elsewhere (Walsh &
Woodley 2010).
As shown in figure 1, the 10 lowest RM energy candidate structures from each
EA run are refined using the SM and standard local optimization techniques,
which are implemented within GULP (Gale & Rohl 2003). At the end of
this second stage, we have a set of stable and low-energy metastable cluster
structures for (Al2 O3 )n and (In2 O3 )n , i.e. the approximate configurations to be
used as initial geometries in an electronic-structure-based approach. Moreover,
as the cluster structures predicted for alumina are found to be similar to those
predicted for india, rather than applying the EA to the other compounds of
interest (boron, gallium and thallium sesquioxides), we simply rescaled our
approximate configurations so that sensible bond lengths (for B–O, Ga–O and
Tl–O, respectively) are obtained. Data mining our results (Sokol et al. 2010) for
alumina and india is far quicker than employing global optimization techniques
for the three other compounds, although we should be aware that important LM
may be missed for larger clusters (higher values of n), especially those of the end
member (B2 O3 )n , where the character of the bonding may differ.
The atom positions of the scaled approximate configurations are relaxed so
as to minimize the PBEsol energy—again we use standard local optimization
techniques, but this time as implemented within the FHI-AIMS code (Blum
et al. 2009; Havu et al. 2009). Consider the PBEsol energy landscape. Ideally,
to make the comparisons between compounds, for each configuration we would
like all five relaxed energy cluster structures. However, if the target configuration
corresponds to an LM within a basin, B, that has a small width, then the
target LM may be difficult to find, e.g. the PBEsol energy point of the initial
approximate configuration may lie outside B and structural relaxations distort
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S. M. Woodley
random configurations
data-mine literature
1
evolutionary
algorithm
[(In2O3)n IP landscape]
low IP energy
(Al2O3)n clusters
low IP energy
(In2O3)n clusters
3
local optimization
(Al2O3)n and (In2O3)n IPs
ranked by IP energies
clusters of (Al2O3)n and (In2O3)n
ranked by hybrid DFT energies
clusters of (B2O3)n , (Al2O3)n,
(Ga2O3)n, (In2O3)n and (Tl2O3)n
+
electronic spectra
fianal calculations
(hybrid DFT energy + GW)
2
construction
new structures
1
evolutionary
algorithm
[(Al2O3)n IP landscape]
4
rescale each cluster to produce
suitable X–O bond lengths for each compound
+
5
minimise energy (DFT local optimization)
5
(B2O3)n, (Al2O3)n, (Ga2O3)n, (In2O3)n and (Tl2O3)n
final DFT optimized configurations
Figure 1. Schematic of how (X2 O3 )n clusters are found. Note that rectangles represent
inputs/outputs of different codes. (1) Our in-house version of the GULP (Woodley et al. 1999;
Al-Sunaidi et al. 2008), (2) Materials Studio (Accelrys Software Inc.), (3) our in-house code
PREGULP (Walsh et al. 2011) and GULP (Gale & Rohl 2003), (4) a simple script and (5)
FHI-AIMS (Blum et al. 2009; Havu et al. 2009). (Online version in colour.)
the cluster towards a different LM. Moreover, the target configuration may not
be stable (or metastable) for all compounds. Where the final structure does not
resemble the initial approximate configuration, we: (i) tried rescaling the PBEsol
minimized cluster structure of the neighbouring compound (where target LM has
been successfully obtained) and (ii) added the new configuration to the set of
approximate configurations if not already included (and thus also targeted for
the other compounds). Rescaling in (i) also reduces the number of optimization
steps.
Finally, to improve the energies and the respective electronic structures, we
recalculated these for each PBEsol-relaxed configuration using PBEsol0 (single
point) and the GW approximation. From the PBEsol0 energies, the smaller (n = 1
or 2) configurations are labelled ‘nR’, where R is the rank, starting from a (the
GM) and increasing alphabetically, of the configuration as predicted for the midsized cation members, (Ga2 O3 )n . For larger configurations, n > 2, R is composed
of either one or four letters that indicate the rank of the cluster for boron oxide
or the ranks for the other compounds, i.e. 5abdc indicates that the n = 5 cluster is
the GM for alumina, ranked second for gallia, ranked fourth for india and ranked
third for thallia.
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(a)
(b)
(c)
1
5.0
1c
4.0
–1
–3
2.0
–5
1.0
1b
–7
0
–9
–1.0
–11
1a
–2.0
–3.0
B
Al
Ga
In
Tl
B
Al
Ga
In
–13
Tl
EHOMO (eV)
energy (eV)
3.0
ELUMO (eV)
2025
Structure of sesquioxide nanoclusters
Figure 2. The relative stabilities compared with configuration 1b (PBEsol unconnected, PBEsol0
connected data points in (a)) and the highest occupied and lowest unoccupied molecular orbitals
(PBEsol0 smaller, GW larger data points in (c)) of the three (X2 O3 )1 configurations (b), for X = B,
Al, Ga, In and Tl. Squares, triangles and diamonds correspond to data points for configurations
1a, 1b and 1c, respectively. (Online version in colour.)
3. Results and discussions
(a) The (X2 O3 )n clusters for B2 O3 , Al2 O3 , Ga2 O3 , In2 O3 and Tl2 O3
The three saddle-point configurations shown in figure 2b were generated for
(X2 O3 )n clusters of size n = 1. The order of stability (or rank) as measured by
PBEsol0 is the same for each cation (figure 2a), i.e. configuration 1a is the GM,
1b the next lowest energy LM and 1c the highest energy configuration. Ignoring
alumina, as the (Al2 O3 )1 configuration 1b is the PBEsol-GM configuration rather
than 1a, the energy differences between the three configurations increases when
PBEsol0 is employed rather than PBEsol. The energy difference is also greatest
between the (B2 O3 )1 configurations, then (Tl2 O3 )1 , i.e. there is a greater stability
of the 1a configuration (compared with 1b and 1c) for the end members. The
PBEsol0 energies for these smallest stoichiometric clusters suggest that shorter
average bond distances are preferred as opposed to a higher average coordination
of the constituent atoms (figure 3).
As reported previously for (In2 O3 )1 (Walsh & Woodley 2010), cluster 1a
relaxes to a linear configuration if an RM is employed, i.e. if polarization effects
are ignored. As polarizability of an ion increases with increasing ionic radius,
we would expect and do find that the X–O–X and O–X–O angles, a and b,
respectively, increase towards 180◦ as the size of the cation decreases. A linear GM
configuration is found for both (B2 O3 )1 and (Al2 O3 )1 . The latter is in agreement
with B3LYP calculations (Sun et al. 2008), which also suggests that larger sized
alumina clusters adopt fullerene structures (which is our finding for boria, and not
alumina—see below). However, a different DFT study suggests that the alumina
cluster should have a = 180◦ (Song et al. 2004).
The 1b clusters are composed of a tetragonal ring and an additional oxygen
atom attached to one of the cations (forming an additional stick). The tetragonal
ring is the (XY)2 GM structure for many XY compounds, for example, (ZnO)2
(Al-Sunaidi et al. 2008), (ZnS)2 (Hamad et al. 2009) and (MgO)2 (Roberts &
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S. M. Woodley
(a)
(c)
180
a
z
3.0
y
g
2.5
y
x
2.0
360º–b
z
1.0
y
y
x
y
x
x
B
Al
Ga
In
Tl
360º–a
1.5
b
x
b
160
b
360º–a
distance (Å)
(b)
y
x
angle (º)
3.5
g
140
120
a
a
a
100
80
b
b
60
B
Al
Ga
In
Tl
Figure 3. (a) Interatomic distances (x, y and z) and (b) bond angles (a, b and g) for (X2 O3 )1
configurations 1a (diamonds), 1b (triangles) and 1c (squares) and X = B, Al, Ga, In and Tl:
(a) Distance (Å) and (c) angle (◦ ). In the stick-and-ball models, the darker spheres represent
the oxygen anions and the lighter ones the cations. (Online version in colour.)
Johnston 2001), whereas the additional stick is the trivial GM for (XY)1 . The
observation that a (X2 O3 )1 cluster can be constructed from a stable (XO)2
cluster provides a possible method for generating (X2 O3 )n clusters from (XO)2n
clusters, i.e. one can construct (X2 O3 )n clusters from (ZnO)2n global minima
configurations found from a previous study (Al-Sunaidi et al. 2008) and decorate
the outer cations with the remaining n oxygen anions (with a maximum of one
additional oxygen anion for each cation) to create potentially low-energy (X2 O3 )n
configurations. Below, such clusters will be noted.
Of the 1b configurations, (In2 O3 )1 has a tetragonal ring that is most like a
square: O–In–O and In–O–In angles are a = 92.2◦ and b = 89.7◦ . Crudely, these
angles gradually change with the size of the cation: for the smallest cation size of
boron, a = 134.0◦ and b = 68.9◦ , and for the largest cation of thallium, a = 82.0◦
and b = 96.0◦ . A similar trend is found for configuration 1c: as the rate of change
of interatomic distance X–X is more rapid than X–O (as clearly seen in figure 3a),
a decreases and b increases with cation size. In contrast, the O–X–O angles, g,
formed by the additional oxygen anion, are approximately 136◦ for all compounds
(which would be 120◦ if there was no second cation). The sum of a and b is not
expected to equal 180◦ due to the presence of this additional oxygen anion. There
are some discrepancies from these trends if we look more closely at the values of
a or b of 1b for alumina and gallia. Although the respective angles were similar,
which relates to the similar sizes of aluminium and gallium cations, a(Al) is in
fact less than a(Ga), and b(Al) is predicted to be greater than b(Ga).
The HOMO and LUMO energies are shown for the (X2 O3 )1 configuration in
figure 2. For each compound, the between these energies, i.e. EGAP = ELUMO −
EHOMO , are greatest for the GM configuration 1a. With the exception of B2 O3
configuration 1b, EGAP as measured using PBEsol is quite small for the metastable
configurations and so smearing occupation of eigenlevels was problematic (ideally,
smearing should be switched off). However, such problems do not occur when the
GW approach is employed as EGAP increases for all configurations, with HOMO
becoming more stable and ELUMO increasing. Note that HOMO is degenerate for
the linear arrangement of configuration 1a.
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Structure of sesquioxide nanoclusters
2d
2e
2l>
2a
2b
2f<
2c
2l<
2h<
2h>
2g<
2m<
2k
2f>
2g>
2i
2j
2m>
Figure 4. Ball-and-stick models of stable, metastable and possibly stationary (X2 O3 )2
configurations, where < and > in the labels indicate the configuration for the smaller and larger
cations, respectively. (Online version in colour.)
(a)
(b)
i
3.2
2
B
m
2.3
–2
–4
Tl
In
–6
Ga
–8
–10
–12
l
1.4
energy (eV)
energy (eV)
0
k
e,f
–0.4
–1.3
Al
j
0.5
dh
g
c
a
–2.2
a b
c
d
e f g h i j k
configuration (X2O3)2
l m n
b
–3.1
B
Al
Ga
In
Tl
X , the binding energy of
Figure 5. PBEsol0 binding energies for (X2 O3 )2 relative to: n lots of E1a
n = 1 GM configuration 1a, in (a), and Ē X , the average of EiX , in (b). (Online version in colour.)
(b) The (X2 O3 )2 clusters for B2 O3 , Al2 O3 , Ga2 O3 , In2 O3 and Tl2 O3
The lowest energy structures (one per unique configuration) for the n = 2
(X2 O3 )n clusters are shown in figure 4. We have labelled them according to their
rank as found using PBEsol0 for gallia. In figure 5a,
X
,
ÊiX = EiX − nE1a
(3.1)
the PBEsol0 binding energies relative to the energy of n non-interacting 1a GM
configurations, are shown for each configuration i. Missing data points correspond
to configurations for which we were unable to find a corresponding local stationary
point on the PBEsol energy hyper-surface, i.e. on local optimization, the
configuration (for example, scaled from that found for a neighbouring compound)
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S. M. Woodley
no longer resembled the initial structure. As already discussed in §2b, the energy
basin containing the LM may either have a very small or zero width, thus making
it hard or impossible to find. For some initial configurations, where structural
changes are probably owing to the cation size, we show more than one relaxed
cluster structure with the same label: the O–X–O–X chain in configuration 2f is
bent if composed of cations larger than Al3+ , whereas there is less bonding (fewer
sticks shown in figure 4) in configurations 2g, 2h and 2l for (B2 O3 )2 and likewise
for 2m composed of cations smaller than In3+ . Although configurations 2d and 2k
have the same connectivity, we labelled them separately as both configurations
were easily found for the larger cations—(Tl2 O3 )2 prefers the planar arrangement
of 2k, unlike the other compounds. In the (B2 O3 )2 example of configuration 2l,
the cluster has actually fragmented into two (B2 O3 )1 clusters.
We now return to our earlier proposal, in §3a, of constructing LM cluster
structures by adding n oxygen atoms to LM (ZnO)2n clusters. Two very different
low-energy configurations for 1–1 compounds are found for compounds like ZnO:
an octagonal ring and a cuboid (Al-Sunaidi et al. 2008). Adding two oxygen
atoms to these, we form configurations 2m and 2n, respectively. Although LM, if
Tl
, both were found to be quite high in energy.
we ignore E2m
For each compound, the energies in figure 5a are arranged using the rank
found for the corresponding configuration for (Ga2 O3 )2 , thus, the energy always
increases for the gallium oxide clusters. The indium oxide clusters have similar
relative energies, between −6 and −2 eV, to those of gallium oxide, likewise,
but between −2 and 0 eV, for boron oxide and thallium oxide. The relative
energies for aluminium oxide are significantly lower, which follows the trend
found for the linear 1a configuration with respect to configuration 1b (see figure 2,
where the energy difference is smallest for Al2 O3 , similar for Ga2 O3 and In2 O3 ,
and much larger for the end members B2 O3 and Tl2 O3 ). Clearly, the linear
configuration is less favourable for aluminium oxide, and as a consequence, there
is a greater driving force for aluminium oxide to form larger clusters (moreover,
the planar configurations 2k and 2m were either not found or much higher
in energy). Although we have not investigated the minimum energy path to
forming an n = 2 configuration from two isolated 1a configurations (and thus
predicting energy barriers, if they exist), we can imagine two linear clusters
coalescing to form configuration 2l, which, for alumina, is approximately 2 eV
lower in energy than the planar configurations and still approximately 2 eV higher
than most other n = 2 configurations and more than 4 eV higher than the GM
configuration.
The energy differences, EiX − Ē X , of the n=2 configurations, where Ē X is the
average of EiX for the set of (X2 O3 )2 configurations, are plotted as a function of
cation (from the smallest, boron, to the largest, thallium) in figure 5b. Examining
this figure, significant changes in the rank of configurations can be clearly seen
(typically associated with a steep gradient). For example, configurations 2a, 2b
and 2i have many crossovers from boron to aluminium. More importantly, we
can see that 2c and 2b are the GM configurations for (B2 O3 )2 and (Al2 O3 )2 ,
respectively, and 2a otherwise. Although much higher in energy for most
compounds, the planar configuration 2m is the second lowest energy configuration
for (Tl2 O3 )2 . A previous DFT study for alumina clusters (Song et al. 2004)
considered a number of configurations including configurations 2a (their GM)
and 2c, we assume that they missed configuration 2b.
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Structure of sesquioxide nanoclusters
bond length (Å)
2.3
360°–a
g
2.1
e
1.9
360°–d
360°–b
1.7
1.5
1.3
B
Al
Ga
In
Tl
(b) 160
150
140
130
120
110
100
90
80
70
bond angle (°)
(a)
a
b
e
d
g
B
Al
Ga
In
Tl
Figure 6. (a) Unique bond lengths (sticks) and (b) bond angles, for (X2 O3 )2 configurations 2a
(diamonds), 2b (triangles) and 2c (squares) and X = B, Al, Ga, In and Tl. In the stick-and-ball
models, the darker spheres represent the oxygen anions and the lighter spheres the cations. (Online
version in colour.)
The ionic radii of the (six-fold coordinated) cations are 0.41, 0.68, 0.76, 0.94
and 1.03 Å. As Al3+ and Ga3+ have similar sizes (difference of less than 0.1 Å),
we might expect both the configurations and PBEsol0 ranking to be more similar
than what our results predict. For example, alumina configurations 2i and 2j
have a much better ranking than that predicted for gallia as they are lower
in energy than configurations 2d–h. The GM configuration for alumina, 2b, is
also different—2a is the GM configuration for X = Ga, In and Tl. Thus, it again
appears that the more compact clusters (with higher coordination numbers) are
more stable for alumina. In a similar way, with a difference of ionic radii of less
than 0.1 Å, we can also find some similarities of configurations and ranking for
(In2 O3 )2 and (Tl2 O3 )2 ; for example, a different 2m configuration is found for other
compounds, and E2n is significantly higher than the energy of other configurations
Al
Al
is lower than E2m
).
reported here (whereas E2n
As already mentioned above, the n = 2 GM configuration changes with cation
size. Bond lengths and angles of the three configurations found are shown in
figure 6. As found for the n = 1 clusters, the bond lengths increase with cation
size, while the tetragonal rings become more square (angles nearer 90◦ ) and angle
a in configuration 2a widens (although always remains below 180◦ ). The bond
angles within the 2b, however, remain constant at approximately 114◦ (120◦ would
be the perfect angle for a three-coordinated site). Interestingly, the search over
the RM energy landscape for alumina generated an additional LM cluster with
the same connectivity as 2b, but flattened with a bond angle, b, of 107.5◦ and
142.7◦ (93.9◦ and 134.5◦ in the SM) that had D2d point symmetry rather than
Td . This second lowest RM energy configuration relaxes to the GM configuration
2b when the PBEsol energy is minimized.
(c) Atomistic structure and interatomic potential energetics for (Al2 O3 )n and
(In2 O3 )n (n = 3−5)
The search over the rigid ion (Al2 O3 )3 energy landscape yields LM structures
that are essentially the same as those previously reported (Walsh & Woodley
2010) for (In2 O3 )3 . Using the labels in figure 7 to distinguish the different
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S. M. Woodley
3aaaa C1
3ggif C2v
3bbbb Cs
3--fd Cs
4caaa D3d
3eccc Cs
3hhg- C2v
4abfc Cs
3cddh C2v
3fihe Cs
4bchi Cs
3dejk D3h
3ij-i Cs
4ddbb C1
3-feg C1
3jkkj Cs
4fegf Cs
3klll Oh
4hf-k C2v
4egeg C2v
4jhde D3h
4gikl Oh
4kjcd C2h
4lkih D3d
4i-jf C2
4a C2v
5aabh C1
5bbs- C3v
5fcaa C1
5ddca C1
5ieed C1
5iffc C1
5egd- C1
5mhki C1
5kigf C1
5pkje C1
5hlnk C1
5qmlj C1
5gn-- Cs
5lqoo C2
5uru- C3v
5tstq C3v
5rtvs D5h
5suw- C3v
5--ij Cs
5---m C3
5a – C1
5c – C2
5d – Cs
5noql Cs
5vvpr C2v
5opm- C1
5wwrp C2
5cjhn C2v
Figure 7. Ball-and-stick models of the PBEsol stationary point (X2 O3 )n configurations for n = 3–5,
with labels indicating size, rank for X = B or X = Al, Ga, In and Tl as measured using PBEsol0,
and its point symmetry. (Online version in colour.)
configurations, 3aaaa was found to be the GM for (Al2 O3 )3 , then, 3bbbb, 3cddh,
3aaaa, 3eccc, 3dejk, 3ij-i, 3fihe, 3–fd, 3∗ ggif , 3klll, 3-feg, 3hhg- and then 3jkkj. The
last three structures were data mined from the configurations generated for india
(so there is a greater chance of finding low-energy LM configurations not shown in
figure 8), and an asterisk is used to indicate that the structure changes symmetry
upon relaxation: C2v to Cs , with atoms in the upper half (as seen in figure 8)
moving to the left so that the initially four-coordinated central oxygen atom is
only three coordinated. The rank, as measured using the RM, for india are similar:
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Structure of sesquioxide nanoclusters
(b) 10
0
5
–10
0
–20
–5
–30
–10
energy (eV)
(a) 10
–40 n = 1
n=2
n=3
n=4
n=5
–15 n = 1
n=2
n=3
n=4
n=5
Figure 8. (a, b) Relative PBEsol0 energies (eV) for the (X2 O3 )n configurations shown in
figure 7 for increasing cation size (X = B–Tl) for each value of n. Black crosses indicate energies
for configurations, which are GM at least for one compound, the thick line connects the five GM
configurations for each n, and the broken lines connect the GM configurations for each compound
(coloured blue, orange, red, green and purple for X = B, Al, Ga, In and Tl, respectively, in online
version for better clarity). Energies are relative to n times the respective n = 1 GM configuration,
in (a), and n/2 times the energy of configuration 2b, in (b). (Online version in colour.)
3aaaa, 3cddh, 3bbbb, 3eccc, 3dejk, 3ij-i, 3fihe, 3∗ ggif , 3–fd, 3-feg, 3hhg-, 3klll and
then 3jkkj. Again, the symmetry of 3ggif is lowered when relaxed, and the last
two configurations were data mined from the LM of alumina. The three lowest
energy LM structures for both compounds resemble a tea cosy with symmetry
point groups C1 , C2v and Cs . The C1 tea-cosy structure is the rigid ion GM for
both compounds, whereas the most symmetric, open tea-cosy structure is ranked
second for india and third for alumina. For alumina, we found more than one C1
tea-cosy LM: ranked first and fourth. Similarly, as reported in §3d, when relaxing
these configurations on the PBEsol energy landscape for all five compounds,
more than one C1 tea-cosy configuration is found. As the bonding (indicated in
figure 8 using sticks) and symmetry are the same, below we only report the energy
of the most stable configuration for the C1 tea cosy. Further similarities are found
between the LM ranking of the two compounds. Considering the order of the india
configurations ranked 4–8, a D3h bubble and four Cs structures (Walsh & Woodley
2010), we only need to reverse the order of the last two to obtain the ranking for
alumina. This change in rank suggests that tetragonal faces are not penalized as
much for alumina. In fact, a rhombic dodecahedron, with Oh symmetry, an oxygen
atom at its centre and not previously reported for india, is the next lowest energy
LM configuration (3klll) found on the (Al2 O3 )3 rigid-ion energy landscape. In
this configuration, the six aluminium atoms form an octahedron about the inner
oxygen atom, and each aluminium atom is also coordinated to four of the eight
outer oxygen atoms, forming 12 edge-sharing Al2 O2 tetragonal faces. For india,
there are at least two more configurations with a better rank than 3klll, including
the more planar, less compact structures, 3-fee and 3hhg-.
The n = 4 and 5 lowest energy stationary point structures found by searching
the RM landscapes for (Al2 O3 )n and (In2 O3 )n after refinement on the PBEsol
landscape are also shown in figure 7 together with a few high-energy structures
already reported in the literature (Sun et al. 2008; Walsh & Woodley 2010).
The initial order, or rank, of these clusters for (Al2 O3 )4 , starting from the GM,
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S. M. Woodley
is 4caaa, 4egeg, 4gikl, 4ddbb, 4jhde, 4kjcd, 4bchi, 4fegf, 4lkih, 4i-jf, 4abfc and
4hf -k, where the last five configurations were initially generated using data
mining, whereas on the (In2 O3 )4 IP landscape, it is 4caaa, 4egeg, 4gikl, 4ddbb,
4bchi, 4jhde, 4kjcd, 4i-jf, 4fegf, 4hf-k, 4abfc and 4lkih, with two of the last
three configurations obtained from data mining our relaxed PBEsol (B2 O3 )4
configurations. As already noted, the configurations in figure 7 were obtained
after minimizing the PBEsol energy for one of the five compounds: alumina
(cations coloured purple in the online version), boron oxide (pink) and indium
oxide (grey). Note that we have chosen to show one particular configuration twice:
configuration 4a has the same connectivity (bonds) as 4hf-k, but is shown rotated
by approximately 90◦ about the vertical axis. As well as a rescaling of bond
lengths and bond angles, which is more significant if the cations are changed,
there are a few more noticeable differences between the IP and PBEsol-minimized
configurations. Ignoring structural changes that involve bond breaking, the largest
difference is found for configurations 4i-jf and 4hf-k: a cation–oxygen–cation bond
angle points inward (towards the centre of the cluster) for PBEsol, but points
outwards for IPs (configuration 4hf-k becomes 4a).
For the largest clusters shown in figure 7, the order, or rank, again starting from
the GM, for (Al2 O3 )5 on the IP-landscape is 5ddca, 5vvpr, 5opm-, 5lqoo, 5ieed,
5cjhn(C2 ), 5hlnk, 5egd-, 5mhki, 5wwrp, 5pkje, 5gnij, 5cjhn(C2v ), 5jffc, 5fcaa, 5fcab,
5kigf, 5qmlg, 5rtvs, 5bbs-, 5tstq, 5aabh, 5suw-, 5c, 5uru-, 5a and finally 5- - -m. The
first 10 configurations were generated from a direct search over the IP-landscape
for (Al2 O3 )5 . And for (In2 O3 )5 , it is 5ddca, 5opm-, 5ieed, 5cjhn(C2 ), 5egd-, 5lqoo,
5mhki, 5hlnk, 5vvpr, 5cjhn(C2v ), 5gnij, 5jffc, 5qmlg, 5wwrp, 5fcaa, 5rtvs, 5kigf,
5aabh, 5bbs-, 5pkje, 5suw-, 5tstq, 5noql, 5uru- and then 5- - -m, with the first eight
configurations generated by a direct search over the IP-landscape for (In2 O3 )5 . Of
these configurations, large structural distortions occur during the relaxations of
5gnij, 5qmlg and 5- - -m (after switching from PBEsol back to IP). Configuration
5cjhn represents two distinct stationary points on the IP landscapes: the higher
symmetry configuration shown in figure 7 has two central four-coordinated oxygen
atoms that are bonded to the lowest eight cations that are part of the four outer
hexagonal faces. The cluster with C2 symmetry can be obtained from the C2v
cluster by displacing atoms vertically so that the coordination of the internal
oxygen atom is lowered to two, and two of the four edge-sharing hexagonal faces
become pairs of edge-sharing tetragonal faces. On the alumina and india IPlandscapes, the higher symmetry configuration is actually a saddle point, and
is higher in energy than the C2 configuration. Configurations 5a and 5- - -m are
lower symmetry versions of configurations 5uru- and 5bbs-. The latter, 5bbs-,
which appears (as viewed in figure 7) to be similar to 5suw-, is in fact an open
cone pointing into the page, whereas configuration 5suw- is a bubble.
Although the IP GM configuration is the same for both alumina and india,
i.e. 5ddaa, there are some noticeable differences in the ranking of the other LM,
especially for 5vvpr and 5wwrp. Both configurations have one oxygen atom within
a cigar-shaped bubble composed of 14 tetragonal and four hexagonal faces. For
5wwrp, the internal oxygen atom bridges two cations, whereas in 5vvrp, five more
tetragons are formed (by the four new X–O bonds). Again, we suggest that
tetragons are penalized more for clusters containing the larger indium atoms,
as 5vvrp is ranked second for alumina and ninth for india, and 5wwrp also falls
four places from 10 to 14.
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Structure of sesquioxide nanoclusters
2033
Upon using the SM or an electronic-based method, i.e. including polarizability
of the oxygen atoms, as reported previously for ZnO clusters (Al-Sunaidi et al.
2008), two-coordinated anions at the surface move outwards (sharper Al–O–Al
bond angles). Also, comparing O–Al–O bond angles for an aluminium atom at
the surface that is coordinated to three oxygen atoms that are also at the surface
(see, for example, 5fcaa, 5noql and 5tstq), a more planar arrangement (sum of
bond angles nearer to 360◦ ) is found in the PBEsol LM configurations than in
the IP LM; typical examples include 354.7◦ and 350.8◦ (PBEsol0), which are
significantly higher than 324.3◦ and 315.8◦ (IPs). This trend is of course only
valid if no bonds are broken or created.
(d) Density functional theory energetics for (X2 O3 )n (n = 1–5 and X = B,
Al, Ga, In and Tl)
As already discussed above, LM configurations on the PBEsol energy landscape
do not always resemble the initial structure (before relaxation) that are data
mined from the LM configurations on the IP energy landscapes. This discrepancy
occurred most frequently for (B2 O3 )n , where the larger clusters relaxed typically
into very different configurations, which are more open as pairs of edge-sharing
faces become a single face (bond, or shared edge, is broken). As such, the
labels used in figure 7 to distinguish the configurations refer to either the rank
found only for (B2 O3 )n or the ranks found for (Al2 O3 )n , (Ga2 O3 )n , (In2 O3 )n and
(Tl2 O3 )n . Of the remaining four, it is expected that the thallia clusters would
produce the greatest structural distortions of the original dataset of structures.
Depending on how configuration 3ggif is initially rescaled, the cluster composed
of the larger cations will relax to 3ggif or 3–fd. For boria, the uppermost tetragon
of 3ggif rotates by 90◦ about the axis that goes through the two uppermost boron
atoms (thus breaking two bonds, while keeping C2v symmetry).
Configuration 3dejk has the point symmetry of D3h and the lowest PBEsol0
energy for (B2 O3 )3 . As the size of the cation is increased, the rank of this
configuration falls: fourth for (Al2 O3 )3 , fifth for (Ga2 O3 )3 , tenth for (In2 O3 )3
and eleventh for (Tl2 O3 )3 . Moreover, the similar ranks match the similarity
of the cation ionic radii of Al3+ and Ga3+ , and In3+ and Tl3+ . A falling (or
improving) rank with cation size is more the exception than the rule: 4bchj
and 4gikl (4ddbb, 4lkih, 5ddca, 5kigf, 5pkje, 5qmlj and 5wwrp) are the other
obvious examples in figure 7. Configurations 3dejk, 4bchj and 4gikl contain large
(greater than hexagonal) faces, however, 4hf-k does not have a fall in rank
between (Al2 O3 )4 and (Ga2 O3 )4 , as might be expected, as it is very similar
to 3dejk. One interesting (X2 O3 )3 cluster is that of the dodecahedron that,
compared with the other clusters in figure 7, is ranked last on the PBEsol energy
landscapes (compared with fourth and second from last on the alumina and india
IP landscape); with the shortest metal-inner oxygen bond length, the (B2 O3 )3
dodecahedron has more of a cubic appearance (outer oxygen atoms are at the
corners of a cube).
Configuration 4a has the point symmetry of C2v and the lowest PBEsol0 energy
for (B2 O3 )4 . The effect of increasing the size of the cation was investigated; upon
relaxation (PBEsol minimized after rescaling), configuration 4hf-k is obtained (an
oxygen atom moves within the cluster—see earlier discussion), with the X–O–X
angle increasing from Al to Tl, although always remaining smaller than that for
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S. M. Woodley
B. A similar change, movement of an oxygen atom from outside inwards, is found
for 4i-jf, in which the bond angle Al–O–Al (pointing outwards) is 158◦ , whereas
for In–O–In and Tl–O–Tl (pointing inwards), it is 150◦ and 152◦ —the direction of
this angle was not found for (Ga2 O3 )4 and is, perhaps, the cause of the difficulty
of finding this configuration for gallia.
Configuration 5a with the lowest PBEsol0 energy for boria can be obtained
from configuration 5uru-, whereas configuration 5b is essentially 5bbs-. Apart from
three tetrahedral boron sites in the latter, the boron atoms in both structures
are coordinated with three oxygen atoms. The only other configurations shown
in figure 7 are 5c and 5d, which were readily found for boria. For the largest
cation, data mining produced the lowest energy configuration twice. Namely,
configuration 5ddca has one bond less than 5fcaa for alumina (Al–O distance
of 3.61 Å compared with 2.00 Å, where a typical bond is 1.77 Å), whereas for
thallia, the two configurations are identical (Tl–O distance of 2.68 Å compared
with a typical bond of 2.03 Å).
The PBEsol0 energies of the configurations shown in figure 7 are plotted twice
in figure 8. Typically, the energy of formation is reported relative to the n = 1 GM
configuration, as in figure 8a. From the predicted structures, we can conclude
that the boria clusters are generally unlike the configurations adopted by the
other compounds, particularly as n increases. In contrast, figure 8a indicates a
similarity between gallia and india, and boria and thallia, with alumina clusters
being much more stable (the thick line has the same profile for each n—a
minimum for Al). As the energies of clusters with similar-sized cations are likely
to be similar, this observation does not meet expectations—in fact, the trend
highlighted by the thick line in figure 8a is merely an artefact of the relative
stability of the n = 1 cluster with respect to typical larger clusters. Therefore, we
plotted the relative PBEsol0 energies again, but this time using the symmetrical
2b configuration (point symmetry Td ), where the cations and anions have a
coordination of three and two, respectively. The mirror image of the earlier profile
is now found for n = 1. For n > 1, the change in the relative energy is significantly
different for the boria GM configurations, as expected. The interesting result is
what happens to the other compounds: initially (n = 1–3), stability of the GM
increases with cation size, but by a decreasing magnitude with increasing n. For
n = 4, similar GM stability (with respect to its smaller sized cluster) is predicted
for alumina and gallia, and similarly for india and thallia, whereas for n = 5, a
greater stability is found for alumina (and by a smaller magnitude india).
The high end of the highest density of the light plus symbols in each column
in figure 8 does not always provide an estimate of the transition between the
highest energy configuration produced using global optimization techniques and
the lowest energy configuration that were obtained by data mining. Although the
worst ranked n = 5 configuration for larger cations, 5rtvs, was generated by data
mining the literature, for smaller cations, the worst ranked configuration reported
here, 5wwrp, was generated by the global search over the alumina IP landscape
(although ranked tenth).
Ignoring clusters of boria, the same GM PBEsol0 configuration is predicted for
n = 3, namely the C1 tea cosy 3aaaa. This structure has also been predicted for
alumina and india on the respective IP landscapes. The tea-cosy structure is also
reported for (Al2 O3 )3 in a study using ab initio total energy linear combination
of atomic orbitals (LCAO) calculations (Fernandez et al. 2003). For n = 4,
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Structure of sesquioxide nanoclusters
2.5
energy (eV)
2.0
1.5
1.0
0.5
0
1
2
3
cluster size (n)
4
5
Figure 9. PBEsol0 energy difference (eV/cluster), Eb − Ea , between the GM and the next lowest
energy configuration for each compound and cluster size: (X2 O3 )n , X = B, Al, Ga, In and Tl and
n = 1–5. Additional points assume that all tea-cosy configurations are equivalent. Solid line, B; filled
squares, Al; filled triangles, Ga; filled circles, In; filled diamonds, Tl. (Online version in colour.)
configuration 4caaa, with point symmetry D3d , is the PBEsol0 GM for gallia,
india and thallia, and can be considered as a fragment cut from corundum (bulk
phase). For alumina, this configuration is the third lowest LM, after 4abfc and
4bchi—both of which rank well for the clusters containing similar-sized cations,
viz. Ga. The corundum fragment was previously reported as a possible GM for
alumina (Song et al. 2004) and has been used when modelling water on (Al2 O3 )n
using ab initio total energy LCAO calculations (Fernandez et al. 2003); however,
configuration 4abfc is also reported as the alumina GM structure by other groups
using EAs to search the DFT (B3LYP) energy landscape (Sierka et al. 2007;
Sierka 2010). For n = 5, cations of a similar size have the same GM PBEsol0
configuration: alumina and gallia adopt 5aabh, whereas for india and thallia, it is
5fcaa—a maximum change in rank that is caused by a cation size of seven (a–h).
In figure 8, more than one black cross is shown in a particular column if the same
GM (X2 O3 )n configuration is not predicted for Al, Ga, In and Tl. The distance,
or energy difference, between the dark plus symbols, therefore, indicates when a
change in configuration can help reduce the strain caused by a change in cation
size. The largest change is found for (In2 O3 )4 between configurations 4caaa and
4abfc, where the change of rank for india is five.
Energy differences between the two lowest energy configurations for each
compound and cluster size are shown in figure 9. The tea-cosy structures of
n = 3 (first four configurations in figure 8) have similar energies and result in the
minimum for the clusters containing the larger four cations; at room temperature
these tea-cosy structures are likely to be part of one ergodic region on the freeenergy landscape, thus we have also included the energy difference between
the GM and the lowest energy non-tea-cosy n = 3 structure. Now the energy
differences in figure 9 show that the relative stability of each GM configuration
with respect to clusters of the same size generally decreases with n. Again, it
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S. M. Woodley
(B2O3)n
(In2O3)n and (Tl2O3)n
(Al2O3)n and (Ga2O3)n
0.09
n=1
0.01
0.06
0.02
0.03
0
0
0
0
0
0
0
0
0
0
0
0
intensity (arb. units)
n=2
n=3
n=4
n=5
–15
–10
–5
0
0
0
–20
0
5 –20
–15
–10
–5
0
5 –20
–15
–10
–5
0
5
energy (eV)
Figure 10. GW spectra (Gaussian smeared with arbitrary magnitude) for the GM clusters for
(B2 O3 )n , (Al2 O3 )n and (Tl2 O3 )n (solid lines), and (Ga2 O3 )n and (In2 O3 )n (broken lines) for
n = 1–5 over the energy range E = −20 to +5 eV. Broken vertical lines are included to aid
comparison of the location of quasi-particle HOMO and LUMO eigenvalues. (Online version
in colour.)
is interesting that there is not more similarities between the results shown for
alumina and gallia; ignoring the boria clusters, the energy difference for (Ga2 O3 )n
is similar to (In2 O3 )1 , (Tl2 O3 )2 , (In2 O3 )3 and (Tl2 O3 )5 .
The complexity of the energy landscapes will typically increase with the
number of variables or atoms. Stability is, of course, not just dependent upon the
energetic barriers (Schön et al. 2003). Kinetic, or entropic barriers will become
increasingly important as the number of atoms, or size of the clusters, increases.
(e) The electronic structure for (X2 O3 )n (n = 1–5 and X = B, Al, Ga, In and Tl)
Analysis of their electronic states (single-particle PBEsol0) reveals that each of
the identified stable and metastable (X2 O3 )n structures has finite HOMO–LUMO
separations, with the highest energy occupied states being dominated by O 2p
and the lowest lying empty states consisting largely of X Ys and Yp orbitals
for (X,Y) = (Al,2), (Ga,3), (In,4), (Tl,5). The density functional calculations are
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0.02
0.04
intensity
Structure of sesquioxide nanoclusters
0.02
energy (eV)
(B2O3)n
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
(Al2O3)n
(Ga2O3)n
(In2O3)n
(Tl2O3)n
–20 –15 –10
–5
0
0
5 –20 –15 –10 –5
0
0
5 –20 –15 –10 –5
0
0
0
5 –20 –15 –10 –5
0
5
Figure 11. GW spectra (Gaussian smeared) for (X2 O3 )n local minimum clusters, 2b (column 1) and
4caaa (column 3), and a Boltzmann-weighted superposition of spectra for clusters 2a–c (column 2)
and 4caaa, 4abfc, 4a (column 4), over the range −20 to +5 eV, for X = B, Al, Ga and Tl, and n = 2
and 4. (Online version in colour.)
therefore consistent with the formal charge states of X3+ (Ys0 Yp0 ) and O2− (2p6 ),
which explain the excellent correspondence between the classical and quantum
energy landscapes for these larger cations. As expected, the eigenvalues resulting
from the semi-local PBEsol functional typically result in a small HOMO–LUMO
gap, whereas the non-local PBEsol0 functional, which counters the self-interaction
error and the energy derivative discontinuity present in the pure DFT functional
(Sham & Schlüter 1983), opens the gap through both a significant lowering of the
HOMO level and a raising of the LUMO level. Previously, we reported that the
application of the GW quasi-particle correction for india GM clusters maintains
the HOMO position of the PBE0 functional (to within 0.3 eV) (Walsh & Woodley
2010). Here, for the PBEsol0 functional, we typically obtained a similar initial
HOMO position, but now the application of the GW quasi-particle correction
lowers the HOMO position; for n = 3–5 by approximately 1.2 eV. This movement
is dependent upon the cation; it is larger for alumina, approximately 2 eV, and
smaller for thallia, approximately 1 eV. For all compounds, the LUMO further
raises towards the vacuum level by approximately 1–2 eV.
The GW spectra near the HOMO (first peak below −5 eV) and LUMO (first
peak above −5 eV) levels of the ground state (GM) clusters for all compounds
and sizes investigated here are plotted in figure 10. The eigenvalue for HOMO,
or quasi-particle ionization energy, for boria is −12.5 eV for n = 1, −10.0 eV for
n = 2, −11.5 eV for n = 3 and 4 and −11.3 eV for n = 5. More interestingly, three
of these clusters have a positive LUMO eigenvalue suggesting that they would
not accept an additional electron from the vacuum, and the electron affinities
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0.004
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Figure 12. GW spectra (Gaussian smeared) for 10 (Al2 O3 )4 local minimum clusters shown next
to their respective ball-and-stick models, the Boltzmann-weighted superposition of these spectra
(top two graphs); and for bulk alumina, experimental data (black lines) for X-ray emission and
absorption spectra taken from Perevalov et al. (2007) (top left graph), ultraviolet photoelectron
spectrum taken from Perevalov et al. (2007) (top right) and X-ray photoelectron spectrum taken
from Tagle et al. (1979) and Balzarotti & Bianconi (1976) (top right, smooth curve). The intensities
are in arbitrary units and energy spans from −20 to +5 eV. (Online version in colour.)
for the other two are small: 0.2 eV for n = 3 and 0.04 eV for n = 5. The quasiparticle ionization energies (electron affinities) in electronvolts for alumina GM
clusters n = 1–5 are −9.9 (1.6), −8.9 (0.8), −10.3 (1.0), −10.3 (1.1) and −10.2
(1.1). Similar values of the ionization energies are obtained for the first four gallia
GM clusters: −10.0 (1.4), −8.9 (1.4), −10.0 (1.7), −9.9 (1.8) and −10.0 (1.9). The
quasi-particle ionization energies (electron affinities) in electronvolts for india GM
clusters n = 1–5 are: −9.1 (1.7), −7.8 (1.5), −8.9 (1.2), −8.9 (2.3) and −9.1 (2.6).
These are more similar to that obtained for thallia: −9.1 (1.5), −7.1 (1.9), −8.5
(2.9), −8.6 (2.9) and −8.6 (3.1). The mean values of the ionization potentials
therefore decrease with cation size (11.4, 9.9, 9.8, 8.8, then 8.4 eV), whereas the
electron affinities increase (0.05, 1.1, 1.6, 1.9 and 2.5 eV). The energy difference
between HOMO and LUMO levels is, for a particular size n, typically largest for
boria and smallest for india and thallia. Moreover, for each n, the spectra for
india and thallia are similar, which is not surprising as they adopt the same GM
configurations, whereas boria adopts very different GM clusters for n > 1, and
the spectra for boria clusters are less like those of the other compounds.
Since the same GM configuration is not found for all compounds, in figure 11
the spectra for two low-energy clusters, configurations 2b and 4caaa, are plotted
for each compound. As expected, the spectra for the same configuration are
similar. The position of the LUMO level is typically constant, whereas the
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Structure of sesquioxide nanoclusters
2039
position of HOMO and the density of states for the clusters containing larger
sized cations typically increase. Now consider the non-GM cluster configurations
that are generated from the GM configurations after changing cations, e.g. 4caaa
for (X2 O3 )4 is one such LM if X = Al. The spectra of these non-GM clusters
can be added to those of the GM clusters. If the spectra are also weighted
with the Boltzmann factor, exp[−(ELM − EGM )/Na kB T ], where Na is the number
of atoms in the GM or LM cluster and kB T = 0.025 eV, then we obtain the
room temperature spectra in columns two and four in figure 11 for n = 2 and 4,
respectively. Ideally, all LM clusters that would add a significant contribution to
the room temperature spectra of the electronic energy levels should be included.
Aligned in figure 12 are the spectra for 10 LM (Al2 O3 )4 clusters, including the GM,
the resulting room temperature spectra for (Al2 O3 )4 , and the observed spectra
for bulk alumina. The room temperature spectra for (Al2 O3 )4 are remarkably
similar to those observed for bulk alumina.
4. Summary
The stable and metastable, as measured using an all-electron DFT approach,
stoichiometric (X2 O3 )n clusters of boron, aluminium, gallium, indium and
thallium oxide have been reported. Initial candidate structures were found
using an EA to search the energy landscape, defined using classical interatomic
potentials, for alumina and india followed by data mining, or rescaling (Sokol
et al. 2010). Electronic characterizations of the refined structures were performed
at the hybrid density functional and many-body GW levels to obtain accurate
predictions of the spectroscopic properties. For n > 1, (B2 O3 )n adopts different
GM configurations than that predicted for the other compounds. With similar
ionic radii, alumina and gallia were expected to adopt similar GM configurations.
However, for n = 2 and 4, the alumina GM configurations were different from
that of gallia, which adopted the same GM configuration predicted for india and
thallia. Comparisons of cluster structures indicate that X2 O2 tetragons are not as
heavily penalized for the smaller Al3+ cation. Also, for the n = 1 clusters, boria
and alumina adopt a collinear structure, whereas the greater polarizability of the
larger cations results in the buckling of this structure. The relative energy of a
cluster is typically either reported with respect to the smallest sized (n = 1) GM
cluster or to the bulk. Here, we show that the use of the former is misleading
when making comparisons of relative energies of different compounds. Finally,
the electronic spectra near the HOMO and LUMO levels were reported for two
example configurations and the GM configurations for all five compounds. The
mean values of the ionization potentials for the GM clusters decreased with cation
size, from boria to thallia, whereas it increased for the average electron affinities.
Furthermore, a Boltzmann (room temperature)-weighted superposition of spectra
for (Al2 O3 )4 LM was found to be similar to the experimental data taken from
the X-ray photoelectron spectrum and the ultraviolet photoelectron spectrum for
bulk alumina.
The author is grateful for valuable input from Alexey Sokol, Aron Walsh, Stephen Shevlin and
Richard Catlow, and for funding from UK’s HPC Materials Chemistry Consortium, which is
itself funded by EPSRC (EP/F067496). In the completion of this work, the author made use
Proc. R. Soc. A (2011)
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2040
S. M. Woodley
of molecular visualizing software from Accelrys and the UCL Legion High Performance Computing
Facility, and associated support services; and the facilities of HECToR (approx. 1 million AUs),
the UK’s national high-performance computing service. The latter is provided by UoE HPCx Ltd
at the University of Edinburgh, Cray Inc and NAG Ltd, and funded by the Office of Science and
Technology through EPSRC’s High-End Computing Programme.
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