Using DFTB to Model Photocatalytic Anatase–Rutile
TiO2 Nanocrystalline Interfaces and Their Band Alignment
Band alignment effects of anatase and rutile nanocrystals in TiO2 powders lead to electron–hole separation, increasing the photocatalytic efficiency of these powders. While size effects and types of possible alignments have been extensively studied, the effect of interface geometries of bonded nanocrystal structures on the alignment is poorly understood. To allow conclusive studies of a vast variety of bonded systems in different orientations, we have developed a new density functional tight-binding parameter set to properly describe quantum confinement in nanocrystals. By applying this set, we found a quantitative influence of the interface structure on the band alignment.
Introduction<!>Methods<!>Parameterization
and Validation of the New DFTB Parameter Set<!><!>Parameterization
and Validation of the New DFTB Parameter Set<!><!>Parameterization
and Validation of the New DFTB Parameter Set<!><!>Parameterization
and Validation of the New DFTB Parameter Set<!><!>Band Alignment<!>Bulk
Band Alignment<!><!>Band Alignment in Nonbonded Nanocrystal Systems<!><!>Band Alignment in Nonbonded Nanocrystal Systems<!><!>Band Alignment in Nonbonded Nanocrystal Systems<!>Band Alignment
in Bonded Nanocrystals<!><!>Band Alignment
in Bonded Nanocrystals<!><!>Band Alignment
in Bonded Nanocrystals<!><!>Conclusions<!>
<p>TiO2 powders
commonly used for photocatalytic applications
such as Degussa P25 (DP25) consist of anatase and rutile nanocrystals
in different compositions (commonly ∼80% anatase and ∼20%
rutile). This composition of the two different TiO2 phases
is shown to have a great impact on photocatalytic efficiency due to
band alignment effects that lead to effective electron–hole
pair separation. There have been several theoretical attempts to predict
the type of band alignment and thus which crystal acts as a hole or
electron trap.1−4 Experiments indicate that for the case of DP25, anatase acts as
an electron trap, while rutile as a hole trap.5 Theoretical works have so far been focused on isolated bulk,1 slab,2 or nanocrystal3,4 systems. While bulk studies predict the trapping of electrons on
anatase and the trapping of holes on rutile in accordance with the
experiment, nanocrystal studies predict a significant size dependence
for the alignment until bulk-like behavior is reached. However, neither
of those approaches considers the potential impact of different types
of additional degrees of freedom in materials such as geometrical
interface arrangements. The role of such interface arrangements was
investigated in ref (2) using slab models. However, such models can only give limited insights
into the interface effects as the slabs of the two phases must be
commensurate, allowing only for a few possible orientations and surface
geometries. This makes it impossible to explore the degrees of freedom
in the interface formation, which are present under experimental conditions
at the microscale.</p><p>The vast phase space of possible nanocrystal
orientations and sizes
thus makes computational studies of anatase–rutile nanocrystal
interfaces very challenging, even for very small nanocrystal models
where a conclusive study on the role of the interfaces would be unfeasible
using density functional theory (DFT)-based calculations alone. Consequently,
we have developed a density functional tight-binding (DFTB) parameter
set to efficiently simulate TiO2 nanocrystals and anatase–rutile
interfaces. DFTB, being 2–3 orders of magnitude faster than
comparable DFT calculations, provides the possibility of studying
a vast number of systems and system sizes. Additionally, with the
right choice of parameters, DFTB retains most of the accuracy of DFT
and thus yields the possibility of making a closer to one-to-one comparison
to experiments. Another benefit of the DFTB approach is its localized
basis set that simplifies the computation of non-periodic structures
such as nanocrystals.</p><p>In the present work, we identify several
issues in describing nanocrystals
with the previously published DFTB parameter sets for titanium dioxide,6,7 e.g., problems in describing the quantum confinement effects correctly
and yielding wrong relaxations when undercoordinated species are involved.
The latter problem has also been reported in ref (8). To remedy these issues,
we have developed a new parameter set, named tio2nano,9 that focuses on the correct description of TiO2 nanocrystals and interfaces. It is based on the 3ob parameter set.10 The electronic parameters of the new Ti species
were tailored to ensure the proper description of the quantum confinement
effects in TiO2 nanocrystals. Instead of the traditional
two-center potentials, we have used a many-body force field11,12 to represent the repulsive contributions to the total energy, following
the same approach as in ref (13). The three-center contributions of the force field allowed
us to obtain improved agreement with DFT-optimized geometries compared
to the results from two-center repulsive potentials only.</p><p>For
the band alignment studies, we considered explicit anatase–rutile
interface structures. We are not aware of any previous attempts to
determine the electronic structure using such models, but similar
CeO2 nanoparticle models were addressed in ref (14). By applying the newly
developed DFTB parameter set on the generated interface structures,
we were able to demonstrate the impact of the geometric alignment
of the crystals on the band alignment.</p><!><p>DFTB calculations
were performed with the DFTB+ code.15 As
described below in detail, the 3ob parameter
set10 has been extended with element Ti
using a density compression radius of 8.5 b and wave compression radii
of 5.609, 3.958, and 7.0 b for the s-, p-, and d-orbitals of the Ti
atom, respectively. The energy and the chemical hardness (also known
as (a.k.a.) the Hubbard U value) of the virtual 4p
orbital of the Ti atom have been set to 0.206 and −0.08 hartree,
respectively.</p><p>The energies were created using the Chebyshev
interaction model
for efficient simulation (ChIMES),11,12 a reactive
many-body molecular dynamics (MD) force field. ChIMES creates many-body
interactions by projecting DFT-computed data (e.g., forces, stress
tensors, and energies) onto linear combinations of many-body Chebyshev
polynomials.16 Briefly, this begins with
an N-body expansion of the total energy for a system.1Here, Ei is the single-atom energy for a given element, Eij and Eijk represent the two-body and three-body interaction
energies, respectively, N is the total number of
atoms in the system, and  corresponds to higher-order terms.</p><p>Specifically, the two-body
interactions are expressed as a linear
combination of Chebyshev polynomials of the first kind2In this
case, Tn(sij) is the Chebyshev polynomial of the
first kind of nth order, ei and ej are the element
types of atoms i and j, and sij is a transformation of the
interatomic distance rij over the Chebyshev interval of [−1,1] using a Morse-like
function. In addition, Θ2 corresponds to the two-body
polynomial order, fCij(rij) is the cutoff function that ensures the potential
and its derivative vary smoothly to zero beyond a specified distance,
and fPij(rij) is a penalty function13,17 that helps to prevent
sampling of interatomic distances below those seen in the training
set (see ref (18) for
further details).  is a set
of permutationally invariant coefficients
of linear combination for a given atom pair type that are determined
via a linear least-squares method.</p><p>Similarly, three-body interaction
energies are expressed as a product
of Chebyshev polynomials for the constituent atom pairs of a given
triplet, yielding an orthogonal three-body polynomial with  total interactions for a given triplet3In
this case, Θ3 is used
to label the three-body polynomial order with a single permutationally
invariant coefficient  for
each set of triplet
atom types. To guarantee that only three-body interactions between i, j, and k are counted
toward the sum, only terms for which at least two of the three m, p, and q indices are
greater than zero are included in the sum (indicated by the prime
in eq 3). In this way,
many-body interactions can be included in the repulsive energy (Erep) by solving a linear least-squares optimization
problem where optimal coefficients of the linear combination are determined
directly. This avoids reliance on iterative approaches that are required
for nonlinear optimization problems (e.g., Levenberg–Marquardt)
that are usually more computationally time-consuming and not guaranteed
to result in the global minimum. ChIMES models have been extended
to include four-body interactions in a similar fashion in reactive
MD simulations,16,19 though truncation of the ChIMES
total energy with the three-body term has proven sufficient for determination
of the DFTB repulsive energy. Note that DFTB in its original formulation
uses a consistent two-center approximation in both the Hamiltonian
matrix elements and the repulsive energy. Our model breaks this consistency
by extending only the latter with three-body terms. Extending the
Hamiltonian with three-body terms would be more involved, as one would
have to deal with the arising pseudopotential-like contributions.20 Therefore, while our approach is not a systematic
extension, it nevertheless represents a simple and pragmatic way of
overcoming some of the deficiencies in the original DFTB formulation.</p><p>Our training set for the ChIMES force field was determined from
DFT-MD simulations of the amorphous TiO2 run at temperatures
of 2250 and 300 K using the Vienna Ab initio Simulation Package (VASP)
code.21 For these calculations, we used
the PBEsol functional22 with an energy
cutoff of 550 eV and Γ-point sampling of the Brillouin zone.
PBEsol was chosen due to its improved description of solids and their
surfaces. The amorphous phase allowed for improved sampling of a wide
range of interatomic distances over the short-time scales of the simulations,
which was enhanced by including data from elevated temperatures. Each
MD simulation was run on a system of 216 atoms for a total of 5 ps,
with configurations taken for Erep training
every 100 fs (to allow for decoupling between training data), resulting
in 50 training configurations taken from each temperature. We also
included additional 10 configurations from an MD simulation run at
40 GPa to improve sampling of close interatomic distances. This yielded
a total of 110 training configurations.</p><p>The training set to
determine Erep was
then computed by subtracting the gradient of the DFTB electronic energy
from our DFT reference property, i.e., F⃗TRAIN = F⃗DFT – F⃗DFTB-elec. In this work, a spline repulsive fitted
to a Ti–Ti dimer was included in the DFTB electronic calculations
to ensure that excessively small interatomic distances were not approached
during calculations. In addition, O–O distances were poorly
sampled in our training set and these repulsive parameters were thus
taken from the 3ob-0-1 parameter set and were not a part of our fit.
ChIMES parameters were determined through linear least-squares fitting
to the resulting ionic forces and the diagonal components of the stress
tensor for each of these MD snapshots. We set the ChIMES two-body
polynomial order to 12 and the three-body order to 8, similar to previous
work,23 and solve for optimal coefficients
using the least-angle regression (LARS)24,25 algorithm
with a least absolute shrinkage and selection operator (LASSO)26 regularization value of 10–4.</p><p>For the DFTB calculations, charges were converged with a
tolerance
of 10–6 au and forces with 10–4 au. Bulk calculations were performed with an 8 × 8 × 8
Monkhorst–Pack27 grid for the primitive
unit cell. The rutile (110) surface was constructed as a 4 ×
2 surface unit cell with six layers, the (100) surface as a 4 ×
2 unit cell with six layers, and the (001) surface as a 4 × 4
unit cell with eight layers. The (001) anatase surface had a 2 ×
2 unit cell with eight layers and the (101) surface had an 8 ×
4 unit cell with five layers. All rutile structures were computed
with a 2 × 2 × 1 k-point mesh, the anatase
(001) surface with a 4 × 4 × 1 mesh, and the (101) surface
with Γ-only calculations. Branching point energy calculations
(discussed below) were performed with the primitive unit cell of the
anatase and rutile crystals using an 8 × 8 × 8 k-point grid. For nanocrystals, we performed Γ-only calculations.</p><p>The nanocrystal interfaces were constructed using the JANUS code,
which is based on a quick hull algorithm for convex hulls.28 This algorithm allows us to rapidly identify
convex hulls from a set of points; in this case, the coordinates of
atoms from the nanocrystals are in the xyz format.
The code then combines simplices of the computed convex hull for all
normals that form an arccos of 0.99 or larger. The largest facets
of each individual nanocrystal are then aligned to each other so that
their surface normals are parallel to each other and to the x-axis with a user-specified distance between them. The
procedure is repeated with the nanocrystals being rotated with respect
to each other around the x-axis for angles between
0 and 180° in steps of 15°. Additionally, the nanocrystal
gets displaced in the yz-plane along the perimeter
of a circle with a user-specified radius. This displacement occurs
with angles between 0 and 360° in steps of 30°. This results
in 156 interface geometries, which are stored in an atomic simulation
environment (ASE)29 trajectory format.
To create the interfaces, we used the two smallest relaxed nanocrystals
of anatase and rutile. The nanocrystals were aligned with a distance
along x of 2 Å, and a radius of 2 Å was
used for the displacement in the yz-plane.</p><p>Interface energies were computed as4where Etot is
the energy of the interface structure, Erut is the energy of the isolated rutile nanocrystal contained in the
interface structure, and Eana accordingly
is the energy of the anatase nanocrystal.</p><p>For comparative DFT
calculations, we used the Vienna Ab initio
Simulation Package21 (VASP 5.4.4) with
the projector augmented wave (PAW) method to describe the cores. For
titanium, the 3p electrons were treated as valence electrons. The
plane wave cutoff was chosen to be 420 eV and the augmentation cutoff
was chosen as 840 eV. Geometry relaxations were performed until energy
differences were converged below an error of 10–3 eV. The k-point meshes were chosen in accordance
with the DFTB calculations.</p><p>As nanocrystal models, we chose
Wulff-type crystal structures for
anatase nanocrystals, which expose (101) facets, and quasi Wulff-type
crystals for rutile, exposing (110), (100), and (101) facets. These
nanocrystal structures were chosen in accordance with ref (3) to allow for direct comparison.
Additionally, for the reasons described below, we also considered
another set of quasi Wulff-type rutile nanocrystals by cutting particles
along the (110) and (101) planes with an even number of (110) layers
across the waist of the particle.</p><!><p>The existing
parameter sets for Ti–O interactions, the tiorg-0-17 and matsci-0-36 sets,
were evaluated only for bulk structures and common surfaces so far.
By applying them to describe anatase Wulff-type nanocrystals, we faced
the issue that the highest occupied molecular orbital (HOMO)–lowest
unoccupied molecular orbital (LUMO) gap did not change with the particle
size as predicted by the theory of quantum confinement3 and as observed in analogous Perdew–Burke–Ernzerhof
(PBE) calculations. Instead of the expected linear function of n–2/3 (with n being the
number of TiO2 units), we observed surface states, which
fall into the band gap and give rise to HOMO–LUMO gap sizes
lower than the bulk value, as shown in Figure 1. To rule out geometrical effects, we also
performed single-point calculations for those nanocrystals using their
PBE geometries but found a similar wrong behavior with both sets.
Efforts to resolve the issue by changing the compression radii of
the Ti atom turned out to be unsuccessful. As a next attempt, we tried
to describe the Ti–O interaction by extending the 3ob parameter
set10 with Ti. The electronic parameters
of the Ti atom were optimized by fitting on the theoretically expected
linear behavior of the band gap with respect to n–2/3. For this fit, anatase nanocrystals with sizes
of n = 84, 165, and 286 and the bulk phase of anatase
had been used, resulting in the compression radii reported in the Methods section.</p><!><p>Calculated HOMO–LUMO gaps of anatase
nanocrystals using
PBE and DFTB with the tiorg-0-1, matsci-0-3, and the newly developed
tio2nano set. The PBE results have been shifted to align with the
values obtained by the new parameter set for the smallest nanocrystal.
The tiorg-SP values were obtained by carrying out DFTB calculations
with the tiorg-0-1 set at the PBE geometries (without relaxing with
DFTB).</p><!><p>After fixing the electronic parameters,
the two-center repulsive
potentials of the Ti–Ti and the Ti–O interactions had
been derived using a symmetrized titanium hexagonal close-packed (hcp)
crystal and a symmetrized rutile bulk crystal as references, respectively.
The symmetrization ensures that the first neighbor Ti–Ti (in
hcp Ti) and Ti–O bonds (in rutile) have identical lengths,
allowing for a simple manual fitting procedure. When relaxing the
nanocrystals with the obtained repulsive potentials, the resulting
geometries exhibited significant deviations in the strongly undercoordinated
tip regions as compared to PBE geometries (see Figure 2 for an example). Unfortunately, those geometrical
changes lead to the appearance of spurious surface states in the gap.
Our attempts to enforce the correct geometry by tuning the two-center
repulsive functions were not successful. This is not surprising as
the incorrect Ti–O–Ti angle seems to be the main driving
force behind the differences in the relaxed geometries. To enforce
correct nanocrystal geometries (and with that also the correct electronic
structure), we decided to represent the Ti–O and Ti–Ti
repulsive functions with the ChIMES force field, which allows the
inclusion of three center terms. The interface to ChIMES software
had been implemented in a development version of the DFTB+ package.
For the Ti–Ti repulsive potential, we kept the two-center one
derived from the symmetrized titanium hcp crystal, with ChIMES corrective
terms. It is important to note that the performance of the Ti–Ti
repulsive potential in describing Ti bulk phases was not evaluated
in this work.</p><!><p>Relaxed geometries of the Ti35O70 nanocrystal.
The DFTB geometry obtained using a two-body repulsive potential for
the Ti–O interaction is shown by balls and sticks, while the
PBE geometry is represented by blue solid lines. The geometries had
been aligned at the top oxygen atom.</p><!><p>The resulting parameterization performs well for anatase and rutile
bulk as well as for their surfaces and is able to describe systems
for which the previous parameter sets have failed. The bulk lattice
parameters given in Table 1 show very good agreement with ab initio counterparts for
rutile. For anatase, we observe a slight overestimation of the a value, while c is underestimated. The
calculated surface energies are listed in Table 2. The current parameterization slightly overestimates
the surface energies for anatase (001) and rutile (110) compared to
the PBE and PBEsol results. The anatase (101) and rutile (001) and
(100) surfaces are underestimated compared to the PBEsol results and
overestimated compared to PBE. The energetic ordering of the surfaces
with respect to each other is reproduced nicely within the tio2nano
set. Overall, the current set shows good agreement to ab initio methods.</p><!><p>Superscripts a,
b, and c denote
values obtained in this work, ref (30), and ref (31), respectively.</p><p>Index a denotes
values obtained
in this work, while values marked with b were taken from ref (32), except the one indicated
by *, which was taken from ref (33).</p><!><p>Figure 3 shows the
PBE gap values of anatase and rutile in comparison to the ones obtained
with the newly developed parameter set. The behavior of both sets
is now comparable to ref (3). The smallest anatase nanocrystal considered shows with
the ChIMES-based repulsive function a relaxation that is very close
to the one obtained with the PBE calculation (see Figure 4). Overall, the new parameters
reasonably describe the various forms and phases of titania. The band
alignments between the phases are described in the following section.</p><!><p>Gap energy
behavior over particle size for anatase (top) and rutile
(bottom). The DFTB values were obtained with the new parameter set
and are displayed in addition to the PBE results. The PBE results
were aligned with the DFTB ones to match the value for the smallest
nanocrystal structure. Note that for anatase, the smallest crystal
size was not included in the linear fit as that structure did not
follow the quantum confinement behavior for the gap states. The straight
solid lines represent a linear fit, while the dashed lines serve only
as guides to the eye.</p><p>PBE Ti35O70 relaxed nanocrystal displayed
by blue lines aligned at the top oxygen atom to the DFTB relaxation
with the new parameter set (shown by sticks and balls). Relaxation
at the tip is so close to the PBE counterpart that the blue lines
are almost not visible in the representation.</p><!><p>The primary aim of the current paper
is to establish a computationally feasible approach that allows us
to study band alignment across a TiO2 anatase–rutile
nanoparticle interface and to determine the role of the interface
in this alignment. The approach is based on the customized DFTB parametrization
described above. The quality of the new method is assessed by comparison
to calculations at the DFT-PBE level of theory.</p><p>First, we present
the bulk band alignment between anatase and rutile using the branching
point energy approach as it was done for TiO2 in ref (1) previously and compare
the PBE and DFTB values. Afterward, following ref (3), the level alignment deduced
from nonbonded nanoparticle structures is discussed. We compare previous
results to our DFT-PBE calculations as well as to results obtained
with DFTB. Finally, we investigate the influence of the interface
on the band alignments by DFTB calculations on bonded nanoparticles.</p><!><p>First, we consider the band alignment
between the bulk phases using the branching point technique. The branching
point energy EBP, or the charge neutrality
level, can be computed from the average of the mid-level states using
the relation5Index k runs over the Nkk-points being
considered. In every k-point, NVB valence band (VB) and NCB conduction
band (CB) states with respective eigenvalues ϵCBk,i and ϵVBk,i are averaged. We chose NVB = 1 and NCB =
1 independently of possible degeneracies of the bands, as the alignment
is known to depend only weakly on the degeneracy and on the number
of included bands.1Figure 5 compares our results for EBP to those in ref (1). The results obtained with the tio2nano set compare well
to the calculated PBE values as well as to the HSE06 values in ref (1). All levels of theory predict
that rutile and anatase act as hole and electron traps, respectively.</p><!><p>Anatase
and rutile band alignments computed by the branching point
technique with DFTB (tio2nano), PBE, and HSE06. The HSE06 values were
taken from ref (1).</p><!><p>Following
ref (3) we performed
DFT-PBE and DFTB calculations for anatase and rutile nanocrystals
of different sizes. We plotted the HOMO and LUMO levels as well as
their gap values as a function of the number of TiO2 formula
units in the structure raised to the factor −α/3. The
parameter α was set to 1.5 as this yielded a better fit to our
data than a value of 1.35 used in ref (3). Note that we did not include the smallest anatase
crystal in the linear fit, as we found it to not follow the particle
in the box model (neither for PBE nor for DFTB). The PBE and DFTB
results for anatase are in good agreement with the PBEx data of ref (3). By extrapolating the DFTB
nanocrystal results to infinite particle sizes using a linear fit
with respect to n–1.5/3, we obtained
a band gap of 3.02 eV as compared to the bulk value of 2.84 eV. A
similar overestimation can be seen in the extrapolation of the PBE
data, which yields a gap of 2.25 eV as compared to a PBE bulk value
of 2.09 eV.</p><p>In the case of rutile nanocrystals, extrapolation
of the band gap is complicated by the presence of surface states.
This leads to the nonlinearity of the band gap caused by the nonlinear
behavior of the VB edge. As shown in Figure 6, although the highest lying occupied core
states clearly follow the confinement linearly, an occupied surface
state with a nonlinear behavior appears in the gap. This leads to
an initial narrowing of the gap with the particle size before reaching
the point where the surface and core states cross and after which
the gap follows a linear behavior again. Similar behavior can be found
in the PBE results, as shown in Figure 7. The observed behavior of the surface states indicates
that the simple textbook particle in a box model can be applied only
to core states but not to surface states.</p><!><p>HOMO and LUMO levels
for anatase (top) and rutile (bottom) calculated
with DFTB. For rutile, both the highest occupied core states (HOMO-core)
and the occupied surfaces states (HOMO-surf) are shown. The corresponding
levels for the second rutile crystal set are indicated with (S2).
The dashed connection lines serve only as guides to the eye. The indicated
gap sizes of the bulk phases are obtained from the linear fits of
the respective HOMO and LUMO states.</p><p>HOMO and
LUMO levels for anatase (top) and rutile (bottom) calculated
with PBE. For rutile, both the highest occupied core states (HOMO-core)
and the occupied surfaces states (HOMO-surf) are shown. The dashed
connection lines serve only as guides to the eye. The indicated gap
sizes of the bulk phases are obtained from the linear fits of the
respective HOMO and LUMO states.</p><!><p>Comparing the extrapolated value of the band gap for rutile, excluding
the surface states, from the nanocrystals with the calculated band
gap of the bulk, we find that the extrapolated value slightly overestimates
the bulk one with DFTB (2.88 vs 2.76 eV) and underestimates it with
PBE (1.62 vs 1.79 eV). We also note that the rutile data indicate
an inverse quantum confinement trend even when excluding the surface
states, in disagreement with the PBEx data presented in ref (3). We are not entirely sure
what the origin of the discrepancy is. We note, however, that both
rutile slabs34 and nanowires35 exclusively exposing (110) surfaces show an
even–odd behavior in the valence and conduction band edges
with the number of layers. For odd numbers, both display an inverse
quantum confinement behavior, while for even numbers, they display
regular quantum confinement behavior. The effect was found to be more
pronounced for the conduction band states.</p><p>Therefore, in addition,
we also considered another set of quasi
Wulff-type rutile nanocrystals by cutting particles along the (110)
and (101) planes with an even number of (110) layers across the waist
of the particle. In the following, we refer to this new set as SET
2. These nanocrystals contain 76, 92, 100, 108, 116, 132, and 316
TiO2 formula units. To make the crystals stoichiometric,
we removed a number of Ti atoms at the (101) facets as opposed to
adding dangling oxygen ions as was done in ref (3). The particles of this
set are found to be more stable compared to those presented in ref (3) (see Figure 8). We also tried building larger particles
with an even number of (110) layers, but these particles became metallic
and also significantly less stable. Using the stable and nonmetallic
particles of the new set, we obtain an extrapolated value (excluding
surface states) for a band gap of 2.89 eV (see Figure 6). We note that the new set displays the
expected quantum confinement trend with a decreasing HOMO-core to
LUMO gap with increasing crystal size. While the HOMO-core states
of the rutile nanocrystals of the new set are located in the same
energy region as those of the original set, the LUMO states are located
at higher energies. This behavior is therefore consistent with the
even–odd behavior with respect to the number of (110) layers
for the VB and CB positions for slabs and nanowires in refs (34) and (35), respectively.</p><!><p>Comparison
of the stability of rutile nanocrystals in the original
set from ref (3) (SET
1) and the new set (SET 2) as a function of size. Energies are given
per TiO2 formula units.</p><!><p>Given that particles of comparable stability may show contrasting
behavior in terms of their HOMO and LUMO positions, suggesting that
we need to be careful in establishing scaling relations—explicit
simulations are clearly required for small rutile nanocrystals.</p><p>Using the HOMO and LUMO level positions obtained above, band level
alignments in combined nanocrystals, consisting of a rutile nanocrystal
and an anatase nanocrystal, can be predicted. However, note that such
predictions would not take the effect of the actual anatase–rutile
interfaces into account, which can have a significant impact (see
the presentation in the Band Alignment in Bonded
Nanocrystals section). We also note that much of such size
dependence in the alignment is driven by the large variation in the
HOMO and LUMO positions in the rutile nanocrystals. The results also
depend on the type of rutile nanocrystals we use. Nevertheless, combining
an anatase nanocrystal with rutile nanocrystals of the type presented
in ref (3), one obtains
a so-called type II rutile alignment (using the nomenclature of ref (3)) for small rutile nanocrystals,
where rutile acts as an electron trap and anatase as a hole trap.
Increasing the size of the rutile nanocrystal results in a transition
to a so-called type I alignment, where both the HOMO and LUMO states
are located within the rutile nanocrystal. Finally, for even larger
rutile crystal sizes, one obtains a so-called type II anatase alignment
(as also predicted by the branching point calculation with the bulk
structures), where rutile acts as a hole and anatase as an electron
trap. The same quantitative behavior was derived for PBEx in ref (3).</p><p>On the other hand,
by combining an anatase nanocrystal with a rutile
nanocrystal from the new set (S2), one can obtain a number of different
alignments. Most notably, using the smallest rutile nanocrystal of
the new set leads to an alignment where both the HOMO and LUMO states
are located within the anatase nanocrystal.</p><p>With the aforementioned
difficulties in establishing robust trends
in the HOMO and LUMO levels in rutile nanocrystals, we conclude that
we are unable to make a robust prediction in the HOMO–LUMO
level alignment between anatase and rutile nanocrystals, at least
for the size range covered by our simulations, ∼1–4.5
nm, without resorting to explicit simulations.</p><!><p>Exploiting the
efficiency of the DFTB method, we have investigated the influence
of the interface on the gap alignment by simulating explicit interface
structures between different orientations of the smallest anatase
and rutile nanocrystals. In the most stable structures, we have obtained
nanocrystals with a rutile (110) and anatase (101) interface, and
the nanocrystals tend to align with an angle of around 30° with
respect to each other along the rutile [001] and anatase [010] directions.
This alignment, as shown in Figure 9, maximizes the interaction area and the bond formation
between the two structures, minimizing also the interface energy that
was computed according to eq 4.</p><!><p>Typical alignment of the most stable rutile–anatase nanocrystal
interface structures. The red and blue structures represent the rutile
and anatase nanocrystals, respectively.</p><!><p>The prediction using independent nanocrystals (neglecting the interface
effects) suggests a type I alignment with mismatches of 0.07 eV for
the HOMO and 1.03 eV for the LUMO. This suggests complete domination
of the gap by rutile states, due to the narrowing of its gap by surface
states. The gap alignments for all 156 investigated bonded arrangements,
their average, and the value obtained from the noninteracting nanocrystals
are displayed in Figure 10. The five configurations with the lowest energy have also
been marked in the figure. It is obvious that although the prediction
using noninteracting nanocrystals delivers the correct alignment type,
the magnitude in the offset is far off from both the average interface
offset and the one obtained from the most stable interface structures.
This suggests that interface effects have a significant influence
on the level alignment and must be accounted for explicitly. One observes
a slight correlation of the LUMO/HOMO alignments along with the indicated
diagonal line, which would correspond to rigid shifts of the band
edges without any change in the band gap. This, therefore, represents
situations where bands in both phases are shifted due to an interface
dipole that varies in magnitude depending on the relative orientation
of the nanocrystals, but where no new interface states are being created.</p><!><p>HOMO
and LUMO alignment of TiO2 nanoparticles composed
of rutile and anatase nanocrystals. The alignment is positive if the
respective edge state in rutile is higher than in anatase. Blue dots
indicate values obtained from bonded nanocrystals, green crosses indicate
their average, and the red cross indicates the value obtained from
the noninteracting nanocrystals. The five most stable structures are
marked with yellow triangles, but since two pairs have very similar
alignment values, only three triangles can be seen. The dashed diagonal
line serves as a guide to the eye and moving parallel to this line
corresponds to alignments with constant gap sizes for both nanocrystals.
In the line shown, the values for these gaps are 2.08 eV for rutile
and 3.18 eV for anatase, corresponding to the HOMO–LUMO gaps
of the isolated particles.</p><!><p>Some typical LUMO and HOMO wave functions of the interfaces are
displayed in Figure 11. We found three different types of LUMO wave functions and one HOMO
wave function. The HOMO wave function is identical to the rutile HOMO
surface state. The most common LUMO wave function is located exclusively
in the rutile nanoparticle but differs considerably from the LUMO
wave function of the isolated rutile nanocrystal. The second most
common is located directly at the interface, while the least common
one, which only occurs in four of the investigated cases, resembles
the original rutile LUMO wave function. The interface in each of the
five most stable structures is of the first type, which is also the
most common among all investigated nanoparticles. This underlines
the necessity to model nanocrystal interfaces explicitly, as the LUMO
edge states might differ considerably from those obtained from the
independent particle model. This aspect would be important to consider,
not least when modeling catalytic reactions at the surface of nanocrystals.</p><!><p>Typical
LUMO and HOMO wave functions in the anatase–rutile
nanocrystals. (a) The least common LUMO wave function type that is
similar to the LUMO of an isolated rutile nanocrystal, (b) the most
common interface wave function, where the wave function is not present
in the interface region, (c) the second most common interface type,
where the LUMO wave function is located at the interface, and (d)
the HOMO wave function, which is similar in all interfaces and corresponds
to the HOMO surface state of the isolated rutile nanoparticles.</p><!><p>We have developed a
DFTB parameter set using the ChIMES force field
with three-body terms as repulsive potentials especially designed
for rutile and anatase nanocrystals. We found that the three-body
term was crucial to predict the correct relaxation around the low
coordinated atoms at the apex of anatase nanocrystals.</p><p>We used
the new DFTB parametrization, the tio2nano set, to investigate
different anatase–rutile band alignment models to predict effective
charge carrier separation in mixed anatase–rutile systems.
While the bulk band alignment model predicts a type II anatase alignment
where anatase acts as an electron trap and rutile as a hole trap in
accordance with the literature,1,2,5 the nanocrystal model indicates the strong dependence of the band
alignment type on the crystal size in accordance to the predictions
in ref (3). The detailed
shape of small rutile nanocrystals also plays a crucial role. We showed
that two types of rutile nanocrystals that predominantly expose the
(110) facets and have similar stability give rise to rather different
behaviors in terms of band alignment. This fact also makes it difficult
to establish robust rules of thumb when it comes to predicting band
alignment in a rutile/anatase nanocrystal mixture and underlines the
necessity for explicit simulations of those systems.</p><p>Using the
efficiency of the DFTB method, we also investigated the
effect of the anatase/rutile nanocrystal interface on the band alignment.
We optimized the geometry of 156 anatase/rutile nanocrystal pairs
where the mutual orientations were systematically varied. While all
interface models show the same type I alignment, the magnitude of
the band offset varies almost 1 eV with different orientations. This
observation suggests that it is important to consider interface effects
on the band alignment and underpins the importance of being able to
extend the reach of electronic structure simulations beyond the realms
spanned by regular DFT.</p><!><p>The authors declare no
competing financial interest.</p>