Structure and Optical Properties of Small - ACS Publications

Article pubs.acs.org/JPCC

Structure and Optical Properties of Small (TiO2)n Nanoparticles, n = 21−24 Richard B. Wang,*,†,∇ Sabine Körbel,‡,§,∇ Santanu Saha,∥ Silvana Botti,‡ and Natalia V. Skorodumova⊥,# †

Department of Physics and the Competence Centre for Catalysis, Chalmers University of Technology, Gothenburg 41296, Sweden Institut für Festkörpertheorie und -optik, Friedrich-Schiller-Universität Jena and European Theoretical Spectroscopy Facility, Max-Wien-Platz 1, Jena 07743, Germany § Institut Lumière Matière, UMR5306 CNRS, Université Claude Bernard Lyon 1, Villeurbanne F-69622 Cedex, France ∥ Department of Physics, Universität Basel, Klingelbergstrasse 82, Basel 4056, Switzerland ⊥ Multiscale Materials Modeling, Department of Materials and Engineering, Royal Institute of Technology (KTH), Stockholm S-100 44, Sweden # Division of Materials Theory, Department of Physics and Astronomy, Uppsala University, Uppsala S-751 20, Sweden ‡

S Supporting Information *

ABSTRACT: Recently, nanostructured TiO2 (“black TiO2”) has been discovered to absorb visible light, which makes it an efficient material for water splitting. Hydrogenization has been proposed to be at the origin of this beneficial electronic structure of black TiO2. Here, we investigate, using ab initio methods, alternative mechanisms related to structure modifications in nanoclusters that could be responsible for absorption in the visible range. To that end, we apply a combination of computational structure prediction using simulated annealing and minima-hopping methods based on density-functional theory to predict low-energy configurations and time-dependent density-functional theory (TDDFT) using a hybrid functional with optimized Hartree−Fock content to obtain optical absorption edges.

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INTRODUCTION Titanium dioxide (TiO2) is mainly used as antireflection coating for solar cells and as photoelectrodes for photochemical energy conversion processes.1 The reasons for the continuous interest in TiO2 for energy conversion are that titanium dioxide is chemically inert and has outstanding corrosion resistance to aqueous solutions.2,3 The photoelectrochemical (PEC) cell consisting of a crystalline TiO2 anode and a Pt cathode can be used for water splitting under irradiation of light. Both optical and catalytical functionalities are required for the PEC cells to obtain maximal absorption of solar energy. Nonetheless, an important drawback for its application as photoelectrode is related to the limited ability for light absorption due to a relatively large band gap value.1,4 TiO2 absorbs only in the ultraviolet part of the solar emission spectra, thus imposing low conversion efficiency. A number of attempts have been made to narrow the band gap of TiO2. Many attempts were focusing on doping of TiO 2 with cation or anion atoms. 5−7 TiO 2 nanostructures have attracted great attention since the prominent discovery made by B. O’Regan who demonstrated that the photovoltaic cell with sintered anatase TiO2 nanoparticles exihibits a commercially realistic energy-conversion efficiency.2 Since then, many studies have been performed on the synthesis, properties, and modifications of TiO2 nanomaterials, which are referred to in recent review papers.8,9 TiO2 nanocrystals (NCs) are mostly synthesized with sol− gel methods, resulting in a high degree of crystallinity.10,11 Although the crystal phases and the morphology are strongly © 2017 American Chemical Society

affected by the synthesis conditions, both experiment and theory proved that smaller samples (with a diameter smaller than 11.2−17.6 nm for hydrothermal samples) show the anatase structure.12,13 State-of-the-art techniques have been employed to characterize the structure of TiO2 nanoparticles, which indicate that the number of distorted octahedral and under-coordinated Ti atoms at the surface of nanoparticles is increased as the particle size decreases.14−17 Hamad et al. performed a theoretical calculation on (TiO2)n clusters (n = 1− 15) with a combination of simulated annealing, Monte Carlo basin hopping, and genetic algorithm methods.18 They claimed that particles with n ≥ 11 have at least one central octahedron surrounded by a shell of surface tetrahedra, trigonal bipyramids, and square base pyramids. Qu and Kroes19 optimized (TiO2)n for n = 1−9 using density-functional theory at the B3LYP/ LANL2DZ level and argued that the lowest-energy (TiO2)n structures contain one or two terminal oxygen atoms. Mingyang Chen et al.20 employed a tree growth (TG) algorithm with a hybrid genetic algorithm (HGA). They found that optimized clusters (for n = 2−13) do not show the character of a TiO2 bulk crystal with 6-fold coordinated Ti. For larger clusters (1−2 nm) synthesized in experiment, the structure of these particles is especially sensitive to the synthesis methods and various treatments and reactions,21−24 which can Received: November 14, 2016 Revised: April 5, 2017 Published: April 6, 2017 9528

DOI: 10.1021/acs.jpcc.6b11461 J. Phys. Chem. C 2017, 121, 9528−9536

Article

The Journal of Physical Chemistry C

continued until the maximum force component of any atom was less than 25 meV/Å. It is well-known that (semi)local functionals such as PBE dramatically underestimate band gaps if the band gap is simply extracted directly from the band structure. Hybrid functionals, which replace a part of the exchange energy by Hartree−Fock (HF) exchange, improve over the (semi)local functionals in this respect. For solids, the fraction of HF exchange is typically chosen to be about 25%. However, it is well-known as well that for wide-band gap semiconductors and especially for molecules, the fraction of HF exchange needed to reproduce experimental gaps is larger than 25% and in general depends on the system considered.44 Luckily, in the case of molecules, the so-called ΔSCF (delta self-consistent field) approach is applicable instead, which is capable of yielding quasiparticle band gaps with a similar quality to more sophisticated many-body methods such as GW,45 at computational costs of DFT calculations, and without the need to use a sophisticated or specially tuned exchange-correlation functional. In the ΔSCF approach, quasiparticle (HOMO−LUMO) gaps are given by

in turn tune their chemical reactivities and electronic and optical properties. Very little work on structure prediction of bare clusters in this size regime has been found in the literature. Studying the structure of these bare nanoclusters is fundamentally important not only to determine their phase stability, but also to gain a basic understanding of structure− property relationships in small clusters.25 Moreover, the combined analysis of electronic and optical (e.g., wide absorption spectrum) properties either of experimentally existing26,27 or of theoretically predicted clusters28,29 has been of great help in the study of both ground-state and excited-state properties of cluster systems. In our study, we therefore investigate nanoparticles with a size similar to that in experiment. Our nanoclusters have diameters of about 1.2− 1.5 nm. Most theoretical studies have focused on passivated nanoclusters in this regime, which assume the clusters having bulklike structure.30 However, passivation usually significantly changes the structure and properties of the nanocluster, which prevents us from understanding the intrinsic properties of the bare clusters. Here, we have performed simulated annealing followed by simulations using the minima hopping method (MHM) for structure prediction of (TiO2)n nanoclusters (n = 21−24). Electronic and optical properties of these clusters are studied with density functional theory (DFT) and time-dependent DFT (TDDFT) using a hybrid exchangecorrelation functional.

0 +1 −1 EgQP = EA − IE = 2Etot − Etot − Etot

(1)

where IE and EA are ionization potential and electron affinity, respectively, and E0tot, E+tot1, and E−1 tot are the total energies of the molecule in the neutral state, after removal and after addition of an electron, respectively, calculated at DFT level. Here, we calculated accurate quasiparticle gaps from ΔSCF using the PBE0 functional,46 which admixes a fraction of 25% Hartree− Fock exchange. Afterward, we adjusted the fraction of HF exchange in the PBE0 functional such that the ΔSCF gaps were reproduced (see Supporting Information), and calculated densities of states (DOS), which are not easily accessible with ΔSCF (in principle, excited states can be accessed using constrained ΔSCF). The resulting fractions of HF exchange ranged from 43% to 48% for all geometries. Optical absorption of the nanoparticles was studied with time-dependent densityfunctional theory (TDDFT), using the hybrid PBE0 functional with the optimized fraction of HF exchange. The calculations of electronic structure and optical absorption were carried out using the program NWChem,47 employing the Gaussian basis set “lanl2dz-ecp” and the atomic pseudopotentials of Dunning and co-workers.48−51 Convergence tests for small TiO2 clusters were performed to ensure convergence of the optical gap with respect to the basis size. Coupling terms were included in the TDDFT kernel.

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COMPUTATIONAL METHODS In the present study, the simulated annealing and structure optimization was carried out within the framework of densityfunctional theory (DFT) using the projected augmented wave (PAW) method31 as implemented in the VASP package.32−34 PAW potentials with the valence states 2s and 2p for O and 3d and 4s for Ti were employed. The exchange-correlation interaction was treated at the level of the generalized-gradient approximation (GGA) using the exchange correlation functional of Perdew, Burke, and Ernzerhof (PBE).35 Periodic boundary conditions were adopted. A large simple cubic cell of 25 × 25 × 25 Å3 was chosen in our calculation to avoid interaction between adjacent cells. All calculations were carried out at the Γ point of the Brillouin zone. Simulated annealing was employed to obtain input structures for the MHM code, which was then used to find low-energy structures. A planewave kinetic-energy cutoff of 400 eV was used for the simulated annealing simulation. After having obtained the low-energy configurations of each size of (TiO2)n (n = 21, 22, 23, and 24), selected structures (five lowest-energy structures for each size) were further relaxed with the conjugate-gradient method with a higher cutoff energy of 600 eV using VASP. The energy landscape of the (TiO2)n clusters (n = 21−24) was explored using the Minima Hopping Method36−39 (MHM) searching for new configurations. This algorithm can efficiently find low-energy structures. The structural exploration was performed at the DFT level using the electronic-structure package BigDFT,40 in which the MHM is implemented. The BigDFT code uses Daubechies Wavelets as its basis set. Again, the PBE functional was used to describe the exchangecorrelation energy35 along with soft norm-conserving Hartwigsen−Goedecker−Hutter (HGH) pseudopotentials including a nonlinear core correction.41,42 The parameters for convergence were set in such a way that energy differences were converged to within 10−4 eV and the relaxation43 of the configuration was

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RESULTS AND DISCUSSION Figure 1 shows the total energy of all cluster geometries after simulated annealing and structure prediction for each size of the TiO2 nanoclusters. The total energy of the lowest-energy configuration is set to zero. It is clear that many metastable configurations are close in energy to the lowest-energy structure for each size. The energy differences between the lowest-energy configuration and the second-lowest one ranging from 86 to 294 meV for all sizes of nanoclusters. In our following study, we have selected from our simulation the five lowest-energy structures for each size to study the relationship between structure and electronic and optical properties. Figure 2 shows the lowest-energy configuration for each size. The Ti atoms sit in oxygen polyhedra that are color-coded according to the coordination number of the Ti atom. We find coordination numbers between 4 (oxygen 9529


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coordinated Ti atoms to that of 5-fold coordinated Ti atoms increases to accommodate the increased number of 2-fold coordinated O atoms. The cohesive energy of a nanocluster is calculated via Ecoh =

E(TiO2)n n

− E TiO2

(2)

where E(TiO2)n is the total energy of the nanocluster, and ETiO2 is the total energy of a TiO2 monomer. The cohesive energies of all of the nanoclusters we have studied are shown in Figure 4. The cohesive energies of all nanoclusters decrease as the size decreases in general; that is, the clusters become more stable with increasing size. However, the cohesive energies of the nanoclusters with n = 23 are higher (their cohesion is weaker) and do not follow the general trend. Quasiparticle and Optical Gaps. After having determined the structures of our nanoclusters, we studied their quasiparticle gaps (HOMO−LUMO gaps, see Figure 5). As can be seen from Figure 5, most of the HOMO−LUMO gaps range from 6.3 to 6.9 eV and do not vary strongly with size in the range studied, except for the clusters with n = 23, for which two of the five structures have a much lower quasiparticle gap of 5.5 and 5.6 eV, respectively. Also, for n = 24 there is one outlier with a larger gap than the other structures. This larger gap of 7.1 eV belongs to the cage structure. The optical gaps are much smaller than the HOMO−LUMO gaps due to electron−hole interaction. They range from 4.2 to 4.6 eV for all geometries except for the cage geometry with n = 24, which has a larger optical gap of 4.8 eV. Structure−Property Relationships. To link the band gap behavior of all nanoclusters to their structures, the gap versus fraction of 2-fold coordinated O atoms in selected nanoclusters is depicted in Figure 6. A fraction of one indicates that all of the O atoms in the nanocluster are 2-fold coordinated, which is the case for the cage-like structure in which all Ti atoms are 4-fold coordinated. From Figure 6, it is evident that the band gap increases as the fraction of 2-fold coordinated O atoms increases in general. There are structures with the same fraction of 2-fold coordinated O atoms but with different gaps. In the previous section, we found that, whereas most clusters have similar gaps, there are two clusters with n = 23 that have smaller gaps, and one geometry with n = 24 that has a larger gap. In the following, we will discuss these outliers and give a more detailed analysis of their electronic structure by means of their density of states (DOS) and HOMO and LUMO levels. Clusters with Deep Levels. Figure 7 shows the quasiparticle DOS and the onsets of the optical absorption spectra of the nanoclusters with n = 23. Structures 1−3 have a DOS that is free of any deep levels, but in the DOS of structures 4 and 5, two occupied states appear high above the continuum of occupied states, as indicated by the arrows. The optical absorption spectra also shown in Figure 7 are the oscillator strengths of the lowest-lying 12−30 optical transitions with a Gaussian broadening by 25 meV (thermal energy at room temperature). The difference between optical absorption onset and quasiparticle gap shows the strong electron−hole interaction present in these small nanoparticles. Like the quasiparticle gaps, the optical ones are smaller in the presence of a dangling bond (structures 4 and 5). The reduction in optical gap is strongly dependent on the Hartree−Fock content (see Figure 9). With the optimized Hartree−Fock content

Figure 1. Total energy of all geometries after structure optimization for each size of the TiO2 nanoclusters with n = 21, 22, 23, and 24. The total energy of the lowest-energy structure is set to zero.

Figure 2. Lowest-energy structure for (a) n = 21, (b) n = 22, (c) n = 23, and (d) n = 24 formula units, respectively. Ti atoms are represented by large blue spheres, and oxygen atoms by small red ones. The coordination number (CN) is indicated by the color of the polyhedra (blue, CN = 4; green, CN = 5 ; yellow, CN = 6).

tetrahedron) and 6 (oxygen octahedron) in each cluster. It can be seen in Figure 2 that higher-coordinated Ti atoms (6-fold coordinated or 5-fold coordinated) are situated in the core of the cluster. The shell mainly consists of 5-fold coordinated and 4-fold coordinated Ti atoms. Next, in Figure 3 we show the structural features of the putative global minima for all sizes of the nanoclusters. In general, all four putative ground-state structures have more or less the same number (9−10) of O atoms whose coordination number (CN) is larger than 3. That is, as the size increases, the number of 2-fold coordinated O atoms increases, while the number of highly coordinated O atoms (CN ≥ 3) remains unchanged. Hence, the ratio of the number of 4-fold 9530


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Figure 3. Histogram of CN of Ti and O in the putative global-minimum structure for (a) n = 21, (b) n = 22, (c) n = 23, and (d) n = 24, using a radial cutoff of 2.6 Å.

Figure 4. Cohesive energies of all of the studied nanoclusters. Figure 6. HOMO−LUMO gap versus fraction of 2-fold coordinated O atoms in the selected nanoclusters.

participation of the dangling-bond states in the optical transitions. Indeed, as shown in Figure 12, at about 40% Hartree−Fock content the HOMO contribution to the lowestlying optical transition drops for the structures with dangling bonds (structures 4 and 5). In Figure 8, where the HOMO and LUMO of two representative clusters with n = 23 are visualized, we can see that the midgap states of structure 4 belong to orbitals localized at the oxygen atom with the dangling bond. The HOMO and LUMO of all structures with n = 23 are depicted in Figure S2; the two structures selected here are representative for the structures 1−3 (without dangling bonds) and the structures 4 and 5 (with dangling bonds), respectively. In Figure 8, we also see that for all structures, the HOMO mainly consists of O 2p states, while the LUMO is mostly made up of hybridized Ti 3d states and O 2p states. For all structures, the LUMO is delocalized over about one-half of the cluster or more; for structures 1−3, this is also true for the HOMO. Structure 3 has the largest band gap, which might be related to its HOMO being more homogeneously distributed over the cluster than in the other structures. We concude that for the two nanoclusters with n = 23 with smaller gaps, the presence of a 1-fold coordinated O atom (dangling bond) contributing occupied localized states above the continuum of occupied states is common to both of them and must be the main reason for band gap narrowing.

Figure 5. Quasiparticle (QP) and optical gaps of all clusters from ΔSCF. For the nanoparticles with n = 24, the third structure (“3”) is the cage geometry.

between 40% and 50%, the optical gap is almost unchanged by dangling bonds, whereas a smaller Hartree−Fock content leads to a strong reduction. As we show in Figure 10, other than the quasiparticle gap, the optical gap depends nonlinearly on the HF content. The position of the deep levels above the continuum of occupied states is approximately independent of Hartree−Fock content (see Figure 11). We attribute the strong dependence of the contribution of the dangling bonds on the optical gap therefore to HF having a strong impact on the 9531


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Figure 9. Optical spectra of all clusters with n = 23 calculated with different Hartree−Fock contents. Figure 7. Quasiparticle DOS (orange) and oscillator strengths (dark blue) of the first six optical transitions (dark blue) for n = 23 from PBE0 with the optimized fraction of HF exchange. The black dashed line is the HOMO level, which is set to 0 for all structures. In other words, all states up to the black dashed line are occupied. Arrows indicate deep levels.

Figure 10. Optical (solid lines) and QP (dashed lines) gaps of all clusters with n = 23 as a function of Hartree−Fock content.

coordinated O atoms and 4-fold coordinated Ti atoms, as depicted in Figure 13. As can be seen in Figure 13, the HOMO is made up of O 2p states, and the LUMO mainly consists of Ti 3d states. This is different from the other, more amorphous structures with n = 23, where the LUMO consists of a mixture of O 2p and Ti 3d states. Both HOMO and LUMO are delocalized almost over the entire cluster. This suggests that the lower gaps of the other clusters result from the higher coordination numbers of either O or Ti atoms. Band Gap and Local Bond Order. In our study, we observe that clusters having the same number of atoms with the same

Figure 8. HOMO (bottom row) and LUMO (top row) orbitals of two of the nanoclusters with n = 23, the lowest-energy structure (“structure 1”) and a structure with a dangling bond indicated by a black arrow (“structure 4”).

Comparison with the Cage Structure. For the clusters with n = 24, we notice that one gap is slightly larger than the others. This large gap (7.1 eV) belongs to a cage-like structure consisting of a network of distorted tetrahedra with 2-fold 9532


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Figure 13. HOMO (bottom row) and LUMO (top row) of the lowest-energy structure (“structure 1”) and the cage-like structure (“structure 3”) for n = 24.

Figure 11. Densities of states (DOS) of all clusters with n = 23 calculated with different Hartree−Fock contents.

third-order invariants are commonly used to distinguish different crystal structures as well as fluids.52,54−56 Here, we consider q4 only because we were unable to find any trend in the data for q6. In the following, the local bond-order parameter q4 is calculated for each cluster. The local bond-order parameter is defined as ⎛ 4π ql = ⎜⎜ ⎝ 2l + 1

l

∑ m =−l

⎞1/2 |qlm(i)| ⎟ ⎠ 2⎟

(3)

where qlm is a (2l+1) dimensional complex vector with the components qlm(i) =

1 Nb(i)

Nb(i)

∑ Ylm(riĵ ) j=1

(4)

Here, Nb(i) is the number of neighboring particles of particles i within a given distance rd. Ylm(r̂ij) are the spherical harmonics calculated for the normalized direction vector r̂ij, where vector r̂ij is determined by the polar and azimuthal angles, θij and ϕij. In Figure 14, we show the remaining band gap variation together with the q4 values of the TiO6 octahedra. Indeed, it appears that there might be a trend to higher band gaps for higher q4; however, there are exceptions, and the statistics here are not sufficient to establish a correlation. Such a systematic study is a work of its own and exceeds the scope of this work.

Figure 12. Contributions of the HOMO to the lowest optical transition for all clusters with n = 23 as a function of Hartree−Fock content.

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CN have different band gaps and cohesive energies. Because we have seen that geometries with 4-fold coordinated Ti have larger gaps than those with 6-fold coordinated Ti, we now concentrate on the octahedra to try to achieve a smaller gap. We will try to relate their distortion in the cluster with the band gap of the cluster. A possible way of measuring the distortion of an atomic environment (such as a TiO6 octahedron) is the local bond-order parameter ql. The advantage of this analysis is that this bond-order parameter is rotationally invariant and independent of the orientation of the clusters.52,53 The distribution of the parameters q4 and q6 together with their

CONCLUSIONS The structure and optical properties of (TiO2)n nanoparticles with n = 21−24 have been studied using advanced structureprediction methods and time-dependent density-functional theory with a hybrid functional. We found a large number of structures that are all based on a few structural motifs, distorted octahedra, pyramids, and tetrahedra connected to each other through oxygen atoms at the corners of the polyhedra. The Ti atoms sit at the center of these polyhedra. In our data set of structures, many differ only by some minor distortion of a 9533


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cohesive energy and band gap. The local bond order parameter appears to be a promising parameter to distinguish between similar cluster structures. In short, to create deep levels and enhance optical absorption in the visible range, one needs to create dangling bonds. We expect that the exact position of deep levels created by dangling bonds may vary with the nanocluster size and geometry. Milder structural distortions are not sufficient for this purpose. An alternative to achieve the same effect could be to work with off-stoichiometric or doped TiO2 clusters.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b11461. Example for the band gap dependence on the fraction of Hartree−Fock exchange and figures of the geometries of the low-energy configurations with n = 23 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Richard B. Wang: 0000-0001-7321-8594 Author Contributions ∇

R.B.W. and S.K. contributed equally to this work.

Notes

The authors declare no competing financial interest.

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Figure 14. q4 (blue-green symbols, right axis) of the TiO6 octahedra and band gaps (black symbols, left axis) for clusters with comparable structures (excluding structures with dangling bonds and the cage structure).

ACKNOWLEDGMENTS We thank SNIC (Swedish National Infrastructure for Computing) for providing computing time. Support from the Swedish Research Council is acknowledged. We would like to thank Prof. Dr. Stefan Goedecker, University of Basel, for his useful discussions and suggestions to this work. Computational resources provided by the Leibniz Supercomputing Centre through the SuperMuk project pr48je is acknowledged. S. Saha acknowledges the support from the Swiss National Science Foundation. Calculations were performed at the CSCS under project s499 and at the sciCORE (http://scicore.unibas.ch/) scientific computing core facility at the University of Basel.

single polyhedron and are nearly degenerate in energy (Figure 1). Moreover, as the size of the nanoparticles increases, the ratio of the number of 4-fold coordinated Ti atoms to that of 5fold coordinated Ti atoms increases. In some structures, a single Ti atom with the bulk coordination number of 6 was also found. The electronic properties of these clusters strongly depend on their structure details. It is impossible to find one single structural parameter to describe their electronic properties. However, some general rules are observed in our study. First, the dangling bonds of one-fold coordinated O atoms will significantly reduce the HOMO−LUMO and optical gap, whereas the effect of the amorphous structure and varying coordination numbers of atoms has a much smaller influence on the gap. Despite the amorphous structure of the clusters, HOMO and LUMO are quite delocalized. Only in the case of singly coordinated O atoms (dangling bonds) have we observed a localized HOMO mainly composed of O 2p states, whose energy eigenvalue is located high above the continuum of occupied states and forms a deep level. The HOMO− LUMO gap and the optical gap qualitatively show the same dependence on the cluster geometry. Because of the small size of our clusters, however, the electron−hole interaction is strong, and the optical gaps are considerably smaller than the HOMO−LUMO gaps. Second, the gaps of our studied nanoclusters generally increase as the fraction of 2-fold coordinated O atoms increases. The local bond order parameter is calculated for structures with the same number of atoms with the same coordination number but with different

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