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Home Search Collections Journals About Contact us My IOPscience Anatase–rutile phase transformation of titanium dioxide bulk material: a DFT + U approach This article has been downloaded from IOPscience Please scroll down to see the full text article 2012 J Phys.: Condens Matter 24 405501 (http://iopscience.iop.org/0953-8984/24/40/405501) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 128.151.244.46 The article was downloaded on 22/04/2013 at 15:59 Please note that terms and conditions apply IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J Phys.: Condens Matter 24 (2012) 405501 (10pp) doi:10.1088/0953-8984/24/40/405501 Anatase–rutile phase transformation of titanium dioxide bulk material: a DFT + U approach Nam H Vu1 , Hieu V Le1 , Thi M Cao1 , Viet V Pham1 , Hung M Le1 and Duc Nguyen-Manh2 Faculty of Materials Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK E-mail: hung.m.le@hotmail.com Received June 2012, in final form 12 August 2012 Published September 2012 Online at stacks.iop.org/JPhysCM/24/405501 Abstract The anatase–rutile phase transformation of TiO2 bulk material is investigated using a density functional theory (DFT) approach in this study According to the calculations employing the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional with the Vanderbilt ultrasoft pseudopotential, it is suggested that the anatase phase is more energetically stable than rutile, which is in variance with the experimental observations Consequently, the DFT + U method is employed in order to predict the correct structural stability in titania from electronic-structure-based total energy calculations The Hubbard U term is determined by examining the band structure of rutile with various values of U from to 10 eV At U = eV, a theoretical bandgap for rutile is obtained as 3.12 eV, which is in very good agreement with the reported experimental bandgap Hence, we choose the DFT + U method (with U = eV) to investigate the transformation pathway using the newly-developed solid-state nudged elastic band (ss-NEB) method, and consequently obtain an intermediate transition structure that is 9.794 eV per four-TiO2 above the anatase phase When the Ti–O bonds in the transition state are examined using charge density analysis, seven Ti–O bonds (out of 24 bonds in the anatase unit cell) are broken, and this result is in excellent agreement with a previous experimental study (Penn and Banfield 1999 Am Miner 84 871–6) (Some figures may appear in colour only in the online journal) Introduction with/without the presence of catalysts [2] In this paper, we present a theoretical investigation of the anatase–rutile phase transformation using first-principles computational methods In solar-cell technology, the phase stability of nanostructured titania has become an important issue since it plays a significant role in the productivity of dye-sensitized solar-cell (DSC) devices [3, 4] Therefore, the understanding of the anatase–rutile phase transition of TiO2 has become a leading concern and been investigated in various experimental aspects over the past few decades The pressure–temperaturedependent anatase–rutile phase transition was examined and reported by Dachille et al [5] and Jamieson et al [6] and it was implied that anatase was metastable with respect to Titania is a well-known material for its wide applications in catalysis, solar-cell devices, and pigmentation in paint manufacturer [1] It is well-established that titania (TiO2 ) exists in nature under ambient condition in three distinct phases, which include anatase, rutile, and brookite In experimental synthesis, anatase is the major product, with a minority amount of rutile and brookite being formed However, it has been proved in many studies that rutile is the most thermodynamically stable phase among those three phases, while anatase and brookite are metastable and usually converted to rutile when the temperature is raised 0953-8984/12/405501+10$33.00 c 2012 IOP Publishing Ltd Printed in the UK & the USA J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al b a Figure Periodic structure of anatase and rutile TiO2 The different Ti–O bonds in each structure are classified (according to [21]) moduli were within 2% and 10% of the experimental data, respectively Nevertheless, their calculations suggested that the anatase phase was more stable than rutile at K and ambient pressure, which was a major disagreement with most reported experimental results Barnard and Zapol [19] reported a theoretical study in which the transition enthalpy of nanocrystalline anatase and rutile were calculated as a function of shape, size, and surface passivation In a subsequent work [20], the surface passivation was further investigated, and it was concluded that surface hydrogenation had a significant affect on the shape of rutile nanocrystals, which led to a dramatic increase in the anatase–rutile phase transition In this study, a potential energy profile of the anatase–rutile transformation is investigated using DFT-based calculations From such a potential energy profile, we will be able to present the transition structure, and details of the transformation process are discussed in terms of Ti–O bond ruptures and charge density analysis rutile A following study was conducted to investigate the kinetics and mechanism of the rutile–anatase transformation under the catalysis of ferric oxide [7] The kinetic scheme of such a transformation was further investigated and reported in a study by Gribb and Banfield [8], in which they suggested that the rate increased dramatically with very finely crystalline anatase In a subsequent study, the anatase–rutile transition was nucleated at anatase [9], and a phase transformation mechanism was proposed that involved seven bond ruptures (out of 24 Ti–O bonds per anatase unit cell) during the transformation process [10] In another experimental study reported by Ha et al [11], the anatase–rutile transformation in titania with alcohol rinsing was investigated using x-ray powder diffraction, FT-IR spectroscopy, and thermogravimetry It was witnessed by Nolan and co-workers that in sol–gel-synthesized TiO2 photocatalysts, the syn-anti binding hindered the cross-linking gel network and caused the anatase–rutile transformation to accelerate at low temperature [12] More details can be consulted in a recent review on the anatase–rutile phase transformation given by Hanaor and Sorrell [13] Much computational effort has been conducted and reported for TiO2 material since the rigorous development of density functional theory (DFT) for condensed matter calculations An abundant variety of aspects have been proposed and studied for years using such theoretical approaches Native point defects in the anatase phase were previously reported using the local density approximation (LDA) [14] and ultrasoft [15] pseudopotential in the Vienna ab initio simulation package (VASP) [16] A two-dimensional approach was conducted by Sato et al [17] as stacked and single-layered lepidocrocite-type titania were investigated using different DFT functionals Phase stability of the body-centered tetragonal and distorted-diamond phases of titania were investigated by Vegas et al [18] In an earlier investigation, Muscat and co-workers conducted first-principles calculations of the crystal structures, bulk moduli, and stability of seven different titania polymorphs, which included anatase and rutile phases [9] In such work, it was reported that the unit-cell volume and bulk Anatase and rutile polymorphs of TiO2 As stated earlier, rutile is the most thermodynamically stable among the phases known under ambient conditions, and has attracted the attention of many investigators The rutile structure, as shown in figure (along with the structure of the anatase phase), adopts the P42 /mnm tetragonal space group The unit-cell parameter for rutile was determined in a previous ˚ and c = 2.954 A ˚ experimental study [21] as a = 4.587 A Each Ti ion in the rutile phase octahedrally connects to six ˚ (for an equatorial O ions, and the Ti–O bond length is 1.95 A ˚ (for an apical bond) This fact implies that the bond) or 1.98 A octahedral coordination of each Ti ion is slightly distorted If we consider a × × rutile supercell (this will be later employed to find the anatase–rutile transition structure), four ¯ plane Ti ions are located on the 110 Anatase, the metastable phase under ambient conditions, adopts the I4/amd tetragonal space group Experimentally, the unit-cell parameters of the anatase phase were determined ˚ (for a) and 9.502 A ˚ (for c) [21] Similarly to the as 3.782 A J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al rutile phase, each Ti ion in the anatase phase octahedrally ˚ connects to six O ions with bond lengths of 1.93 or 1.98 A From the experimental data, it can be seen that the rutile unit cell is more compact although the Ti–O bond lengths in both structure are almost similar As suggested by Hanaor and Sorrell [13], anatase can be converted to rutile under the affect of pressure rather than temperature (they observed no phase transformation as the temperature was raised up to 2500 K) Interestingly, it is found that nano-particles of the anatase phase have wide commercial applications in photocatalyst solar cells In the anatase unit cell, the four Ti ions are not located on the same plane, and a dihedral angle of 76.3◦ is observed according to our DFT optimizations 3.2 The DFT + U method Alternative approaches have been developed to improve the treatments for exchange–correlation effects in DFT formation energies, which include the use of the DFT + U method [25–27] and the Heyd–Scuseria–Ernzerhof (HSE) hybrid functional [28] The use of the HSE functional was previously employed by Janotti and co-workers to investigate the defects and oxygen vacancies in TiO2 [29, 30] The Becke 3-parameter Lee–Yang–Parr (B3LYP) functional [31, 32], originally fitted to reproduce molecular properties, has also been used to make a correct description of defect states on reduced TiO2 surfaces [33] All these calculations using hybrid functional methods have improved the bandgap description, but so far they have been applied for studying the rutile phase of TiO2 only In this study, we use the DFT + U method to investigate the phase transformation from rutile to anatase phases in TiO2 , and therefore provide a correct prediction of phase stability between these two structures as well as their semiconductor gaps Within the DFT + U implementation, the total energy of a system is expressed as follows: Computational details 3.1 DFT method DFT calculations presented in this study are performed using the Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional [22, 23] with the Vanderbilt ultrasoft pseudopotential [15], as implemented in the Quantum Espresso package [24] The ultrasoft pseudopotentials include 2s and 2p electrons for O, and 3s, 3p, 3d, and 4s electrons for Ti The total energy convergence criterion for self-consistent field (SCF) calculations is 1.36 × 10−8 eV per unit cell (or 2.27 × 10−6 meV/atom) For optimizing anatase and rutile cells, the energy convergence criterion is 1.36 × 10−6 eV per unit cell (or 2.27 × 10−4 meV/atom) and the ˚ −1 per gradient convergence criterion is 2.57 × 10−4 eV A ˚ −1 /atom) In most DFT studies, unit cell (or 0.043 meV A the value of cut-off energy plays a very important role in calculation accuracy and computational cost In this work, the cut-off energy is determined by performing a series of total energy calculations with respect to cut-off values from 20 to 50 Ryd When the cut-off energy reaches 40 Ryd, satisfaction in total energy convergence for both anatase and rutile phases has been found, and we therefore select this cut-off level to conduct DFT calculations The k-point mesh is another important factor that has a significant effect on the total energy In this work, with the affordability of our computational resource, a (4 × × 2) k-point grid is sufficient to provide satisfactory accuracy of the DFT calculations of titania As mentioned earlier, the relative phase stability of anatase and rutile was evaluated in a previous computational work [9] The reported results showed that anatase was more thermodynamically stable than rutile, and such results are somewhat contradictory to the experimental thermodynamics data [13] In our case, when the PBE functional is applied to compute the total energy of anatase and rutile, we also confront similar computational uncertainties, i.e., the anatase phase is shown to be more energetically stable than rutile Thus, a reliable correction method must be employed to deal with the failure of the conventional DFT calculations for TiO2 EDFT+U = EDFT + EHub − EDC (1) where EHub is the Hubbard Hamiltonian contribution representing the Hartree–Fock-like interaction This term replaces the DFT on-site due to the fact that one subtracts a double counting energy EDC which supposedly equals the on-site DFT contribution to the total energy [34] Since the introduction of DFT + U, this method has been successfully employed in various condensed matter systems, which include a case study for electronic structure and structural stability of uranium dioxide (UO2 ) [35] The extended Hubbard model was previously employed to study three various systems [34], which included two typical semiconductors (Si and GaAs) Although the DFT + U method is designed for strongly-correlated Mott–Hubbard insulators, the use of DFT + U to describe the origin of semiconductor band gaps is still believed to be valid It has been implied by Campo and Cococcioni [34] that both types of material (Mott insulators and band insulators) have the same nature, i.e the electronic defects in the exchange–correlation feature that are missing from conventional DFT; and the DFT + U method is capable of including the missing effects of electron–hole exchange interaction in wide bandgap semiconductors For the TiO2 case, although the +4 oxidation state is dominant, its bandgap has a high degree of covalent bonding between 3d-states of Ti with 2p-states of O As we will demonstrate in section 4.1 of this paper, the origin of the bandgap cannot be described purely by the ionic nature of a full shell ion (3d0 ) of titanium even within the PBE functional for the exchange–correlation interaction In fact, the method DFT + U has been employed to deal with TiO2 polymorphs in many previous studies [36–38] Therefore, we believe that it is appropriate to choose the DFT + U method to deal with titania bulk material In particular, DFT + U has been applied to take into account the strong electron correlation effect in the TiO2 J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al problem, and the reported results are very consistent with the experiments Morgan and Watson [38] demonstrated such a method for oxygen vacancies at the rutile (110) surface, and found that with U ≥ 4.2 eV, the electronic charge on Ti atoms was strongly localized Jedidi et al [37] employed the DFT + U method to explore the electronic absorption of several gaseous molecules on the rutile (110) surface with different values of U and DFT functionals, and it was concluded that the value of U depended strongly on the DFT functional used The defects in bulk rutile TiO2 were investigated using the DFT + U approach with U = 2.5 eV, which provided fair agreement with the experiments for the energetic gap states Recently, Dompabo et al [36] conducted a pressure-induced study of TiO2 polymorphs using various values of U (from to 10 eV), and concluded that with an approximate U value of eV, the best agreement was found for the transition pressures of anatase-columbite and rutile-columbite phase transformations In nature, the columbite polymorph is formed under extremely high pressure [9], and it can be considered as a defect structure Similar to anatase and rutile, each Ti ion in columbite octahedrally connects to six O ions; however, the Ti cation is distorted from the octahedron center From the reported studies, it can be seen that the value of U depends strongly on the choice of DFT functionals and pseudopotentials Thus, it is significant in our study to determine the appropriate value of Hubbard potential (U) based on the selected DFT functional and pseudopotential In this paper, we apply the Hubbard U correction within the PBE functional approximation for both Ti and O sites in order to improve the semiconductor bandgap description in TiO2 Note that the on-site Coulomb correction applied to both Ti and O sites has also been applied previously in order to investigate the oxygen vacancy effect in reduced rutile TiO2 [39], although these calculations were performed within the LDA + U approximation convergence criteria are attained, and the intermediate image (crystal configuration) with highest energy is considered as the transition state (saddle point) The transformation of the anatase to the rutile phase in our study not only requires numerous changes in the coordination of Ti–O bonds, but the transformation process also requires a significant change in the unit-cell volume In other words, at the intermediate transition state, the transformation involves bond ruptures (of Ti–O bonds [10] according to Penn and Banfield) together with a simultaneous change of the conventional unit cell To be more specific, the intermediate cell vectors are adjusted based on the analysis of stress tensors with respect to cell vectors given by the Quantum Espresso code [24], which is available for use in our computational facilities The idea of finding such a transition pathway with volume-change has been recently adopted to deal with the solid–solid transformation of CdSe material Sheppard et al [41] proposed modifications to improve the NEB method for solid-state transition, and the new method was termed the ss-NEB method In the traditional NEB method, the intermediate transition states are optimized by adopting a constant volume assumption Such a traditional algorithm is only valid for systems that not involve volume changes during the transformation process (gas-phase reaction in vacuum, gas–surface interaction) On the other hand, the ss-NEB method take the changes in cell vectors into account by analyzing the stress tensors [41] In this study, we have implemented an algorithm that performs ss-NEB force analysis and calls the Quantum Espresso package to execute DFT calculations The rutile–anatase transition pathway is investigated using this implemented procedure 3.3 Solid-state nudged elastic band (ss-NEB) calculations of the transition state The inspection of U is conducted in our study by examining the band structures of rutile TiO2 with respect to U (from to 10 eV) Indeed, with the inclusion of U = eV, PBE calculations produce a theoretical bandgap of 3.12 eV for the rutile phase, which is in excellent agreement with the listed experimental value of 3.1 eV [42] In order to numerically determine the bandgap at U = eV, we have analyzed the band structure (figure 2(a)) of the rutile phase The density of state (DOS) is explored as a result of our DFT + U calculations, as shown in figure 2(b) It is shown clearly in the DOS plot that the valence band and conduction band are well separated by a gap of 3.12 eV, which reveals the fact that rutile TiO2 is an insulating material More interestingly, the partial density of state (PDOS) plot in figure 2(b) also reveals that both valence and conduction bands have been formed by a high degree of covalent bonding between 3d-states of Ti and 2p-states of O We also note that the valence band in rutile titania is mostly constituted by the 2p electrons from O, while the conduction band mostly accommodates the 3d electrons from Ti Figure 2(c) shows the similar DOS and PDOS for the rutile phase of TiO2 calculated within the Results and discussion 4.1 Electronic structures and total energy calculations A variety of computational methods have been developed in order to locate the intermediate structures (which include the transition states or saddle points) of chemical reactions and phase transformation processes Among those methods, the nudged elastic band [40] (NEB) method has been proved to be powerful and robust in locating transition structures The operating principle of a such method is simple The crystal configurations of reactant and product are considered as the first and last images, and temporary intermediate configurations (intermediate images) are initially assigned on the transition pathway While the first and last images are fully relaxed (vanishing of Cartesian forces), the NEB forces of temporary intermediate images are computed in order to energetically relax these images on the pathway An intermediate image in the relaxing process is referred to as an ‘NEB image’ in this text A procedure to calculate the NEB forces was presented in more detail by Sheppard et al [40] The relaxing process is executed iteratively until the energy J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al a b c Figure (a) Band structure of the rutile phase produced by DFT + U (U = eV) calculations This structure plot is produced with the (4 × × 2) k-point mesh, and a bandgap value (Eg ) of 3.12 eV can be numerically extracted from the plot (b) Density of state (DOS) and partial density of state (PDOS) plots of the rutile phase according to DFT + U calculations (U = eV) The Fermi level is positioned at eV In the DOS plot, the valence band and conducting band are well separated, which suggests that rutile is insulating with a bandgap Eg = 3.12 eV The PDOS plots clearly show that Ti 3d electrons constitute the conduction band, while O 2p electrons constitute the valence band (c) Density of state (DOS) and partial density of state (PDOS) plots for the rutile phase resulting from conventional PBE calculations (U = 0) The Fermi level is positioned at eV The total DOS shows that the bandgap is underestimated (about 1.9 eV) seen that the Ti 3d and O 2p orbitals are shifted linearly with respect to U, and consequently results in a linear relationship between Eg and U When calculations are executed with U = 3–7 eV, the slope coefficient of Eg versus U is about 0.31 (see figure 3) When U becomes larger than eV, the linear relationship is still retained, but there is a change in the slope of Eg versus U In a previous study, Dompablo and Morales-Garcia conducted bandgap calculations for rutile TiO2 with respect to various values of U [36], and their theoretical evidence provided a similar linear dependence as observed in our study For the anatase phase, the experimental bandgap is reported as 3.2 eV, which is slightly wider than that of the rutile phase [42] Similarly to the rutile case, the Eg for anatase using straight PBE calculations is underestimated (2.15 eV), and the use of DFT + U is highly recommended to predict the bandgap of anatase From the theoretical evidence of the anatase band structure and DOS at U = 3.5 eV, we find the conventional PBE functional approximation Comparing these results with those calculated by the PBE + U method in figure 2(b), it is clearly seen that the semiconductor bandgap has been underestimated in the PBE calculations as expected However, the hybridization character of bonding between Ti 3d and O 2p orbitals in the formation of valence and conduction bands remains almost the same as in the PBE + U calculations These results justify a correct description of the semiconductor gap in TiO2 within the DFT + U method in which the Hubbard U correction is applied for both Ti and O sites Anisimov et al [43] have suggested the energy εi of the ith orbital as follows: εi = εDFT + U( 12 − ni ) (2) where ni is either (the orbital is occupied) or (the orbital is unoccupied), which makes the energy shift of the ith orbital either −U/2 or U/2, respectively From equation (2), it can be J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al Table Total energies of anatase and rutile (1 × × supercell) with respect to various U Total energy (eV/four-TiO2 ) Hubbard U (eV) 10 Rutile −9860.858 9683 −9843.691 4691 −9838.397 1662 −9833.317 9703 −9828.449 6023 −9823.788 8757 −9819.320 4099 −9815.034 6147 −9810.918 4443 Anatase E = ERutile − EAnatase (eV) −9861.249 5997 −9843.761 7820 −9838.326 5078 −9833.093 9713 −9828.066 4328 −9823.242 7784 −9818.618 5421 −9814.186 6366 −9809.938 5905 0.390 6314 0.070 3129 −0.070 6585 −0.223 9990 −0.383 1695 −0.546 0973 −0.701 8678 −0.847 9781 −0.979 8538 Table Cell parameters and volumes of anatase and rutile resulting from PBE and DFT + U calculations Anatase Rutile a PBE DFT + U Expa PBE DFT + U Expa ˚ a (A) ˚ c (A) ˚ 3) V (A 3.789 3.820 3.782 4.622 4.610 4.587 9.613 9.764 9.502 2.952 3.021 2.954 138.012 142.501 135.912 63.058 64.208 62.154 Reference [42] with the use of higher Hubbard potentials (U ≥ eV), the electronic defects are handled well and the phase energies become more consistent Therefore, the DFT + U (with U = eV) method is sufficient to be employed to investigate the transformation pathway from anatase to rutile The resulting errors in cell parameters are within a few per cent compared to the experiment From our calculations, we observe the overestimation of cell parameters, which is clearly revealed in table The addition of Hubbard U results in small extensions of cell parameters in both phases, thus leading to the growth of unit-cell volumes With straight PBE calculations (U = 0) and the Vanderbilt ultrasoft pseudopotential [15], we observe that the unit-cell volumes in for both anatase and rutile are about 1.5% larger than the experimental volumes, with small extensions in the lattice parameter a For lattice parameter c, the overestimations for anatase and rutile phases are 1.2% and 0.1%, respectively It can be observed that with the utilization of the Hubbard potential term to correct electronic defects in TiO2 , the cell parameters are even more overestimated than the original PBE calculations For the anatase phase, the lattice parameter c is overestimated by 2.8% compared to the experimental value [42], and thereby results in a 4.8% extension of the anatase volume The extension in the rutile volume when employing DFT + U is 3.3%, which is somewhat less than the error in the anatase case Overall, the overestimations in all calculations are less than 5%, and such uncertainties are still within the acceptable range With the use of Hubbard potential U = eV in titania calculations, the rutile phase proves to be more energetically stable, and the obtained bandgap is in good accordance with the experimental value However, compared to the Figure Bandgap resulting from DFT + U calculations with various values of U (from to 10 eV) At eV, we have successfully generated a bandgap that is close to the experimental measurement best agreement with the experimental result (Eg = 3.18 eV) Nevertheless, at U = eV (which provides the most corrected bandgap for rutile), Eg for anatase is calculated as 3.74 eV, which is 16.9% higher than the experimental value From this case study, it is observed that the band gaps of two TiO2 phases (rutile and anatase) can only be predicted correctly with different values of Hubbard U However, in the transformation process, the calculations can only be executed with a chosen value of U We therefore choose to favor the product side in this research, and subsequently select U = eV to proceed to the next stage of our DFT + U calculations The total energy difference of anatase and rutile phases is critical and plays an important role in the relative structural stability as well as the phase transition process It is obligated for us to testify the energy levels of the two phases, and verify that the choice of Hubbard U has provided satisfactory sensibility In table 1, the total energies for a fully-relaxed anatase and rutile phase with respect to different values of U are reported It is observed that without the use of DFT + U (or equivalently U = 0), anatase is more energetically stable than rutile, which does not reveal the experimental reality This is also the case when U = eV is applied, the anatase phase is more stable than rutile, as shown in table However, J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al original PBE calculations, the DFT + U method does show some particular limitations, i.e., the unit-cell parameters and volumes resulting from DFT + U calculations are further overestimated compared the experimental results Nevertheless, such limitations are negligible, and we believe that the DFT + U method is appropriate to explore the anatase–rutile transition state pathway In this study, we also test a hybrid functional to examine the method validity Besides PBE calculations with the introduction of Hubbard U, we also employ the BLYP [31, 32] hybrid functional implemented in the Quantum Espresso package to optimize the titania crystal and calculate the theoretical bandgap From our BLYP calculations, the calculated bandgap of rutile is reported as 2.20 eV The resulting data from BLYP calculations not provide good agreement with experimental bandgap data; hence, we decided to exclude the use of the BLYP functional in this study Figure Energy barrier resulted from NEB optimizations of the anatase–rutile transformation pathway In the NEB optimization, we have generated 15 transition images, and the optimization progress takes 191 iterations until convergence 4.2 Transition state resulting from NEB calculations The intermediate transition state is located by employing the solid-state NEB [41] algorithm as stated earlier in this paper In order to locate the transition state, a series of 15-image optimizations (excluding the relaxed anatase and rutile images) are performed The NEB forces of image ith are computed based on the analysis of the current Cartesian forces and energies of the surrounding images During the NEB calculations, a (1 × × 2) supercell of rutile (four Ti and eight O ions) is considered as the product size, while the conventional unit cell of anatase is employed as the reactant size The total energy of an anatase unit cell is 0.224 eV per four-TiO2 above the total energy of the (1 × × 2) rutile supercell In the atomic scale, we can consider that the relative difference in energy per atom between the two phases is 0.018 eV/atom In the previous experimental investigations [13, 44, 45], H for the anatase–rutile transformation at 298 K has been reported over the past few decades with a large variance among the reported values (1.7–11.7 kJ mol−1 or 0.002–0.012 eV/atom), which are lower than our calculated result (0.018 eV/atom) As indicated in table 2, the unit cell of the (1 × × 2) rutile supercell is more compact than the unit cell of anatase (about 9.88% volume) It can be clearly seen that there is significant change in the unit-cell shape when the anatase bulk undergoes the phase transformation In order to find the relaxed intermediate cells, we must initialize the ss-NEB procedure by assigning temporary cell vectors to each transition structure based on the linear cell variation of the initial image (anatase) and last image (rutile) Once the intermediate temporary cells are initialized, the numerical analysis of fractional crystal coordinates and cell vectors is performed to locate the saddle points In the ss-NEB optimization, the convergence criterion is set to 10−3 eV per unit cell (four TiO2 ) Overall, 190 iterations are required to find the converged saddle point, which is 9.794 eV per four-TiO2 above the total energy of anatase, as illustrated in figure Theoretically, such an energy barrier can be concluded to be the activation energy for the anatase–rutile TiO2 phase transformation at K according to our DFT + U calculations Interestingly enough, we also observe an inflection point in the NEB plot, as indicated at the 8th image along the transition pathway The phase transition of anatase–rutile comprises not only of a simple re-orientation of ions and geometry, it also involves the destruction and formation of several ionic Ti–O bonds According to Penn and Banfield [10] in a previous experimental study, such a transformation process involves the rupture of seven Ti–O bonds out of 24 bonds in an anatase unit cell Such a conclusion has encouraged us to explore the dissociation scheme in more detail In the next stage, the Ti–O bonding interaction in the transition state is examined carefully in order to verify the interesting statement suggested by Penn and Banfield [10] Recall that at equilibrium, the Ti–O bond in rutile is a bit more stretched than that in anatase In the rutile phase, ˚ the experimental Ti–O bonds range from 1.95 to 1.98 A, while in anatase, the experimental Ti–O bonds range from ˚ [42] The equilibrium bonds resulting from 1.93 to 1.98 A our calculations are slightly overestimated when they are compared to the experimental values According to our DFT + U calculations, for anatase, the equilibrium Ti–O ˚ while the equilibrium Ti–O bonds range from 1.96 to 2.00 A, ˚ bonds in rutile lie in the range of 1.98 and 1.99 A The transition state structure (figure 5) is obtained with ˚ , which is slightly smaller a cell volume of 139.561 A than the volume of the anatase cell The Ti–O bonds in the transition state structure are then calculated with periodic considerations, and we find that seven Ti–O bonds are heavily stretched, with the range varying from 2.33 ˚ while the remaining Ti–O bonds are in the to 2.70 A, ˚ which can be considered to fluctuate range of 1.77–2.15 A, around the Ti–O equilibrium position The fact that we J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al Figure Transition structure of the anatase–rutile transformation It is observed in this transition state that seven Ti–O bonds are broken with an activation energy of 9.794 eV per four-TiO2 The volume of the intermediate transition state (saddle point) is about ˚ 139.6 A Figure Electron charge density of three Ti–O bonds (with different bond lengths) of the transition state and the inflection (8th) image In each plot, we choose three Ti–O bonds: a ˚ a stretched bond (2.33 A), ˚ and a near-equilibrium bond (1.94 A), ˚ Such analysis allows us to completely-dissociated bond (2.7 A) conclude the ruptures of Ti–O bonds in the transition state Similarly, the charge density evidence at the inflection point proves that six Ti–O bonds are broken observe seven Ti–O bonds to be highly stretched away from the equilibrium position agrees very well with an experimental implication [10] as stated earlier in the paper To achieve the saddle point (transition structure), 9.794 eV per four-TiO2 are required to break seven bonds as well as allow translation of ions within the unit cell On an average, we can conclude that approximately 1.399 eV (per two atoms) is required to break the Ti–O bond as the unit cell undergoes a phase transformation from anatase to rutile In a previous experimental study [46], an investigation of the phase transformation of nanocrystalline anatase–rutile was conducted, and the activation energy for interface nucleation and surface nucleation for pure anatase at 620–690 ◦ C were measured as 167 kJ mol−1 (1.731 eV per anatase unit cell) and 466 kJ mol−1 (4.830 eV per anatase unit cell), respectively The sum of these two energies would be 6.561 eV per anatase unit cell, which can be considered as the activation energy for the surface–interface anatase–rutile transition in the temperature range of 620–690 ◦ C The resulting value (9.794 eV per anatase unit cell) from our study has been obtained by the DFT + U calculations at T = K for investigating the structural and bonding transformation Therefore, it is understandable that the theoretical value of such a barrier is higher than the energy required for surface and interface transformation at higher temperature, which was also confirmed by Zhang and Banfield [46] We also examine the Ti–O bonding interaction for the inflection structure (8th NEB image) According to the bond distance calculations, we observe Ti–O distances of at least ˚ while the remaining 18 Ti–O distances are shorter than 2.37 A, ˚ Most of those Ti–O 18 bonds are near the equilibrium 2.28 A position; however, there are two Ti–O distances of 2.28 and ˚ It is suspected that these two Ti–O bonds remain 2.23 A unbroken Overall, in this intermediate structure, we suspect that six Ti–O bonds are broken To clarify the statement that we consider the Ti–O bond ˚ and above, charge suffers dissociation at a distance of 2.33 A density calculations in Quantum Espresso are performed for three Ti–O bonds at the transition state (whose lengths are ˚ and three Ti–O bonds at the 8th NEB 2.33, 1.94, and 2.70 A) ˚ The plots image (whose lengths are 2.28, 2.23, and 2.37 A) of charge density along the chosen Ti–O bond vector (with normalized bond lengths) are shown in figure In figure 6(a), we analyze the electron density of the saddle image A near-equilibrium bond is first considered with ˚ the electron density mostly locates a bond distance of 1.94 A, on the middle of the bond to form bonding interaction In the other case, where we considered a stretched Ti–O bond ˚ the electron density locally resides on the Ti and (2.33 A), O ions This in fact proves the weak bonding interaction between the two ions because they nearly dissociate In the ˚ we observe almost last case (with Ti–O bond being 2.70 A), no interaction between the two ions, and this bond can be considered completely destroyed In conclusion, we find that seven Ti–O bonds are broken in the saddle-point structure By employing a similar method, we analyze three Ti–O ˚ of the 8th image as distances of 2.28, 2.23, and 2.37 A ˚ shown in figure 6(b) With distances of 2.28 and 2.23 A, the Ti–O interactions still remain bonding, while in the other ˚ we observe a Ti–O bond rupture In this local case (2.37 A), intermediate structure, it is observed that six Ti–O ionic bonds dissociate As the system proceeds to the next transformation step, another Ti–O bond undergoes dissociation, which brings the number of Ti–O bond ruptures to seven, and this result is in excellent agreement with an experimental result [10] The theoretical evidence from our DFT + U study with the use of U = eV strongly suggests excellent agreement J Phys.: Condens Matter 24 (2012) 405501 N H Vu et al reached, and 15 converged transition images are obtained According to the energy barrier shown in figure 4, the theoretical activation energy of anatase–rutile transition at K is approximately 9.794 eV per four-TiO2 (one anatase unit cell) This energy level is higher than the surface–interface anatase–rutile transformation activation energy (6.561 eV per anatase unit cell) [46], but we believe that our result is sensible for a transformation in bulk material, which heavily involves structural change, bond breaking, and translation of ions To confirm the bond breaking assumption, the bonding interaction of Ti–O connectivity is explored at the saddlepoint structure The Ti–O ionic distances between Ti and O ions are computed with periodic consideration; consequently, we have found that seven Ti–O bonds are extended far away ˚ from the equilibrium bond distance (which is around 2.00 A) Those seven Ti–O bonds can be considered broken according to the charge density analysis Therefore, we conclude that at the transition state, the anatase–rutile transformation undergoes both ionic displacements and seven bond ruptures (out of 24 Ti–O bonds) Such a conclusion in this theoretical study agrees excellently with previous experimental evidence reported by Penn and Banfield [10] with previous experimental data [10] The choice of U is made, based on the criteria of finding good agreement with the previously reported experimental bandgap of the rutile phase A different choice of U (3.5 eV) is also available, which allows us to find good agreement with the experimental bandgap of the anatase phase The choice of U = 3.5 eV may accordingly reduce the total energy difference E between the two phases to be insignificant (according the table 1), and get closer to the reported experimental energy difference (0.002–0.012 eV/atom) [13, 44, 45] However, all theoretical energy differences reported in this study are less than the amount kB T for room temperature (0.026 eV/atom), at which the previous experimental measurements for energy difference were conducted, and we recognize that such an amount of energy is insignificant Therefore, the theoretical investigation using DFT + U with U = 3.5 eV is not performed in this study Summary and conclusions The computational investigation of the anatase–rutile transformation in bulk material is conducted in this study All calculations are performed using the PBE exchange functional [22, 23] with the Vanderbilt ultrasoft pseudopotential [15] as implemented in the Quantum Espresso package [24] With the straight use of the PBE functional, the anatase phase is proved to be more energetically stable than rutile, which is contradictory to experimental results Therefore, the calculations purely based on the PBE functional are not reliable, at least in the energetic point of view Hence, we employ the particular DFT + U method [25–27] with Hubbard U correction applied to both Ti and O sites in order to treat the electron–hole exchange–correlation effects as an alternative treatment to the DFT calculations of TiO2 bulk material When the DFT + U method is employed, it is significant to determine the value of Hubbard potential U by examining the resulting bandgap From various TiO2 studies using the DFT + U method [29, 30, 36–38], different values of U have been determined and reported The determination of U in our study is also conducted in the same manner With U = eV, the bandgap of rutile is computed as 3.12 eV, which agrees excellently with the experimental Eg (3.1 eV) [42] Recall that in a previous TiO2 study using the DFT + U method [36], the most appropriate Hubbard U was also determined as eV [36] The total energies of anatase and rutile phases are then validated, and the DFT + U calculated results have proved that rutile is more energetically stable than anatase, with an energy difference of 0.224 eV per formula units of TiO2 Therefore, the anatase–rutile transition state investigation is proceeded using the DFT + U method with U = eV The key objective in this research is finding the energetic transformation pathway and exploring the changes in crystal structure and bonding at the transition state The newly-developed solid-state NEB algorithm [41] is then applied to search for the transformation pathway 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rate increased dramatically with very finely crystalline anatase In a subsequent study, the anatase–rutile transition was nucleated at anatase [9], and a phase transformation