DSpace at VNU: Anatase–rutile phase transformation of titanium dioxide bulk material: a DFT C U approach

Hence, we choose the DFT + U method with U = 5 eV to investigate the transformation pathway using the newly-developed solid-state nudged elastic band ss-NEB method, and consequently obta

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)

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

View the table of contents for this issue, or go to the journal homepage for more

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 Le1and

Duc Nguyen-Manh2

1Faculty of Materials Science, University of Science, Vietnam National University, Ho Chi Minh City,

Vietnam

2EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK

E-mail:hung.m.le@hotmail.com

Received 5 June 2012, in final form 12 August 2012

Published 6 September 2012

Online atstacks.iop.org/JPhysCM/24/405501

Abstract

The anatase–rutile phase transformation of TiO2bulk 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 3 to 10 eV At U = 5 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 = 5 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-TiO2above 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)

1 Introduction

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

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 nano-structured 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 TiO2has become a leading concern and been investigated in various experimental aspects over the past few decades The pressure–temperature-dependent 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

1 0953-8984/12/405501+10$33.00 c 2012 IOP Publishing Ltd Printed in the UK & the USA

b

Figure 1 Periodic structure of anatase and rutile TiO2 The different Ti–O bonds in each structure are classified (according to [21])

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 initiosimulation 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

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 0 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

2 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 figure1(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 experimental study [21] as a = 4.587 ˚A and c = 2.954 ˚A Each Ti ion in the rutile phase octahedrally connects to six

O ions, and the Ti–O bond length is 1.95 ˚A (for an equatorial bond) or 1.98 ˚A (for an apical bond) This fact implies that the octahedral coordination of each Ti ion is slightly distorted If

we consider a 1 × 1 × 2 rutile supercell (this will be later employed to find the anatase–rutile transition structure), four

Ti ions are located on the 1¯10 plane

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

as 3.782 ˚A (for a) and 9.502 ˚A (for c) [21] Similarly to the

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 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

pseudopo-tential [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

gradient convergence criterion is 2.57 × 10−4 eV ˚A−1 per

unit cell (or 0.043 meV ˚A−1/atom) In most DFT studies,

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 × 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

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 TiO2only 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:

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 TiO2polymorphs

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 3

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 3 to 10 eV), and concluded

that with an approximate U value of 5 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

3.3 Solid-state nudged elastic band (ss-NEB) calculations of

the transition state

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

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 7 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 do 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

4 Results and discussion 4.1 Electronic structures and total energy calculations The inspection of U is conducted in our study by examining the band structures of rutile TiO2 with respect to U (from

3 to 10 eV) Indeed, with the inclusion of U = 5 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 = 5 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 figure2(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

Figure 2 (a) Band structure of the rutile phase produced by DFT + U (U = 5 eV) calculations This structure plot is produced with the (4 × 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 = 5 eV) The Fermi level is positioned at

0 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 0 eV The total DOS shows that the bandgap is underestimated (about 1.9 eV)

conventional PBE functional approximation Comparing these

results with those calculated by the PBE + U method in

figure2(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(1

where ni is either 1 (the orbital is occupied) or 0 (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

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 7 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 Egfor 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

5

Table 1 Total energies of anatase and rutile (1 × 1 × 2 supercell) with respect to various U.

Total energy (eV/four-TiO2) Hubbard U (eV) Rutile Anatase 1E = ERutile−EAnatase(eV)

0 −9860.858 9683 −9861.249 5997 0.390 6314

3 −9843.691 4691 −9843.761 7820 0.070 3129

4 −9838.397 1662 −9838.326 5078 −0.070 6585

5 −9833.317 9703 −9833.093 9713 −0.223 9990

6 −9828.449 6023 −9828.066 4328 −0.383 1695

7 −9823.788 8757 −9823.242 7784 −0.546 0973

8 −9819.320 4099 −9818.618 5421 −0.701 8678

9 −9815.034 6147 −9814.186 6366 −0.847 9781

10 −9810.918 4443 −9809.938 5905 −0.979 8538

Figure 3 Bandgap resulting from DFT + U calculations with

various values of U (from 3 to 10 eV) At 5 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 = 5 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 TiO2phases (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 =

5 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 = 3 eV is applied, the anatase

phase is more stable than rutile, as shown in table1 However,

Table 2 Cell parameters and volumes of anatase and rutile resulting from PBE and DFT + U calculations

a( ˚A) c( ˚A) V( ˚A3) Anatase PBE 3.789 9.613 138.012

DFT + U 3.820 9.764 142.501 Expa 3.782 9.502 135.912 Rutile PBE 4.622 2.952 63.058

DFT + U 4.610 3.021 64.208 Expa 4.587 2.954 62.154

aReference [42]

with the use of higher Hubbard potentials (U ≥ 4 eV), the electronic defects are handled well and the phase energies become more consistent Therefore, the DFT + U (with U =

5 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 2 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 = 5 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

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 do not provide

good agreement with experimental bandgap data; hence, we

decided to exclude the use of the BLYP functional in this

study

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 × 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 × 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],1H 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−1or 0.002–0.012 eV/atom), which

are lower than our calculated result (0.018 eV/atom)

As indicated in table2, the unit cell of the (1 × 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 figure4 Theoretically, such an energy

Figure 4 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

barrier can be concluded to be the activation energy for the anatase–rutile TiO2 phase transformation at 0 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 1.93 to 1.98 ˚A [42] The equilibrium bonds resulting from 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 bonds range from 1.96 to 2.00 ˚A, while the equilibrium Ti–O bonds in rutile lie in the range of 1.98 and 1.99 ˚A

The transition state structure (figure5) is obtained with

a cell volume of 139.561 ˚A3, which is slightly smaller 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

to 2.70 ˚A, while the remaining Ti–O bonds are in the range of 1.77–2.15 ˚A, which can be considered to fluctuate around the Ti–O equilibrium position The fact that we

7

Figure 5 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 ˚A3

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 = 0 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 6 Ti–O distances of at least

2.37 ˚A, while the remaining 18 Ti–O distances are shorter than

2.28 ˚A Most of those Ti–O 18 bonds are near the equilibrium

position; however, there are two Ti–O distances of 2.28 and

2.23 ˚A It is suspected that these two Ti–O bonds remain

unbroken Overall, in this intermediate structure, we suspect

that six Ti–O bonds are broken

Figure 6 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

near-equilibrium bond (1.94 ˚A), a stretched bond (2.33 ˚A), and a completely-dissociated bond (2.7 ˚A) Such analysis allows us to conclude the ruptures of 7 Ti–O bonds in the transition state Similarly, the charge density evidence at the inflection point proves that six Ti–O bonds are broken

To clarify the statement that we consider the Ti–O bond suffers dissociation at a distance of 2.33 ˚A and above, charge density calculations in Quantum Espresso are performed for three Ti–O bonds at the transition state (whose lengths are 2.33, 1.94, and 2.70 ˚A) and three Ti–O bonds at the 8th NEB image (whose lengths are 2.28, 2.23, and 2.37 ˚A) The plots

of charge density along the chosen Ti–O bond vector (with normalized bond lengths) are shown in figure6

In figure 6(a), we analyze the electron density of the saddle image A near-equilibrium bond is first considered with

a bond distance of 1.94 ˚A, the electron density mostly locates

on the middle of the bond to form bonding interaction In the other case, where we considered a stretched Ti–O bond (2.33 ˚A), the electron density locally resides on the Ti and

O ions This in fact proves the weak bonding interaction between the two ions because they nearly dissociate In the last case (with Ti–O bond being 2.70 ˚A), we observe almost

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 distances of 2.28, 2.23, and 2.37 ˚A of the 8th image as 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 case (2.37 ˚A), we observe a Ti–O bond rupture In this local 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 = 5 eV strongly suggests excellent agreement

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 1E

between the two phases to be insignificant (according the

table1), 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 kBTfor 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

5 Summary and conclusions

The computational investigation of the anatase–rutile

transfor-mation in bulk material is conducted in this study All

calcula-tions 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 =5 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 5 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 4 formula

units of TiO2 Therefore, the anatase–rutile transition state

investigation is proceeded using the DFT + U method with

U =5 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 and saddle

point After 191 iterations, the convergence criterion has been

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 0 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 saddle-point 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]

Acknowledgments The authors thank the Faculty of Materials Science, University

of Science, Vietnam National University in Ho Chi Minh City for their computing support We acknowledge the cogent discussion from Viet Q Bui during the course of this research This work is funded by Vietnam National University in Ho Chi Minh City under grant B2012-18-10TD

References

[1] Mo S-D and Ching W Y 1995 Electronic and optical properties of three phases of titanium dioxide: rutile, anatase, and brookite Phys Rev B51 13023–32 [2] Hu Y, Tsai H L and Huang C L 2003 Phase transformation of precipitated TiO2nanoparticles Mater Sci Eng A

344 209–14 [3] Lee C Y and Hupp J T 2009 Dye sensitized solar cells: TiO2 sensitization with a bodipy-porphyrin antenna system Langmuir26 3760–5

[4] Yun H-G, Park J H, Bae B-S and Kang M G 2011 Dye-sensitized solar cells with TiO2nano-particles on TiO2 nano-tube-grown Ti substrates J Mater Chem.21 3558–61 [5] Dachille F, Simons P Y and Roy R 1968 Pressure–temperature studies of anatase, brookite, rutile and TiO2-II Am Miner

53 1929–39 [6] Jamieson J C and Olinger B 1969 Pressure–temperature studies of anatase,brookite rutile, and TiO2(II): a discussion

Am Miner.54 1477–81 [7] Heald E F and Weiss C W 1972 Kinetics and mechanism of the anatase/rutile transformation, as catalyzed by ferric oxide and reducing conditions Am Miner 57 10–23 [8] Gribb A A and Banfield J F 1997 Particle size effects on transformation kinetics and phase stability in nanocrystalline TiO2Am Miner.82 717–28 [9] Muscat J, Swamy V and Harrison N M 2002 First-principles calculations of the phase stability of TiO2Phys Rev.B

65 224112

9