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Modern Physics Letters BVol. 27, No. 15 (2013) 1350113 (8 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0217984913501133

LARGE HOLE POLARONS IN Sc-DOPED TiO2 CRYSTALS

ARVIDS STASHANS

Grupo de Fisicoquımica de Materiales, Universidad Tecnica Particular de Loja,Apartado 11-01-608, Loja, Ecuador

arvids@utpl.edu.ec

YESSICA BRAVO

Escuela de Ingenierıa Quımica, Universidad Tecnica Particular de Loja,Apartado 11-01-608, Loja, Ecuador

ymbravo@utpl.edu.ec

Received 21 February 2013Revised 17 April 2013Accepted 18 April 2013Published 15 May 2013

First-principles calculations based on the density functional theory (DFT) withinthe generalized gradient approximation (GGA) have been used to study Sc-dopedTiO2, rutile and anatase, crystals. Local defect microstructure, electronic and electricalproperties have been obtained and discussed in the present work. Large radius holepolaron state found here points out to the possibility of p-type electrical conductivity inSc-doped titania.

Keywords: TiO2; Sc-doping; hole polaron; microstructure; DFT+U .

PACS Number(s): 61.72.-y, 61.72.S-, 71.15.Mb, 71.38.Fp

1. Introduction

Titanium dioxide, TiO2, is a fascinating material proving its usefulness in a

wide range of applications. It is a key material for clean energy production by

photocatalytic water splitting.1 TiO2 also has applications in coatings2 and sensors3

and is gaining more interest as a material for novel electronic memory devices, such

as memristors.4 Nanostructured TiO2 electrodes5–8 are essential in solar energy

applications. Unfortunately, relatively wide band-gap width of TiO2, which lies in

the ultra violet region, is too large for efficient use of solar energy. Therefore, a

tremendous amount of effort has been devoted for experimental and theoretical

modelling strategies to reduce the band-gap width towards the visible region.9–12

The most widely used approach has been to dope TiO2 with another species. A

single dopant introduces new states into the energy band gap, in principle leading

to a reduction of the band-gap width of the host oxide.

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A number of trivalent dopants have been studied in TiO2 crystals. Few examples

include Al-doping investigated both experimentally13–15 and theoretically,9,16–18 as

well as Ga19 and In9 incorporation studied computationally. Sc atom, which is also

a trivalent dopant in the TiO2 systems, to our knowledge has not been studied

so far sufficiently at the fundamental quantum scale. Sc atom incorporation in

titania can produce so-called hole polaron state in both rutile and anatase crystals.

Quantum-chemical description of such systems could be of interest for scientific

reasons since polarons affect electrical transport and the other important features

in a given material.

2. Computational Methodology

Present study has been carried out using first-principles DFT approach as it

is implemented in the Vienna ab initio simulation package (VASP)20 computer

code and the generalized gradient approximation (GGA).21 Within the current

framework valence electronic states are expanded in a set of periodic plane

waves, and the interaction between the core electrons and the valence electrons

is implemented through the projector augmented wave (PAW) method.22 Finally,

Perdew–Burke–Ernzerhof (PBE)23 GGA functionals have been exploited to

describe exchange-correlation interactions. Calculations of anatase structure were

carried out using a cut-off kinetic energy of 500 eV by converging the total energy of

the system to less than 1 meV/atom. Γ-centered Monkhorst–Pack (MP) grid with

a 0.03 A−1 separation has been applied, which corresponds to a k-point mesh of

9×9×4 for the 12-atom primitive unit cell of the tetragonal (space group I41/amd)

anatase. For the rutile structure, a cut-off kinetic energy of 450 eV was implemented

to converge the total energy of the system to less than 1 meV/atom. Γ-centered MP

grid with a 0.045 A−1 separation has been applied, which corresponds to a k-point

mesh of 6× 6× 9 for the 6-atom primitive unit cell of the tetragonal (space group

P42/mnm) rutile. The previously mentioned parameters were obtained through the

atomic relaxation until all the forces were found to be less than 0.008 eV/A, and

the equilibrium state of the system was achieved.

Standard DFT has difficulties to describe correctly the strong correlation of Ti-d

electrons, and, in order to take into account these issues, an intra-site Coulomb

repulsion U -term has been included. Such an inclusion results in the so-called

DFT+U method.24 Performed test calculations showed that U = 3.5 eV is the

most appropriate value for our systems. This parameter has a strong effect on the

band-gap width, which experimentally is 3.2 eV for anatase25 (calculated value in

the present work is 2.72 eV) and 3.0 eV for rutile26 (computed value is 2.52 eV).

Larger values of parameter U would allow obtaining band-gap width even closer to

the experimental findings but would have a negative impact on the other properties

of these systems, e.g. lattice parameters, so we did not enlarge further the magnitude

of U . The obtained lattice parameters were found to be equal to a = 3.81 A

and c = 9.70 A for anatase and a = 4.65 A and c = 3.01 A for rutile crystals,

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Large Hole Polarons in Sc-Doped T iO2 Crystals

respectively, which are in very close concordance to the available experimental

data.25,26

In order to enable a proper description of localized defect states in oxides, we also

used U = 7.0 eV (Ref. 27) for some test calculations. This U value was attributed to

the O 2p states. The same U value was employed to compute free self-trapped hole

polarons (positive holes trapped by negative oxygens) in pure TiO2 crystals. The

latter was done to check out if plane wave implemented DFT is able, in principle,

to reproduce such a quasi-particle.

3. Results and Discussion

In order to study Sc-doped rutile and anatase crystals, the respective primitive unit

cells have been expanded to enable the incorporation of a single Sc point defect.

96-atom supercell (sixteen times, 2×2×4 expansion) has been exploited to describe

the rutile crystal. 108-atom supercell (nine times, 3 × 3 × 1 expansion) has been

used to study Sc dopant effects in anatase structure. The corresponding k-point

meshes of 3× 3× 2 and 3× 3× 4 for rutile and anatase systems, respectively, have

been employed throughout the dopant studies. These particular k-point meshes

were utilized to preserve the established k-point separation in the reciprocal space

for the crystals under study.

One of the Ti atoms situated in the central part of the corresponding supercell

model has been replaced by a Sc impurity. Such an acceptor-type doping produces

a hole since a Sc atom has smaller charge than a Ti atom which it substitutes for,

so effectively it is a negative charge with respect to the perfect titanium dioxide

lattice. Using Bader charge analysis,28 the calculated impurity charge q(Sc) =

2.04 e in both rutile and anatase doped crystals whereas for a Ti atom in perfect

TiO2 q(Ti) = 2.36 e; therefore, the effective defect charge is −0.32 e. That is why

Sc impurity might act as a trap for a positively charged hole. Electron density

maps generated around the impurity atom are shown in Figs. 1 and 2 for rutile

Fig. 1. Electron density map around the Sc dopant in rutile crystal.

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Fig. 2. Electron density map around the Sc dopant in anatase crystal.

Fig. 3. Microstructure of hole polaron in Sc-doped rutile crystal.

and anatase materials, respectively. Sc atom appears to have larger electron cloud

than that of the replaced host Ti atom. This fact implies certain preferences for

augmentation of the covalent bonding in the defect-surrounding region.

Formation of hole polaron, as expected, has been accompanied by atomic

distortion in the dopant neighborhood. Equilibrium atomic configuration for

hole polaron in the rutile crystal is shown in Fig. 3. As shown by arrows six

defect-neighboring O atoms tend to move away regarding the impurity. Two oxygen

atoms situated within the same xy plane as Sc atom move by 0.08 A while the

other four oxygens displace themselves by about 0.11 A leading to an asymmetric

distortion. No movements for the defect-closest Ti atoms have been observed.

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Large Hole Polarons in Sc-Doped T iO2 Crystals

Fig. 4. Microstructure of hole polaron in Sc-doped anatase crystal.

Case of atomic distortion for the Sc-doped anatase is depicted in Fig. 4. Two

O atoms situated along the crystallographic z-axis move away with respect to the

defect by approximately 0.13 A whereas four oxygens sharing the xy plane with the

impurity shift themselves by 0.08 A outwards with respect to the defect. Again, an

asymmetric solution for lattice distortion has been encountered. The displacements

of Ti atoms situated within the defective region are found to be around 0.01 A. Our

results on enlarging Sc–O distances compared to the original Ti–O distances are

supported by the time-differential perturbed-angular-correlation measurements,29

demonstrating up to 10% larger Sc–O distances that those of Ti–O ones. We

consider that origin of just described atomic shifts is change in the electrostatic

interaction due to the defect incorporation.

Despite the encountered lattice distortions, one cannot argue for the formation

of small radius or small hole polarons since the hole charge has been shared by the

majority of O atoms forming the supercell. Thus, we propose large polaron (another

term is Frohlich polaron) model in the Sc-doped rutile and anatase compounds.

As a next step, extra computations have been carried out applying U = 7.0 eV

for the O 2p states. Nevertheless, yielded lattice distortion as well as hole charge

distribution is almost identical to that of the previously discussed case. That implies

once more that Sc-doped TiO2 crystals are not suitable for localized small polarons,

instead one has to expect large Frohlich polarons in these compounds. Such an

outcome is contrary to results for the other trivalent dopants (Al, Ga, In) in TiO2

materials.9

Partial density of states (PDOS) for the Sc-doped rutile and anatase materials

are depicted in Figs. 5 and 6, respectively. It is clearly visible that Sc impurity

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Fig. 5. PDOS for Sc-doped rutile crystal for energy interval between −8 eV to 10 eV. Verticalline denotes the Fermi level. Contribution of Sc dopant can be seen in the CB region only. Largehole polaron band is due to the O 2p states exclusively.

Fig. 6. PDOS for Sc-doped anatase crystal for energy interval between −8 eV to 10 eV. Verticalline denotes the Fermi level. Contribution of Sc dopant can be seen in the CB region only. Largehole polaron band is due to the O 2p states exclusively.

generates unoccupied band at the valence band (VB) maximum. That is true

for both rutile and anatase materials, Figs. 5 and 6, respectively. PDOS analysis

specifies O 2p atomic orbitals as responsible for these empty bands. Therefore,

one can expect large hole polarons in the Sc-doped TiO2 materials with a wave

function mainly due to the O 2p atomic orbitals. It appears that the polarons

are not localized being shared by a number of O atoms in spite of our attempts

to localize this quasi-particle. Large or Frohlich-type polarons might contribute

to the p-type electrical conductivity, i.e. the solid can be treated as a polarizable

continuum with conduction holes moving through the lattice. However, one has to

take into consideration that mobility of large polaron is often limited (at least in

certain temperature range) by scattering due to the optical phonons, as a result

of strong electro-optical-phonon coupling in some materials. Comparison of Figs. 5

and 6 allows one to argue that number of conducting hole polarons at the top of the

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Large Hole Polarons in Sc-Doped T iO2 Crystals

VB are larger for Sc-doped anatase compared to that of the rutile material. That

is true since the area of band from the Fermi level to the VB maximum is larger in

Fig. 6.

Our attempts to localize a positive hole in otherwise pure TiO2 crystals were not

successful suggesting that DFT even with the added U -term is not suitable for such

a task. Obviously, screened hybrid functionals such as the Heyd–Scuseria–Ernzerhof

functional30 has to be used to describe properly hole self-trapping in the perfect

materials. Similar computations, e.g. carrier self-trapping, has been already carried

out31,32 in TiO2 using this technique. Work on iodine vacancy centers in SrI2(Ref. 33) might be also very useful in order to compare directly DFT and nonlocal

hybrid functionals applicability for charge localization in crystals. Nevertheless,

DFT+U is sufficient to depict very well Sc dopant behavior in TiO2 as well

as resulting electronic features of these materials. Discovered p-type electrical

conductivity in Sc-doped TiO2 could be important in band-gap engineering for

visible-light photocatalysis34 and transparent electronics as well.

4. Conclusions

Using first-principles DFT+U electronic structure calculations Sc-doped rutile

and anatase crystals have been studied. Equilibrium state geometry suggests

asymmetric atomic distortion in the neighborhood of impurity with O atoms

displacing themselves by 0.08 A to 0.13 A with respect to the dopant. Only six

defect-nearest O atoms have non-negligible shifts whereas the displacements of the

other atoms situated within the defective region can be neglected. Formation of

large hole polaron has been proposed in the present work with a hole being shared

by the majority of O atoms constituting the supercell. We also expect enhancement

of p-type electrical conductivity due to the large hole polarons in Sc-doped TiO2.

Acknowledgments

We sincerely thank Richard Rivera for the assistance in managing some of the VASP

files. Help of Henry P. Pinto in drawing Figs. 5 and 6 are greatly appreciated.

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