Ab initio calculations of electronic properties of pure

and Ge doped anatase TiO2

Qiang Chen*, Hong-Hong Cao

School of Science, Beijing University of Aeronautics and Astronautics, 37 Xue Yuan Road, Haidian District, 100083 Beijing, People’s Republic of China

Received 18 May 2004; accepted 1 October 2004

Available online 5 April 2005

Abstract

In this paper we compared the electronic properties of the Ge doped anatase TiO2 with that of the pure host lattice. The ab initio

calculations were performed by using the full potential-linearized augmented plane wave method (FP-LAPW). The fully optimized structure

and the relaxation introduced by the impurity were obtained by minimizing the total energy and atomic forces. The resulted band structure,

density of states maybe instructive to the understanding of exceptional behavior in this system.

q 2004 Elsevier B.V. All rights reserved.

Keywords: Ab initio; DFT; FP-LAPW; Anatase; TiO2; Electronic properties

1. Introduction

Titanium dioxide (TiO2) has received intensive attention

as a promising material for photochemical applications and

as a photocatalyst because of its excellent functionality,

long-term stability, and non-toxity [1]. Rutile, anatase, and

brookite are the three distinct polymorphs of TiO2. Anatase

is less stable than rutile, but is more efficient and more

widely used in catalysis and photoelectrochemistry. TiO2 is

a wide-gap semiconductor: the band gap energy of anatase

is 3.2 eV, which means that it is mainly activated by

ultraviolet (UV) lights. Many studies [2–8] such as cation-

doped TiO2 have been carried out in the attempt to shifted

the absorption edge to a lower energy, thereby increasing

the photoreactivity in the visible-light region. Recently,

Fromknecht [9,10], etc. reported experimentally the excep-

tional behavior of Ge- and Pb-implanted TiO2 single

crystals in diffusion effects and conductivity. In this paper

we investigated the electronic structure of the Ge doped

anatase TiO2, which should be helpful to understand some

of the experimental results.

This paper presents the ab initio calculations of TiO2

in the anatase structure using FP-LAPW method.

0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2004.10.008

* Corresponding author.

E-mail address: qchen@buaa.edu.cn (Q. Chen).

The results include the fully optimized ground-state

structure obtained with total energy and atomic forces,

band structure, charge densities and densities of states.

The article is organized as follows: the computing model

systems and methods are described in Section 2, the

results are reported and discussed in Section 3, a short

section of conclusions closes the paper.

2. Calculation models and methods

Anatase is one of the two most common and widely used

polymorphs of TiO2. Its structure can be described in terms

of chains of TiO6 octahedra. In the anatase structure each

octahedron is in contact with eight neighbors (four sharing

an edge and four sharing a corner) [11]. The conventional

unit cell of TiO2 in the doped anatase structure is shown in

Fig. 1, which the doped Ge atom replace the Ti at the center

of unit cell. Besides, two lattice parameters, a, c, the

parameter dap is the apical Ti–O or Ge–O bond length, deq is

the equatorial bond length correspondingly [12].

The atomic configuration of titanium is [Ar] 3d2, 4s2 and

that of germanium is [Ar] 3d10, 4s2, 4p4 while for oxygen it

is [He] 2s2, 2p4. In many cases it is desirable to distinguish

three types of electronic states, namely core, semi-core

and valence states. In our work we treat the Ti 1s, 2s and 2p

Journal of Molecular Structure: THEOCHEM 723 (2005) 135–138

www.elsevier.com/locate/theochem

Fig. 1. The conventional unit cell of anatase TiO2. Dark gray and white

balls correspond to oxygen and titanium atoms, respectively, assuming that

Ge (black) replaces Ti atom at the center of unit cell (for interpretation of

the references to colour in this figure legend, the reader is referred to the

web version of this article).

Q. Chen, H.-H. Cao / Journal of Molecular Structure: THEOCHEM 723 (2005) 135–138136

states, Ge 1s, 2s, 2p, 3s, 3p together with the O 1s as core

states. The semi-core states are Ti 3s, 3p, O 2s and Ge 3d,

which correspond to local orbitals (LO). This leaves the Ti

3d, 4s, O 2p and Ge 4s, 4p as valence states, respectively.

The calculations were carried out by using FP-LAPW

method within density-functional theory (DFT) [13–15].

We worked in generalized gradient approximation (GGA)

[16]. In this method, the unit cell is divided into two

regions: non-overlapping atomic spheres with radii and an

interstitial region. Taking the structure optimization into

account, the atomic sphere radii adopted for Ti, O and Ge

were 0.95, 0.79 and 1.06 A, respectively. We took for the

parameter RKMAX the value of 7. The basis sets consist

of 2978 LAPW functions. We also introduced local orbitals

(LO) to include Ti 3s, 3p, O 2s and Ge 3d orbitals.

Integration in a reciprocal space was performed by the

tetrahedron method taking 84 k-points in the irreducible

Brillouin zone (IBZ). The calculation was carried out self-

consistently and stopped by the force convergence criterion.

Our aim is to calculate from first principles the structural

relaxations produced in the anatase TiO2 when a Ge

Table 1

The relaxed structure of the doped anatase TiO2 and the optimized structural par

theoretical results (only for pure)

a (A) c (A)

Doped 3.814 9.591

Pure 3.823 9.612

FPLAPW [12] 3.692 9.471

PHF [22] 3.763 9.851

B3LYP [23] 3.7923 9.8240

TM [24] 3.744 9.497

Exp. 1a [21] 3.78479 [3] 9.51237 [12]

Exp. 1b [21] 3.78216 [3] 9.50226 [12]

Exp. 2 [25] 3.7842 [13] 9.5146 [15]

The Exp. 1a [21] is obtained at 295 K, while the Exp. 1b [21] is at 15 K (exp., e

impurity replaces a Ti atom and the electronic structure of

the resulting system. The FP-LAPW method that we

employed, for reliability, treats all electrons and has no

shape approximations for the potential and charge density

[17]. The implementation of the FP-LAPW method includes

total energy [18] and atomic-force calculations [19], which

allow structure optimization [20]. The optimized structure

of pure anatase and the relaxed structure of the doped for a

set of V and c/a were decided when the total energy was

minimized and the force on each atom was smaller than

1 mRy/a.u. All results in this work were obtained under

these conditions.

3. Results and discussion

To calculate the optimized structure of the pure anatase

TiO2 and the relaxed structure when one Ti atom has been

replaced by a Ge atom, we use the forces acting on the atoms

to move them to the equilibrium positions, where the forces

are smaller than the criterion and the corresponding total

energy has a minimum. The following steps are performed:

(i)

amete

xperim

Volume optimization: begin with the experimental

ratio of c/a, calculate the total energy as the function of

the unit cell volume, and obtain the lattice parameters

corresponding to the lowest energy.

(ii)

Geometry minimization: upon the structure obtained

above, perform movements of the atomic positions

according to the calculated forces. The iteration was

stopped when the force on each atom was smaller than

1 mRy/a.u., and repeat step (i).

(iii)

c/a Optimization: on the basis of (i) and (ii), calculate the

total energy with the various ratios of c/a, and find the

relaxed structure corresponding to the lowest energy.

In Table 1, we present the relaxed structure of the doped

anatase and the optimized structure of the pure one. For

comparison, we also list other experimental and theoretical

results. In general, the agreement of the calculated

structures with experiment is good. In particular, our

rs for pure anatase in this work, compared to experiments and other

dap (A) deq (A) (c/a)

1.990(Ti–O) 1.950(Ti–O) 2.515

1.906(Ge–O) 1.948(Ge–O)

1.997 1.954 2.514

1.948 1.893 2.566

1.995 1.939 2.618

1.9972 1.9509 2.590

1.967 1.916 2.536

1.9799 [5] 1.9338 [1] 2.51332

1.9788 [4] 1.9322 [1] 2.51239

1.9797 [23] 1.9338 [5] 2.5143

ent). The italic values are derived from the experimental data.

Fig. 2. The band structure of the pure and doped anatase TiO2 as described in text. The top of the valence band is taken as the zero of energy.

Q. Chen, H.-H. Cao / Journal of Molecular Structure: THEOCHEM 723 (2005) 135–138 137

calculated pure structure is in better agreement with

experiment for deq and c/a than other theoretical outcomes.

There are following two aspects in the comparison of the

pure and doped structures that attract our attention:

First, the size of the doped unit cell (a, c), is smaller than

that of the pure one, which can be seen in Table 1.

Fig. 3. Total and projected DOS for the doped anatase TiO2. The origin of the en

Mainly of Ti states, which show two distinct structures, below and above 4.52 eV.

valence bands (VB).

Second, the apical and equatorial Ti–O bond length of

the pure structure is 1.997 and 1.954 A, while for the doped

structure the correspondences of Ge–O decrease to 1.906

and 1.948 A, respectively.

With the pure and doped structure, we investigated the

band structure of the anatase TiO2 using GGA [16],

ergy scale is taken at the Fermi level, as the vertical dotted line indicates.

The upper CB is mainly introduced by Ge states. The O states contribute the

Q. Chen, H.-H. Cao / Journal of Molecular Structure: THEOCHEM 723 (2005) 135–138138

respectively. The calculated band structure along the high-

symmetry directions of the BZ is presented in Fig. 2. The

origin of the energy scale is taken at the Fermi level. The

band structure is similar to that of the rutile and is very flat

as expected for a mostly ionic compound. The present

calculations of the pure anatase TiO2 is found to be similar

to that of the B3LYP calculations [23], while that of the

TM calculations [24] is not consistent with ours in the range

of Z to G.

With the pure structure, our calculation yields in the pure

anatase an indirect transition of 2.12 eV from the top of the

VB near X to the conduction band (CB) at G, while for the

doped unit cell, the corresponding band gap decrease to

1.96 eV.

Although both of them are smaller than that experimen-

tally observed due to the well-known shortcoming of the

GGA [26], but the narrowing trend of the energy gap should

be paid attention to in both the experimental and theoretical

studies.

To identify the contributions from various orbitals after

doping we also calculate the total and partial densities of

states (DOS), as shown in Fig. 3.

There are seven inequivalent atoms in the unit cell,

two Ti, one Ge and four O atoms. It can be seen that the

conduction bands (CB) below 7.41 eV consist mainly of

Ti states, which show two distinct structures, below and

above 4.52 eV. The upper CB is mainly introduced by

Ge states. The O states contribute the valence bands

(VB).

4. Conclusion

We have presented an ab initio study of the electronic

structure of the Ge doped anatase TiO2 using the highly

precise FP-LAPW with the exchange-correlation functional

of GGA.

The work can be summarized as follows:

(i)

The fully optimized and relaxed structures were

obtained by using the forces acting on the atoms to

move them to the equilibrium positions, where the

forces are smaller than the criterion and the correspond-

ing total energy has a minimum.

(ii)

The band structures of both pure and doped anatase

TiO2 were calculated. The density of states and the

electron density of the relaxed structure were also

obtained, which will be helpful to analyzing and

interpretation.

References

[1] D.F. Oills, H. Al-Ekabi, Photocatalysis Purification and Treatment of

Water and Air, Elsevier, New York, 1993.

[2] M. Anpo, Y. Ichihashi, M. Takauchi, Res. Chem. Intermed. 24 (1998)

143.

[3] W. Choi, A. Termin, M.R. Hoffmann, J. Phys. Chem. 98 (1994)

13669.

[4] D. Morris, Y. Dou, J. Rebane, C.E.J. Mitchell, R.G. Egdell,

D.S.L. Law, A. Vittadini, M. Casarin, Phys. Rev. B 61 (2000) 13445.

[5] J. Claverie, J. Verniolle, G. Campet, J.P. Doumerc, P. Hagemuller,

Mater. Res. Bull. 16 (1981) 1091.

[6] K. Kobayashi, M. Tanaka, Y. Fujimura, S. Okamoto, J. Appl. Phys. 60

(1986) 4191.

[7] T. Umebayashi, T. Yamaki, H. Itoh, K. Asai, Appl. Phys. Lett. 81

(2002) 454.

[8] T. Umebayashi, T. Yamaki, T. Sumita, S. Yamamoto, S. Tanaka,

K. Asai, Nucl. Instrum. Methods Phys. Res. B 206 (2003) 264.

[9] R. Fromknecht, T. Wiss, I. Khubeis, O. Meyer, Surf. Coat. Technol.

128-129 (2000) 364.

[10] R. Fromknecht, I. Khubeis, O. Meyer, Nucl. Instrum. Methods B 161–

163 (2000) 528.

[11] A.L. Linsebigler, G.Q. Lu, J.T. Yates Jr., Chem. Rev. 95 (1995) 735.

[12] R. Asahi, Y. Taga, W. Mannstadt, A.J. Freeman, Phys. Rev. B 61

(2000) 7459.

[13] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, WIEN2k, An

Augmented Plane WaveCLocal Orbitals Program for Calculating

Crystal Properties, Karlheinz Schwarz, Techn. Universitat Wien,

Austria, 2001, ISBN: 3-9501031-1-2.

[14] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864.

[15] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133.

[16] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.

[17] E. Wimmer, H. Krakauer, M. Weinert, A.J. Freeman, Phys. Rev. B 24

(1981) 864 (and references therein); E. Wimmer, H. Krakauer,

M. Weinert, A.J. Freeman, Phys. Rev. B 26 (1982) 4571;

H.J.F. Jansen, A.J. Freeman, Phys. Rev. B. 30 (1984) 561.

[18] M. Weinert, E. Wimmer, A.J. Freeman, Phys. Rev. B 26 (1982) 4571.

[19] R. Yu, D. Singh, H. Krakauer, Phys. Rev. B 45 (1991) 8671.

[20] W. Mannstadt, A.J. Freeman, Phys. Rev. B 55 (1997) 13298.

[21] J.K. Burdett, T. Hughbanks, G.J. Miller, J.W. Richardson Jr.,

J.V. Smith, J. Am. Chem. Soc. 109 (1987) 3639.

[22] A. Fahmi, C. Minot, B. Silvi, M. Causa, Phys. Rev. B 47 (1993)

11717.

[23] A. Beltran, J.R. Sambrano, M. Calatayud, F.R. Sensato, J. Andres,

J. Surf. Sci. 490 (2001) 116.

[24] M. Mikami, S. Nakamura, O. Kitao, H. Arakawa, X. Gonze, Jpn. J.

Appl. Phys. 39 (2000) 847.

[25] M. Horn, C.F. Schwerdtfeger, E.P. Meagher, Z. Kristallogr. 136

(1972) 273.

[26] C. Stampfl, C.G. Van de Walle, Phys. Rev. B 59 (1999) 5521.