Electronic band structures

Chapter 1: Introduction

1.5. Properties of TiO 2

1.5.1. Electronic band structures

Since the nature of the individual dopant as well as its concentration and distribution determine the defect structure of the crystal, it will also influence the electronic and optical properties of the resulting TiO₂ crystal. In this study we have chosen to look at the doping of Fe, Cr and Co in TiO₂ nanostructures for their influence on the optical and magnetic properties. The average cation radii of Ti, Fe, Cr and Co in (Å) unit for the coordination number 6 of octahedral coordination system are shown in Table 1.2.

Table 1.2. Average cation radii of Ti, Fe, Cr and Co in (Å) unit for the coordination number 6 of octahedral coordination system. “NA (not applicable)” denotes this ion does not exist.

Valence →→→→

Element ↓↓↓↓ +6 +5 +4 +3 +2

Ti NA NA 0.75 0.81 1.0

Fe NA NA 0.73

0.69 0.79 (high spin)

0.75 0.92 (high spin)

Cr 0.58 0.63 0.69 0.76

0.87 0.94 (high spin)

Co NA NA 0.75 (high

spin)

0.69 0.75 (high spin)

0.79 0.86 (high spin)

gap is 3.05 eV (Γv - Γc).⁹⁵ An indirect band gap of 2.92 eV is also obtained from Γv to M_c.⁹⁵ Ekuma and Bagayoko⁸³ reported similar direct band gap of 3.05 eV (Γv - Γc) and indirect band gap of 2.95 eV (Rv - Γ^c) for rutile TiO2 using Bagayoko – Zhao – Williams (BZW) based on self-consistent ab initio calculation. For anatase, the indirect band gap of 3.0 eV (Γv

- Mc) and direct band gap of 3.26 eV (Γ^v - Γ^c) is reported.⁹⁵ Zhang et al.⁹⁶ used hybrid DFT based on Dirac-Slater exchange, the HF (Hartree-fock) exchange, and the LDA (local density approximation) correlation to calculate the band structure of brookite TiO₂. A direct band gap of 3.1 eV was reported at Γ point. Yahia et al.⁹⁷ calculated the indirect band gap (M- Γ) of 2.68 eV for TiO2(B). The band structures of rutile, anatase and brookite are displayed in Fig.

1.6.

Fig. 1.6. Band structures of rutile, anatase, brookite and TiO2(B) phases of TiO2. The arrow in each case indicates the band transition from valence band maximum to conduction band minimum.

Adapted from Ref. [95, 96, 97].

The electronic density of states (DOS) for rutile, anatase and brookite phases calculated by Landmann et al.⁹⁸ are shown in Fig. 1.7. The partial density of states feature shows predominantly O 2p-like valence band states and Ti 3d-like conduction band states around the band edges for all three major phases. As indicated in DOS, the broad Ti 3d-like bands show a distinct separation into two sub-bands which are assigned to e_g and t_2g states.

Due to the different Ti-O bond lengths, Ti 3d orbitals will split into two sets of t_2g and e_g orbitals. In an octahedral-type crystal field, the five unoccupied d-states of the central Ti ion

are split into the twofold-degenerate eg-like states with dx2 -y2

and dz2

character and the energetically lower lying threefold-degenerate t2g-like dxy, dyz and dxz type states.⁹⁸

Fig. 1.7. Electron density of states in rutile, anatase and brookite TiO2. Adapted from Ref. [98].

The electronic band structures are modified by the presence of native defects and/ or dopants in TiO2 materials. The missing of an oxygen atom in TiO2 from the bulk or surface results in one or two electrons localized in an oxygen vacancy state. As a result, the place occupied by the O^2- anion in the regular lattice is taken by one or two “free” electrons in the defective crystal. These electrons located on the oxygen vacancy states have a direct effect on the electronic structure of TiO₂ by forming a donor level below the conduction band.⁷⁴ From the ultraviolet photoemission spectroscopy and electron energy loss spectroscopy studies, it is known that the energy level of localized donor states originating from oxygen vacancies lie at about 0.7 – 1.0 eV below the conduction band of TiO₂.^99-101 Moreover, the removal of neutral oxygen atoms to form oxygen vacancies can also cause the redistribution of the excess electrons among the nearest neighboring Ti atoms around the oxygen vacancy site, and form shallow donor states below the conduction band originating from Ti 3d orbits (i.e. Ti³⁺).¹⁰² The formations of singly ionized oxygen vacancy states is also demonstrated as donor states in rutile TiO2.⁸⁰ The electrons in Ti interstitial defects occupy localized states at the bottom of the conduction band and act as shallow donors in TiO2.⁸⁰ These localized donor states are reported theoretically^{78, 80, 81} and also probed experimentally.⁸² The density of states of the defect sites for the neural oxygen vacancy, singly ionized and doubly ionized oxygen vacancy, and Ti interstitial defects in rutile TiO2 are calculated by screened-exchange hybrid

density functional method, which are shown in Fig. 1.8. It has been demonstrated that for the neutral oxygen vacancy the defect state is occupied by two electrons while for the singly ionized oxygen vacancy the defect state is occupied by one electron. However, there is no gap state for doubly ionized oxygen vacancy because the excess electrons at the vacancy site have been ionized.⁸⁰

Fig. 1.8. Density of states for oxygen vacancy and Ti interstitial defects in rutile TiO2: (a) neutral, (b) singly ionized and (c) doubly ionized oxygen vacancy; (d) Ti interstitial defects. The arrows indicate the defect states within the band gap. For singly ionized oxygen vacancy and Ti interstitial the spin-up and spin-down states are shown, respectively, above and below the abscissa. The top of the valence band is set to be the zero energy and is denoted by the vertical dash line. Adapted from Ref. [80].