TiO , as one of the wide band-gap semiconductors, with stable, nontoxic, and efficient photocatalytic activity, has been widely used in photocatalytic, solar battery, sensor, self-cleaning materials, and so on.^[1–4] TiO thin films usually exist in two polymorphs after high-temperature annealing, i.e., the anatase and the rutile, shorted as a-TiO and r-TiO , respectively.^[5–9]

In experiments, optical properties of thermally annealed TiO have shown intrinsic birefringence and anisortropy in some samples.^[10] For example, the fundamental optical properties of optically anisotropic materials such as r-TiO have been measured at room temperature with linearly polarized light of wavelength longer than 110 nm.^[7] Polarized reflection spectra of single-crystal a-TiO have been obtained in a photon energy range from 2 eV to 25 eV by using synchrotron orbital radiations, and the optical properties show anisotropy.^[8] Spectroscopic ellipsometry measurements were made on thin-film and single-crystal a-TiO using a two-modulator generalized ellipsometer. The results show that the complex refractive indices and dielectric function in a- and r-TiO are quite different. Below the band edge, compared with a-TiO , r-TiO has a higher refractive index, as well as greater value of birefringence.^[9]

Theoretically, the crystal structure, the band structure, and the density of states of a-TiO have been analyzed by using a first-principles local density approximation (LDA) approach.^[11] The resulting band-gap value is 2.25 eV. Near the absorption edge, it shows a significant optical anisotropy between the components with directions parallel and perpendicular to the c axis.

In the case of r-TiO , the structure and electronic properties have been calculated by using “soft-core” ab-initio pseudopotentials constructed within the local density approximation (LDA), and the resulting band-gap value is 2.0 eV. The obtained dielectric function and reflectivity of r-TiO for polarization vector also display difference between the directions parallel and perpendicular to the c axis.^[12]

The self-consistent orthogonalized linear-combination-of-atomic-orbitals method in LDA has been used to study the electronic structure and optical properties of the three phases of TiO .^[13] The obtained band-gap values of a-TiO and r-TiO are 2.04 eV and 1.78 eV, smaller than the experimental data, 3.2 eV and 3.0 eV, respectively. The theoretical calculations show a little difference in the optical properties of the three phases in TiO .

Regardless of the above computational efforts, the physical mechanism for the intrinsic birefringence and anisotropy of the optical properties of TiO found in some samples remains to be understood. Note that the band gaps obtained by the previous methods are significantly lower than the experimental values. So, it is important to investigate the electronic structures of a-TiO /r-TiO and clarify the difference of optical properties based on the first-principles band structure calculations with electronic correlations to be taken into account effectively.

The LDA+U method has been employed to investigate the properties of r-TiO . The results are dramatically improved when additional correlation corrections are introduced on the O 2p orbitals in the LDA approach.^[14] The experimental band-gap value (3.20 eV) can be reproduced when eV and eV. In a similar approach, the influence of oxygen defects upon the electronic properties of Nb-doped TiO was studied, and the effective U parameter eV has been used to correct the strong Coulomb interaction between 3d electrons localized on Ti in anatase models.^[15] The calculated results show that the anatase NbTi cells are degenerated semiconductors with a typical n-type degenerated characteristic in their electronic structure, which is in good agreement with the experimental evidence that anatase Nb:TiO film is an intrinsic transparentmetal. Other researchers set up a model for Zn Ag O ( , 0.0278, 0.0417) to calculate the geometric structure and energy via the method of generalized gradient approximation (GGA ), showing that the absorption spectrum in these systems all coincide with experimental data.^[16]

In this paper, we use the GGA+U scheme formulated by Loschen et al.,^[17] to calculate the electric structures and optical properties of r-TiO and a-TiO . The on-site Coulomb interactions of 3d orbitals on Ti atom ( and of 2p orbitals on O atom ( are determined so as to reproduce the experimental value of the band gap for the two phases of TiO . The comparison of the electronic structure, birefringence and anisotropy between the two phases of TiO is also presented. Finally, the adaptability of the GGA+U approach has been discussed.

Density functional theory (DFT) calculations are performed with plane-wave ultrasoft pseudopotential, by using the GGA with the Perdew–Burke–Ernzerhof (PBE) functional and the GGA+U approach as implemented in the CASTEP code (Cambridge Sequential Total Energy Package).^[18] The ionic cores are represented by ultrasoft pseudopotentials for Ti and O atoms. For the Ti atom, the configuration is [Ar] 3d 4s , where the 3s , 3p , 3d , and 4s electrons are explicitly treated as valence electrons. For the O atom, the configuration is [He] 2s 2p , where 2s and 2p electrons are explicitly treated as valence electrons. The plane-wave cut off energy is 380 eV and the Brillouin-zone integration is performed over the 24 × 24 × 24 grid sizes using the Monkorst–Pack method for geometry structure optimization. This set of parameters assures the total energy convergence of 5.0 × 10 eV/atom, the maximum force of 0.01 eV/Å, the maximum stress of 0.02 GPa and the maximum displacement of 5.0 × 10 Å. We calculated the electronic structures and the optical properties of a-TiO and r-TiO by means of the GGA method without U and GGA after having optimized the geometry structure. The details of the calculation have been shown elsewhere.^[19]

The space group of a-TiO is /amd and the local symmetry is C4h-19, and r-TiO is /mnm and D4h-14. The lattice constants a and c are experimentally determined to be a = 0.3785 nm and c = 0.9515 nm, a = 0.4593 nm and c = 0.2959 nm,^[20,21] respectively. The GGA calculation of the perfect bulk a-TiO and r-TiO is performed to determine the optimized parameters in order to check the applicability and accuracy of the ultrasoft pseudopotential. The optimized parameters are a = 0.3795 nm and c = 0.9837 nm, and a = 0.4645 nm, and c = 0.2968 nm for a-TiO and r-TiO , respectively, in good agreement with experimental and other theoretical values.^[10–13] However, the value of the band gap in a-TiO and r-TiO is around 2.16 eV and 1.85 eV, respectively, smaller than the experimental value of 3.23 eV^[22] and 3.0 eV.^[23] This is due to the fact that the DFT results often undervalue the energy of 3d orbitals of the Ti atom, lowering the bottom level of conduction bands. As a result, of TiO obtained by GGA is lower than the experimental one.

In order to reproduce the band gap, we first introduce for 3d orbitals of the Ti atom. Using the experimental lattice parameters, we optimize the geometry structure and calculate the band structure and density of state (DOS) of r-TiO . The band gap obtained from the band structure is shown in Fig. 1(a) as a function of . It can be seen that firstly increases, and then drops with increasing , showing a maximum value (2.46 eV) at eV, where the lattice parameters of the optimized structure are a = 0.4664 nm and c = 0.3086 nm. The maximum value is smaller than the experimental one (3.0 eV). The saturation of with may be related to the approach of 3d states toward 4s and 3p states, though the microscopic mechanism is not yet fully understood. Next, we introduce for 2p orbital of O atom, while keeping eV. The results in Fig. 1(b) show that monotonically increases with . When eV and eV, the calculated band gap of r-TiO is 3.0 eV, well consistent with the experiment one, where the lattice parameters of the optimized structure are a = 0.4664 nm and c = 0.3082 nm. For a-TiO , when eV and eV, the calculated band gap of a-TiO is 3.229 eV, well coinciding with the experiment one, where the lattice parameters of the optimized structure are a = 0.38891 nm and c = 0.9870 nm.

Fig. 1.

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	Fig. 1. (color online) Calculated band gap of r-TiO as a function of (a) and (b) .

By adopting these U values: eV and eV, ( eV and eV for a-TiO , we perform the GGA+U calculation for r-TiO . The band dispersion is shown in Fig. 2(a). The bottom of the conduction band is located at the G/B point. Since the bottom shifts to higher energy with accompanied by the reconstruction of the conduction band, the separated density of states at 3.03 eV and 6.26 eV obtained by the generalized gradient approximation without U (not shown here) merges to one sharp structure at 3.74 eV as shown in Fig. 2(b). The conduction band is predominantly constructed by Ti 3d states, while the valence band is by O 2p states (as shown in Fig. 2(c) and Fig. 2(d)). Therefore, the excitations across the gap are mainly from the O 2p states to the Ti 3d states.

Fig. 2.

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	Fig. 2. (color online) The band structure and DOS of r-TiO obtained by GGA ( eV, eV). (a) Band structure. The total DOS, the partial DOSs of Ti and O atoms are shown in panels (b), (c), and (d), respectively.

Figure 3 shows the comparison of the total DOSs between a- and r-TiO obtained by GGA and GGA ( eV, eV, and eV, eV for r-TiO and a-TiO , respectively). From the wide region (shown in Fig. 3(a)), the total DOSs of r-TiO and a-TiO obtained by GGA and GGA look quite similar; while zooming into the region around the Fermi surface (Fig. 3(b)), we can see that compared with the results of GGA without U, a significant change after including U is that the conduction bands are pushed to higher energies, which accordingly produces larger band gaps.

Fig. 3.

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	Fig. 3. (color online) Total DOSs of the two phases of TiO by GGA without U, and GGA ( eV, eV, and eV and eV for r-TiO and a-TiO , respectively) in a wide region (a), and zoom of the region near Fermi surface (b).

Figure 4 shows the dielectric function of r-TiO obtained by GGA ( eV and eV) (a) parallel and (b) perpendicular components. The parallel and perpendicular components of the real part have maximums of 12.53 at 3.86 eV and 10.98 at 3.94 eV, respectively. The calculated static dielectric constants are 6.298 and 5.485 along the different directions, coinciding with the experimental value 6.33.^[24] The parallel and perpendicular components of the imaginary part show the maximums of 10.28 at 6.74 eV and 9.50 at 4.90 eV, respectively. Other optical properties can be computed from the complex dielectric function.^[25] For example, we can obtain the refractive coefficient of r-TiO , whose parallel and perpendicular components are and . It is obvious that the values by GGA are closer to the experimental values and ,^[7] or and .^[26]

Fig. 4.

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	Fig. 4. (color online) Dielectric function of r-TiO obtained by GGA ( eV and eV). (a) The parallel and (b) the perpendicular components.

Figure 5 shows the dielectric function of the parallel and perpendicular component in a-TiO obtained by GGA ( eV and eV). From Fig. 5, we can see that the maximums of the parallel and perpendicular components of the real part are 11.79 at 3.75 eV and 12.56 at 3.75 eV, respectively. The static dielectric constants of the two components are 6.137 and 5.995, coinciding with the experimental value 5.62.^[24] The maximums of the parallel and perpendicular components of the imaginary part are 10.29 at 4.74 eV and 11.22 at 4.52 eV, respectively. Other optical properties can be computed from the complex dielectric function.^[25] The parallel and perpendicular components of the refractive coefficient of r-TiO are and . It is obvious that the values by GGA are closer to the experimental values of and .^[27]

Fig. 5.

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	Fig. 5. (color online) Dielectric function of a-TiO obtained by GGA ( eV and eV). (a) The parallel and (b) the perpendicular components.

From Fig. 4 and Fig. 5, we can notice the optical anisotropies in both a- and r-TiO between parallel and perpendicular components. The rutile form of TiO has comparatively high anisotropy, both above and below the band gap, which makes r-TiO a very useful optical material.^[9] Below the band gap, r-TiO has a large birefringence (difference between refractive indices of the parallel and perpendicular components) in our calculation, which is in good agreement with the experiment data (Δn = 0.27^[28] or 0.30^[7]).

To study the adaptability of the GGA+U approach, we further calculate the electronic structure of a titanium-vacancy and of hafnium-substitution solid solutions in rutile TiO . For titanium-vacancy solid solutions of rutile TiO , we first set 2 × 2 × 2 supercell. Namely, in one super cell, there are 16 titanium and 32 oxygen atoms. We delete only one Ti atom in the center of the super cell, whose vacancy rate is 6.25%, and we obtain the electronic structure of titanium-vacancy solid solutions of rutile TiO by GGA or by GGA+U ( eV and eV for Ti 3d and O 2p, respectively) after geometry optimization. The calculation results are shown in Fig. 6(a).

Fig. 6.

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	Fig. 6. (color online) (a) The DOS of r-TiO with 6.25% vacancy by GGA and GGA+U; (b) the DOS of the pure r-TiO and r-TiO with 12.5% Hf doped by GGA+U.

The band gap of vacancy solid solutions by GGA and by GGA+U is 1.695 eV and 2.593 eV, respectively. As everyone knows, DFT (either by GGA or by LDA) usually undervalues the band gap. From Fig. 6(a), there are two clear sharps by GGA (in red color). If the GGA+U is used, the two sharps will merge together and the sharp height becomes bigger than those by GGA, much similar to our other works like cubic HfO ^[19] and monoclinic HfO .^[29] For hafnium-substitution solid solutions in rutile TiO , we first set 2 × 2 × 1 super cell. Namely, in one super cell, there are 8 titanium and 16 oxygen atoms. We substitute only one Ti atom by hafnium in the center of the super cell, whose substitution rate is 12.5%, and we obtain the electronic structure of hafnium-substitution solid solutions and rutile TiO by GGA+U ( eV, eV, and eV for Hf 5d, Ti 3d, and O 2p, respectively) after geometry optimization. The calculation results are shown in Fig. 6(b). The band gap of the pure and Hf-doped rutile TiO by GGA+U is 3.0 eV and 2.895 eV, respectively. From Fig. 6(b), we can see that the two lines are much similar between the pure and Hf-doped rutile TiO by GGA +U. Considering the similar electronic configuration of Ti and Hf in the fourth subgroup with 4 valence electrons, it is reasonable that the pure r-TiO has a very similar DOS with the doped one by 12.5% Hf. Besides, the calculated results of Mulliken atomic population in r-TiO and the doped r-TiO by 12.5% Hf are much similar, and the transfer charge of the O atoms in r-TiO and the doped one by 12.5% Hf is −0.645 and −0.657, respectively.

The electronic structures and optical properties of anatase and rutile TiO are calculated by means of first-principles generalized gradient approximation GGA and GGA) +U approaches. By GGA, the resulting band gaps in a-TiO and r-TiO are around 2.16 eV and 1.85 eV, smaller than the experimental values. Introducing the Coulomb interactions of 3d orbitals on the Ti atom ( and of 2p orbitals on the O atom ( , we can reproduce the experimental values of the band gap for a- and r-TiO . The best values for and are eV and eV, eV and eV for r- and a-TiO , respectively. The complex dielectric functions and refractive index of r-TiO and a-TiO are calculated. The results show optical anisotropy in both a-TiO and r-TiO . The r-TiO has relatively large anisotropy with birefringence Δn = 0.29, which makes it a useful optical material. We further calculate the electronic structure of the titanium-vacancy and of hafnium-substitution solid solutions in rutile TiO , and the calculation results show that the GGA+U approach has wide adaptability.