Author: Jeremy Hu

Abstract

Density functional theory (DFT) calculations of TiO₂ anatase, a commonly used catalyst support, typically requires a DFT+U correction term to account for the electron self-interaction error in Ti. DFT as implemented in the Vienna Ab Initio Simulation Package (VASP) was used to determine the minimum value of the Hubbard’s U parameter to accurately represent the electronic density of states of TiO₂ and partially-reduced TiO₂. This study determined that a minimum Hubbard’s parameter of U = 3 eV is sufficient to accurately determine the density of states of TiO₂ and partially-reduced TiO₂.

Introduction

Titanium dioxide (TiO₂) is a common metal oxide for depositing catalytically-active metals onto, such as those used in biochemical production [1].  The metal oxide support provides structure for the active metal sites to be deposited onto. TiO₂ supported-catalysts for biochemical syntheses generally undergo reduction conditions, with the formation of oxygen vacancy sites in the presence of H₂ [2]. The anatase polymorph of TiO₂ most readily forms oxygen vacancies, with the (001) facet widely considered the most catalytically active [1]. Understanding the chemistry of TiO₂ using DFT typically requires a DFT+U correction term to account for the self-interaction error between electrons. This correction term is typically necessary for Ti due to its numerous d-orbital valence electrons. Thus, the DFT+U correction penalizes the delocalization of d-orbital electrons in Ti [3]. Previous literature suggests that the U parameter can range anywhere from 2-10 eV, with the U parameter “sufficient” when the density of states of TiO₂ behaves as an insulator with KS orbitals appearing in the band gap in the presence of oxygen vacancies [3]. Thus, understanding the effects of the U parameter on the density of states of TiO₂ could bring insight into the minimum value of U necessary to confirm the localization of electrons in bare TiO₂ and TiO₂ under oxidation conditions.

Methods

Electronic Methods

Density functional theory (DFT) analysis of TiO₂ anatase (001) was calculated using the plane-wave basis set in the Vienna Ab Initio Simulation Package (VASP) [4]. The Perdew–Burke-Ernzerhof (PBE) exchange correlation functional was used [5]. PBE+D3 was used for dispersion corrections and the projector augmented-wave method (PAW) corrected for core-valence interactions [6] [7]. The Hubbard’s parameter (U) for the DFT+U correction was iterated for Ti from U = 0-3 eV [8]. Each structure was reoptimized for each value of U before calculating the density of states. The forces on the atoms from the geometric optimizations used a convergence criteria of < 0.05 eV/Å. The self-consistent field tolerance for all calculations was 10^-5 eV.

Similar work on TiO₂ suggest that a Monkhorst-Pack k-point mesh of 3 x 3 x 1 and a cutoff energy of 450 eV are above the minimum for convergence [9] [10]. The valence electrons considered for each atom type were O (2s² 2p⁴) and Ti (3s² 3p⁶ 4s² 3d²).

A vacuum space of 15 Å between slabs was used to minimize dipole interactions in the z-direction (i.e., normal to the surface). A 2 x 2 supercell of anatase TiO₂ (001) was used for the DFT calculations, with the bottom three atomic layers fixed while the rest of the atoms were allowed to freely relax (Fig. 1). The slab’s thickness was two layers thick, with a single layer defined as the minimum thickness of atoms to have a stoichiometric ratio of Ti_xO_2x. Additionally, the termination of the surface was chosen to be the same as the termination on the bottom of the slab (i.e., oxygen atoms on the top and bottom) to minimize any large dipole moments that would otherwise occur through asymmetry.Figure 1. A 2 x 2 supercell of TiO₂ anatase (001) used for the density of states calculation.

The second-coordinated surface oxygen atom on the surface, which required the lowest amount of energy to remove, was removed for the density of states calculations of anatase TiO₂ with an oxygen vacancy (Fig. 2)

Figure 2. A 2 x 2 supercell of TiO₂ anatase (001) with a second-coordinated bridging oxygen (O_2C ) removed used for the density of states calculation. The symmetrically-equivalent O_2C behind the removed atom was removed in this figure for clarity.

Results and Discussion

The density of states for the bare TiO₂ anatase (001) was plotted for values of U from 0 to 3 (Fig. 3a-d).

Figure 3a. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=0. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 3b. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=1. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 3c. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=2. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 3d. The total density of states vs. the Fermi level subtracted from the energy (eV) for U=3. The blue line represents the spin up states and the black line denotes the spin down states.

There was slight noise and variation in the density of states when U was iterated from 0 to 3. However, the band gap of approximately ~2 eV stayed relatively constant in all cases, which relatively agrees with the experimentally determined band gap of ~3 eV in TiO₂ [11]. The nuances in the density of states at U=3 (i.e., the clearer differentiation of peaks) suggest that a higher U value may be more appropriate for the calculation of TiO₂ density of states.

The density of states for the partially-reduced TiO₂ (i.e., TiO₂ with a surface oxygen vacancy) was plotted as well from U = 0 to 3 (Fig. 4a-d).

Figure 4a. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=0. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 4b. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=1. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 4c. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=2. The blue line represents the spin up states and the black line denotes the spin down states.

Figure 4d. The total density of states vs. the Fermi level subtracted from the energy (eV) for TiO₂ with a surface oxygen vacancy (O_2C) at U=3. The blue line represents the spin up states and the black line denotes the spin down states.

As expected, there was noise and variation in the density of states ranging from U = 0 to 3. The peak in the band gap only clearly appeared in the cases where U=0 and U=3. As mentioned previously, the density appearing in the band gap for TiO₂ with an oxygen vacancy likely represents occupied KS orbitals following reduction. Recent experimental and DFT studies of TiO₂ with an oxygen vacancy suggest a band gap of approximately 3.0 eV with a gap state appearing around 0.7 eV below the conduction band [12].  In the case where U was set to 0, the gap state was ~0.50 eV below the conduction band, whereas when U=3 the gap state was ~0.70 eV below the conduction band. Since the peak in the band gap more closely matches experimental results in the case where U=3, it is likely that a U parameter of 3 and above is necessary to correctly determine the density of states of TiO₂ with an oxygen vacancy.

 Conclusion

The density of states was calculated for anatase TiO₂ (001) and anatase TiO₂ (001) with a surface oxygen vacancy at varying U correction values from 0 to 3. The data suggest that for bare TiO₂, a Hubbard’s U correction of 3 eV and above may prove appropriate for clearer peak density. However, the band gap of around 2 eV stayed relatively constant in all four cases. In the case of TiO₂ (001) with an oxygen vacancy, the appearance of a gap state near the Fermi level correlates to occupied KS orbitals following the reduction of TiO₂. Although gap states appeared in both cases where U=0 and U=3, the gap state peak which occurred when U=3 more closely agreed with experimental data. Thus, the results suggest that a Hubbard’s U correction of 3.0 eV and above is appropriate for calculating the density of states of TiO₂ (001) and TiO₂ (001) with an oxygen vacancy.

Citations

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