Origin of the deep band-gap state in ${\mathrm{TiO}}_2$(110): $dd\sigma$ bonds between Ti-Ti pairs

Defects play crucial roles in the photonic and chemical activities of ${\mathrm{TiO}}_2$. The origin of the deep band-gap defect state $S_{bg}$ in the rutile ${\mathrm{TiO}}_2$(110) surface has remained controversial for quite a long time. Using many-body Green's function theory, we believe that $S_{bg}$ can be attributed only to $\sigma$ bonds formed between $3d$ orbitals at the Ti interstitial, while the nonbonded Ti $3d$ defect states from the oxygen vacancy and polaron, which are held to be responsible for $S_{bg}$ by the present prevailing view, are shallow regardless of their spatial distribution. Especially, we discover the defect-induced appreciable downshift of unoccupied Ti $3d$ bands which should be the key for accurately describing the electronic structure of ${\mathrm{TiO}}_2$ but was missed in previous studies based on density functional theory. Our model could more consistently and more reasonably account for various experimental phenomena on rutile (110) than the current model based on the oxygen vacancy and polaron.

Figure 1

Structures of defects in rutile ${\mathrm{TiO}}_2$. (a) Oxygen vacancy (marked by the black circle), (b) OH pair, and (c) subsurface Ti interstitial at the rutile (110) surface. (d) Small polaron in rutile bulk yielded after structural optimization by $\mathrm{LDA}+U$ ($U=6.0\mathrm{eV}$). (e) Length variation of the six nearest Ti-O bonds around a small polaron in (d) with respect to intact ones. Yellow isosurfaces in (d) and (e) indicate the charge distribution of polaron. Gray, red, and white balls represent Ti, O, and H atoms, respectively. The bottom O-Ti-O trilayer is saturated by pseudohydrogen atoms (see the context for details).

Figure 2

$GW$ band structures and DOSs for defects in the rutile ${\mathrm{TiO}}_2$(110) (a)–(d) surface and (e) and (f) bulk. Filled Ti $3d$ defect levels are highlighted in red. The VBM is set to zero. The hatched area below (above) the VBM indicates the valence band of the defective system (projected conduction band of the defect-free system). The DOS peak arising from the defect state (red curve) is magnified 5 times for a clearer view.

Figure 3

Charge distribution for filled defect states in the rutile ${\mathrm{TiO}}_2$(110) surface. (a) Side and (b) top views for bridging the oxygen vacancy (${\mathrm{O}}_{br}$-vac; marked by a black circle). (c) Side and (d) top views for deep defect state of the Ti interstitial (Ti-int). Yellow isosurfaces in (a)–(d) represent charge distribution. $\sigma$ bonds are formed between $d$ orbitals of Ti-int and two lattice Ti atoms. The inset of (c) present charge density contours in the plane cutting through these three Ti atoms, manifesting formation of $\sigma$ bonds. Gray, red, and white balls represent Ti, O, and H atoms, respectively. Simulated STM images for filled defect states of (e) ${\mathrm{O}}_{br}$-vac and (f) Ti-int. (f) is the sum of contributions from both shallow delocalized and deep localized defect states of Ti-int. Positions of ${\mathrm{O}}_{br}$-vac and Ti-int in (e) and (f) are indicated by arrows. STM images in (e) and (f) correspond to an $(8\times 4)$ supercell replicated from the $(4\times 2)$ one applied in real calculations. ${\mathrm{Ti}}_{5c}$ (${\mathrm{O}}_{2c}$) indicates the position of surface Ti (O) atom rows.

Figure 4

Comparison between the $GW$ method and DFT on the electronic levels of filled polaronic states in the rutile ${\mathrm{TiO}}_2$(110) surface and bulk. (a) Small polaron in a $(4\times 2)$ surface cell optimized by $\mathrm{LDA}+U$ ($U=6.0\mathrm{eV}$). (b) Small polaron in a $(3\times 1)$ surface cell optimized by HSE06. (c) Small polaron in the bulk optimized by $\mathrm{LDA}+U$ ($U=4.0\mathrm{eV}$). Spin polarization is considered for (b) and (c) but not for (a). $GW, \mathrm{LDA}+U$ [$U=6.0\mathrm{eV}$ for (a) and (b), $U=4.0\mathrm{eV}$ for (c)], and HSE06 are utilized to compute the electronic levels of the polaron for each of these three polaron configurations. Columns with different colors represent the gap at the $\mathrm{\Gamma }$ point between specific bands. Light yellow: between the valence and conduction bands (VB-CB) for the intact rutile surface or the bulk supercell without a polaron. Lavender: between the valence and polaronic bands [but the lower spin manifold of the polaronic band in (b)]. Red in (a) and (c): between the conduction and polaronic bands. Red in (b): between two spin manifolds of the polaronic band. Orange in (b): between the conduction and the highest spin manifold of the polaronic bands. In the case of (c), the number of electrons in the system is odd (see Sec. 2 for details); therefore, there is just one spin manifold of the filled polaronic level between VB and CB. The sum of the columns of different colors in each category is the VB-CB gap for the rutile-containing polaron. The VB-CB gap is appreciably narrowed in $GW$ after the polaron is introduced but not in DFT.

Figure 5

Ti interstitial in the deep of rutile ${\mathrm{TiO}}_2$(110) surface. (a) Relative stability of the Ti interstitial (Ti-int) in different layers of rutile (110). The ground-state total energy of Ti-int trapped between the first and second O-Ti-O trilayers (1-2) is set to zero. With Ti-int trapped between the second and third trilayers (2-3) as the example, the inset of (a) shows the surface model used for stability investigation (see Sec. 2 for details). (b) $GW$ band structure for Ti-int in the rutile bulk. Filled Ti $3d$ defect levels are highlighted in red. VBM is set to zero.