Fig. 1. (Color online) (a–c) Variations of surface energies and surface stresses, as well as (d–f) the electronic structure properties with the slab thickness. Black solid curves and red dashed curves are for the unrelaxed and relaxed surfaces, respectively. The valence band maximum (VBM,(e)) and the conduction band minimum (CBM,(f)) are referenced to the vacuum level. These figures are adapted from Ref. [14].

Fig. 2. (Color online) (a) The area percentage of anatase (001) facets with variation of the applied strain. The inset displays the equilibrium shape of the anatase crystallite. (b –d) The surface energy ratio and the ratio of with the variation of the applied strain. The anatase (001) surface appears only when the surface energy ratio is smaller than . These figures are adapted from Ref. [13].

Fig. 3. (Color online) Schematic of the rutile TiO₂(110) surface. The bridging oxygen (BO), the subbridging oxygen (SBO) and the in-plane oxygen (IPO) atoms are indicated by the arrows. and denote the atomic rows in [110] direction, with surface terminated by the bridging oxygen atoms or the five-coordinated titanium atoms, respectively. This figure is adapted from Ref. [12].

Fig. 4. (Color online) (a) The formation energy of OV and (b) the changes of the surface stresses and surface elastic constants induced by the OV as a function of the OV depth. The depth is defined as the distance below the unrelaxed 1-bridging O atoms. The dashed horizontal line in (a) denotes the bulk vacancy formation energy. Solid or dashed curves in (b) are to guide the eyes. and are in units of eV/ . These figures are adapted from Ref. [10].

Fig. 5. Phase diagram of the type of the energetically most favorable OV as a function of the external strain and . The dashed curves are predicted according to the surface elasticity properties shown in Table 3. It indicates that BOV, IPOV, or SBOV can be energetically most favorable within different range of the external strain. This figure is adapted from Ref. [11].

Fig. 6. (Color online) Energy profiles of OV pathways within the primitive cell of rutile TiO using different size of supercell, namely , and . The inset shows the schematic of a primitive cell. Large light spheres and small dark spheres denote the O and Ti atoms, respectively (the same below). The numbers label different oxygen atoms in order for reference of OVs. The notations I, II and III correspond to the paths from OV to OV , from OV to OV and from OV to OV , respectively. This figure is adapted from Ref. [12].

Fig. 7. (Color online) (a) Schematic of bulk OV diffusion along the [110] and directions in rutile TiO₂. The numbers label different oxygen atom sites for reference in (b–f). (b–f) The energy profiles of OV diffusion in rutile TiO₂, when (b) isotropic strain or anisotropic strain applied in directions along (c, d) and (e, f) [110]. These figures are adapted from Ref. [12].

Fig. 8. (Color online) The strain-dependent diffusion barrier of the surface OV along [001] (Path I) and along (Path II, directive hopping between the bridging site; Path III, concerted mechanism mediated by the in-plane OV). Upper panels: The predicted energy profiles of the OV along Path III under external strain along (a) and (b) [001]. The solid lines are predicted according to the surface elasticity, and the symbols denote the calculated energy profiles using NEB method when (discrete circles), (discrete triangles) and (discrete squares) are also shown for comparison. Lower panels: the diffusion barriers of different pathways as a function of the external strain along the (c) direction and (d) [001] direction, with discrete points and dashed lines denoting the calculated and predicted values, respectively. These figures are adapted from Ref. [11].

Fig. 9. (Color online) Adsorption energies per water molecule in (a) molecular state and (b) dissociative state on the surface with different supercell size and different thickness . The differences between the adsorption energies of the dissociative state and the molecular state are shown in (c). The x and y direction of the surface is along [001] and direction, respectively. These figures are adapted from Ref. [14] by the permission of PCCP.

Fig. 10. (Color online) The change of the total energy of the s-TiO₂(110) surface induced by the in-plane polarization under strain along as a function of the doped charge. Negative charge and positive charge correspond to electron doping and hole doping, respectively. This figure is adapted from Ref. [15] by the permission of PCCP.

Table 1. The elastic constant , bulk modulus (B), Young’s modulus along the axis ( ), along the in-plane r direction ( ), and along the axis ( ) of TiO₂ rutile and anatase phases, in units of GPa. This table is adapted from Ref. [13].

Table 2. The calculated surface energy and surface mechanical properties of rutile and anatase phases. , , and denote surface energy, surface stress, and surface elastic constant, respectively. Subscripts 11 and 22 denote the directions along [100] and [010] for (001) surface, [010] and [001] for (100) surface, and [010] for (101) surface, and for (110) surface. The units are in J/m². This table is adapted from Ref. [13].

Table 3. Changes of surface energy, surface stress and surface elasticity induced by different types of oxygen vacancies. The units are eV/ . The subscripts 11 and 22 denote the directions along and [001] direction, respectively. This table is adapted from Refs. [10, 11].