Titanium dioxide is a promising photocatalysis material, which also has wide applications in heterogeneous catalysis, solar cells, gas sensor, etc^[1-5]. The surface of TiO₂ has been extensively studied for many years due to its essential role in the applications. Especially, surface defects have been proved to be vital for surface reactivity. For instance, surface oxygen vacancies are reactive sites for water and other molecules^[6]. It is desirable to use external field to control the surface properties in order to improve the surface reactivity of TiO₂.

Strain field is unavoidable in fabrication of nanostructures and thin film^{[7, 8]}. It has been demonstrated that the band structure of TiO₂ can be changed by using strain^[9]. Besides, the surface properties can also be engineered by strain. We have paid our attention on the strain effect on the surface of TiO₂ since 2008, and found that the strain can change many aspects of the surface properties, such as the surface energy, the distribution of surface defects, surface adsorption and dissociation^[10-15].

In this paper, we review our work on the strain engineering of the surface properties of TiO₂. The following section is arranged as follows. In the second section, we will first introduce the simple theoretical background on the strain engineering. In the third section, we will discuss how the strain can be used to improve the relative percentage of the reactive surface. In the fourth and fifth sections, the thermodynamic and kinetic properties of oxygen vacancies under strain are reviewed, respectively. In the sixth and seventh sections, the effect of strain on the surface adsorption on stoichiometric and reduced TiO₂(110) are reviewed, respectively.

The structure and thus the properties of a bulk material respond to the external strain differently according to its bulk elastic properties. The bulk elastic properties of TiO₂ are described by six independent elastic constants. The bulk modulus and Young’s modulus can be further calculated by these independent elastic constants. The results are presented in Table 1, including our own calculations based on the density functional theory (DFT)^[13].

Similar to the case of a bulk material, the response of a surface to the external strain depends on the surface elastic properties. Generally speaking, different surface structures also have different surface elasticity. Therefore, the thermodynamic equilibrium surface structures would adjust themselves according to the external strain, and the surface properties change with the strain correspondingly.

The surface elastic properties can be described mainly by the surface stresses and surface elastic constants, which are defined as follows^[19],

$ \begin{array}{l} \sigma_{\alpha \beta} = \dfrac{1}{A_{\rm{L}}}\dfrac{\partial (A_{\rm{L}} \gamma_{\rm{L}})}{\partial \epsilon_{\alpha \beta}} = \dfrac{\partial \gamma_{\rm{L}}}{\partial \epsilon_{\alpha \beta}}, \\ S_{\alpha \beta\alpha' \beta'} = \dfrac{\partial \sigma_{\alpha \beta} }{\partial \epsilon_{\alpha' \beta'}} = \dfrac{\partial^2 \gamma_{\rm{L}}}{\partial \epsilon_{\alpha \beta} \partial \epsilon_{\alpha' \beta'}}. \end{array} $ (1)

Here and are the area of the surface and the Lagrangian surface energy per unit area in an unstrained state, respectively. The indices are 1 or 2, denoting the two orthogonal directions of a surface. To investigate the interplay between external strain and the surfaces, whether intrinsic or defective, we first of all calculated the surface energies of stoichiometric or defective surfaces with variation of external strain, from which the surface elasticity in different situations are derived according to Eq. (1). Then, the surface energy under arbitrary small strain can be calculated according to the continuum elastic theory. For instance, if the in-plane strains are and , the surface energy can be calculated as follows,

$ \begin{align} \gamma(\epsilon) =\; & \gamma(0)+ \sigma_{11} \epsilon_{11}+ \sigma_{22} \epsilon_{22} \\ & +\frac{1}{2}\epsilon_{11}^2S_{1111}+\frac{1}{2}\epsilon_{22}^2S_{2222} +\epsilon_{11}\epsilon_{22}S_{1122}. \end{align} $ (2)

Under a certain strain, by comparing the surface energies of different surface structures, one can expect to investigate the effect of strain on the surface thermodynamic or kinetic properties.

The surface elastic properties of TiO₂ are shown in Table 2, the calculation details of which can be found in Ref. [13]. The surface elastic constants of rutile TiO₂ surface are always negative, whereas for anatase they are either positive or negative. Note that the significance of elastic constants of a surface are different from those in bulk. The elastic constants of a stable material must be positive, whereas negative surface elastic constants can occur, which means that the presentation of the surface makes TiO₂ in nanoscale softer than its bulk phase^[13].

It is worthy to note that there is an odd–even oscillation of the surface properties of the relaxed TiO₂ in the thin film slab. For instance, for rutile (110) surface, although the surface energy and the band gap converge in a slab within about ten trilayers, the surface stresses in the direction are still in obvious oscillation even when the thickness of the slab is 14 trilayers, as shown in Fig. 1. The thick-dependent surface stresses can be applied to change the relative stability of a thin film with different thickness, according to Eq. (2), by the strain along direction. It suggests that one can selectively grow a thin film with a specific atomic layers by applying a suitable external strain.

Generally speaking, there is an intrinsic controversy between stability and reactivity in nature. The more reactive a surface is, the less stable it is, or the higher surface energy it is. According to the Wulff's theorem, it means that the percentage of a reactive surface is usually low. For example, in equilibrium anatase TiO₂, the most reactive (001) surface has a surface energy much larger than the (101) surface, as shown in Table 2. As a result, the area proportion of anatase (001) surface is no more than 3%, which strongly limits the reactive efficiency of TiO₂ in application as a photocatalyst^[20-22].

Fortunately, the anatase (001) surface possesses the largest surface stress among all the surfaces, which is positive and occurs along the [100] direction. We therefore expect that the relative stability of (001) surface can be significantly improved by applying an compressive strain along the [100] direction, and thus its area percentage of anatase (001) according to Wulff construction. By first principle calculations, we have verified this^[13]. As displayed in Fig. 2(a), under 5% biaxial compressive strain or compressive strain along [100], the area proportion of anatase (001) surface increases more than five times. In this way the external strain can be used to enhance the reactivity of TiO₂ catalyst.

Oxygen vacancies are the most common defects in transition metal oxide. The reduced materials usually show different physical properties compared to the stoichiometric one. In TiO₂, the oxygen vacancy (OV) has a low formation energy, and it has been experimentally found reactive in the surface reaction^[23-30]. In order to investigate the influence of strain on the bulk and surface properties when different OVs are introduced, we first of all need to understand the responses of the formation of OVs to different external field^{[31, 32]}. By using DFT calculations, we have found that external strain can modify the type of oxygen vacancy, which further influences the surface reactivity. Hereafter we refer the stoichiometric TiO₂ as s-TiO₂, while the TiO₂ with oxygen vacancies as r-TiO₂.

There are three types of oxygen atoms, bridging, in-plane and subbridging oxygen atoms, as shown in Fig. 3. Correspondingly, three different types of OV can be introduced by removing corresponding different types of oxygen atoms, namely BOV, IPOV and SBOV. We named different types of OVs to include the site of the defect. For example, 2-in-plane OV represents the OV formed by removing an in-plane oxygen atom in the second trilayer.

The OV formation energy can be calculated as follows,

$ E_{\rm{OV}} = E_{\rm{r}}-E_{\rm{s}}+\frac{1}{2} E_{{\rm{O}}_{2}} , $ (3)

where is the energy of the isolated O₂ molecular, and and stand for the energies of r-TiO₂ and s-TiO₂, respectively.

The OV formation energy as a function of OV depth is shown in Fig. 4(a). Obviously the 1-bridging OV is the most easily formed OV defect because of the lowest formation energy. In addition, the in-plane OV formation energy remains almost constant at different depth, while the bridging and subbridging OVs formation energies appear a kind of odd-even oscillation behavior. With the position of OV going deeper, both of them tend to approach the formation energy of bulk OV^{[33, 34]}.

To get the influence of oxygen vacancies on the surface elasticity, we also calculated the vacancy-induced change of surface energy,

$ \Delta \gamma = \frac{E_{\rm{r}}-E_{\rm{s}}} {A_{\rm{L}}} = \theta \frac{ E_{\rm{OV}} - \frac{1}{2} E_{{\rm{O}}_{2}}}{A_{\rm L0}}, $ (4)

where is the density of vacancies and is the area per surface unit. The data of the vacancy-induced changes of surface stress and surface elastic constants were thus obtained according to Eq. (1). In our calculations, a supercell is used, and the corresponding defect density is . Lower defect density does not change the OV formation energies evidently. However, the values of , and are proportional to OV density even when the density dependence of is ignorable, as indicated by Eq. (4).

The value of , and induced by oxygen vacancies with are listed in Table 3 and and are shown in Fig. 4(b). It can be seen that the subbridging OV induces more tensile stress in and less compressive stress in [001], as well as smaller elastic constants in both direction than that in the bridging one.

By substituting the data in Table 3 into Eq. (2), a phase diagram can be obtained for the most stable oxygen vacancy in the first trilayer^[11]. As shown in Fig. 5, the sub-bridging OV and the in-plane OV are the most favorable surface OV under the compressive and tensile biaxial strain, respectively. In contrast, the bridging OV is most favorable when a tensile strain is applied along and meanwhile a compressive strain is along [001]. Calculations on the formation of OV in bulk TiO₂ also suggest that the OV is more easily formed when it sits in the titanium-contained (110) plane under tensile strain along the direction, consistent with the surface situation^[12].

It is worthy to mention that we did not realize until recently that the incipient ferroelectricity of TiO₂ is also quite important for its surface properties. Under tensile strain, it tends to be more stable in the ferroelectric polarization state with lower symmetry^[35-37]. However, we calculated the surface elasticity of OVs without intentionally reducing the symmetry. Therefore only if an OV could automatically break the symmetry, the tensile strain may drive the system to a polarization state^[15]. We recently found that this occurs only when introducing an in-plane OV, whereas in the other cases the polarization state cannot be spontaneously reached although it is thermodynamically more stable under tensile strain. Therefore in more accurate calculations of the surface elasticity, one should take the polarization state into account.

The distribution of defects can also be influenced by the diffusion kinetics. We have extensively studied the diffusion properties of OVs in TiO₂ and the effects of strain on the kinetics^[12]. We found that the barrier is very high for an OV to directly hop along [001] direction, therefore the global energy barrier for an OV to diffuse either along [001] or [110] is determined by the path from the base planes to the apex positions of the Ti-O octahedrons. As shown inFig. 6, this barrier decreases with increasing density of OV in the [001] direction. It indicates that the interaction between neighboring OVs facilitates the mobility of OVs in the bulk of TiO₂.

When a strain is applied along [001] direction or biaxially along [100] and [010], it would not break the symmetry. In this case, we found that the diffusion barrier is reduced by tensile strain and enhanced by compressive strain. If, however, the strain is just applied in [110] and , it would increase the diffusion barrier of OVs whatever the strain is compressive or tensile. It also results in anisotropic diffusion behaviors along [110] and . Among the two otherwise equivalent directions, the bulk OV prefers to diffuse along the one under more compressive or less tensile strain, as indicated in Fig. 7.

The diffusion barrier of OV on the surface is much lower than that of bulk OV. However, a highest diffusion barrier of OV occurs between the second trilayer and the third one, which hinders the OVs in bulk from diffusing to the surface^[12]. Externally applied tensile strain can reduce the energy barrier of the subsurface OV diffusion, and help to improve the diffusion of OVs from bulk to surface^[12].

The diffusion of OVs within the (110) surface is also anisotropic. As shown in Fig. 8, the surface bridging OV diffuses more rapidly along , mediated by the in-plane OV, with an energy barrier much smaller than in the bulk. The in-plane tensile strain lowers the diffusion barrier along both direction. With increasing tensile strain along , the barrier along the decreases more significantly than along the [001] direction. Therefore the tensile strain along the direction further enhances the anisotropy in the surface diffusion^[11].

Since the hydroxyl ions and protons introduced by water dissociation play key roles in most relevant surface reactions, the adsorption and dissociation of water on TiO₂ surfaces have been extensively studied in the past years. We have found that the adsorption and dissociation of water on stoichiometric rutile (110) surface depend on both the coverage and the thin film thickness^[14]. The adsorption energy was calculated as follows,

$ E_{\rm{M(D)}} = (E_{\rm{w}}- E_{\rm{s}})/2 - E_{{\rm{H}}_2{\rm{O}}}, $ (5)

where , and stand for the total energy of water-adsorbed system, total energy of stoichiometric surface and the isolated water molecular respectively. The adsorption energies and correspond to the molecular and dissociation configurations, respectively. The dissociation energy is defined as .

As shown in Fig. 9, both the adsorption and dissociation of water are energetically more favorable when the thin film has an odd O–Ti–O trilayers. Especially, when the adsorbed water has no neighboring one along the [001] direction, the dissociative water is more favorable on the TiO₂(110) surface of a thin film with three trilayer. This can be easily understood by referring to Fig. 1(a), since the slab containing an odd number of layers is less stable and thus more reactive than the one of even number of layers. With increasing thickness of the thin film or increasing coverage of the water, both the adsorption and the dissociation of water become less favorable.

The adsorption of water is more energetically favorable on thin films under tensile strain. Besides, the dissociation barrier of the water also decreases with increasing tensile strain^[14]. Furthermore, the odd–even oscillation of the dissociation energy is enhanced by compressive strain, while reduced and even reversed by tensile strain. It means that under moderate tensile strain, the dissociation of water is more favorable on thin films containing even numbers of trilayers. This can be understood from the thickness-dependent surface stress. As shown in Figs. 1(b) and 1(c), the surface stress of the (110) thin film are positive. Therefore, tensile strain should increases their surface energy and thus their surface reactivity.

As mentioned above, the rutile TiO₂ is a well-known incipient ferroelectric material. Previous theoretical studies suggested that the transition can occur under negative pressure or uniaxial strain. For the most stable TiO₂(110) surface, spontaneous in-plane polarization is predicted to occur under tensile strain. This strain-induced in-plane polarization is expected to play roles in the surface reactivity of the TiO₂(110) surface, which we have just explored^[15].

By studying the adsorption of a series of molecules on the reduced TiO₂(110) surface under tensile strain, we found that the stabilities of the polarized structure relative to the non-polarized one are determined by the surface doping caused by the charge transfer between the adsorbates and the surface. The polarized structure favors the surface adsorption, and vice versa, if the adsorbate captures extra electrons of the reduced TiO₂ surface and introduces no additional holes. It is because the free carriers would screen the long-range Coulomb interactions and suppress the spontaneous polarization. The electrostatic interaction between the polar molecule and the in-plane polarization is of less importance relative to the effect of the surface doping.

The adsorbates we calculated were a series of chalcogen-containing molecules, including H₂O, H₂S, O₂ and CO₂, in molecular state and dissociative state. Since the molecular adsorption of H₂S is unstable against partial dissociation, we only considered the adsorption of H₂S in a partial dissociative or complete dissociative state, denoted respectively as (D ) and (D₂). The other molecules in molecular and dissociative states are hereafter denoted as (M) and (D), respectively.

We calculated the total energies of reduced TiO₂(110) after surface adsorption. The energies are denoted as and for the stable strain-induced polarization state and the metastable non-polarization state, respectively. The difference was calculated to demonstrate the polarization induced changes of the total energies of r-TiO₂ after surface adsorption. We found that the adsorbates can be clearly divided into two categories. For H₂O(M, D), H₂S(D , D₂) and CO₂(M), follows the variation of , , the energy difference of r-TiO₂(110). For CO₂(D) and O₂(M), equals approximately , , the energy difference of s-TiO₂(110). The two categories of adsorbates are hereafter denoted respectively as and . O₂(D) is intermediate between and since the total energy difference after adsorption of O₂(D) falls between and .

In order to understand the difference between the two different categories of adsorbates, the density of states (DOS) of r-TiO₂(110) with different adsorbates were compared. We found that the structures have similar if only they have similar Fermi levels, independent of the specific adsorbates. The bader analysis shows a consistent variation for the electronic doping of the surface by surface adsorption as the shift of the Fermi level does. In brief, the charge transfer between adsorbates and r-TiO₂(110) is trivial, so that the surface remains doped with electrons by the oxygen vacancy. Instead, the adsorbates heal the reduced surface and result into a neutral surface. Moreover, the adsorption of O₂(D) further dopes holes after healing the reduced surface.

In order to further verify the conclusion, we charged the polarized and non-polarized s-TiO₂(110) surface by simply adding and removing electrons, and then calculated the energy difference under the strain applied in direction without further structural relaxation. As shown in Fig. 10, has a minimum when the surface is neutral, and increases with the amount of the charge ( ) approximately linearly. We then plotted the polarization-induced energy difference of r-TiO₂(110), without adsorption and after adsorption, at a preset charge of s-TiO₂(110): for or with adsorbates , for with adsorbates , and for after adsorption of O₂(D). We found that these energy differences match quite well with at the preset values of . It confirms that the interaction between the in-plane polarization of TiO₂ surface and the adsorbate depends mainly on the surface doping by the adsorbate.

In this paper, we have reviewed our work within recent ten years on the strain engineering of TiO₂ surface properties. We found that when suitable external strain is applied in specific directions, the surface and bulk properties undergo a significant change, due to the difference in the elastic properties of different surface structures. We thus propose that the external strain is a good way to engineer the physical and chemical properties of TiO₂ and to further improve the catalytic reactivity. We hope our work will stimulate more experimental studies.

The numerical calculations were carried out at the High Performance Computing Center of Nanjing University. This work was supported by the National Natural Science Foundation of China (Grants No. 11974164).