Dissimilar anisotropy of electron versus hole bulk transport in anatase ${\mathrm{TiO}}_2$: Implications for photocatalysis

Recent studies on crystal facet manipulation of anatase ${\text{TiO}}_2$ in photocatalysis have revealed that reduction and oxidation reactions preferably occur on (100)/(101) and (001) facets, respectively; however, a fundamental understanding of their origin is lacking. Here, as a result of first-principles calculations, we suggest that a dissimilar trend in the anisotropy of electron vs hole bulk transport in anatase ${\text{TiO}}_2$ can be a dominant underlying mechanism for the difference in photochemical activity. Photoexcited electrons and holes are driven to different facets, i.e., electrons on (100)/(101) and holes on (001), leading to the observed preference for either reduction or oxidation. This trend of electrons vs holes found in pure ${\text{TiO}}_2$ applies even for cases where a variety of dopants or defects is introduced.

Figure 1

Crystal and energy-band structures of ${\mathrm{TiO}}_2$. (a) Unit cell of ${\text{TiO}}_2$. Blue and red circles signify a Ti-centered octahedronlike ${\text{TiO}}_6$ unit and an O-centered $\mathsf{T}$-shape-like ${\text{OTi}}_3$ unit, with selected orbitals (Ti $3d_{xy,yz}$, O $2p_{\pi ,\sigma }$) drawn at the right. (b) Calculated energy band structures along high-symmetry $k$ directions. Orbital-resolved PDOSs are aligned at the right.

Figure 2

Anisotropy of electron vs hole effective mass in ${\text{TiO}}_2$. Circular plots of the (a) electron ($m_e^{*}$) and (b) hole ($m_h^{*}$) effective mass, as a function of the propagation directions through the lattice on the $xz$ plane. The distance measured from the origin (O) to the embedded surface offers relative comparisons of the effective masses between crystal directions. (c) Exemplary schematic of a ${\text{TiO}}_2$ particle with dominant surface facets. The color code on the facet in indicates the ratio $R$ ($m_e^{*}/m_h^{*}$), which represents the degree of the reductive and oxidative nature of the exposed facets.

Figure 3

Origin of opposite anisotropy of electron vs hole effective mass. (a) The lowest CBs along the [100] or [010] (circles) and [001] (triangles) $k$ directions. The color code indicates the ratio O $2p_{\pi }$/Ti $3d_{xy}$. (b, c) Two-dimensional cross sections of the square of wave functions of the CBs at $k=\mathrm{CBM}+(0.03,0,0)$ ($2\pi /\text{Å}$) (b) and $k=\mathrm{CBM}+(0,0,0.03)$ ($2\pi /\text{Å}$) (c). (d) The highest VBs along the aforementioned directions in (a). The color code indicates the ratio Ti $3d$/O $2p_{\pi }$. (e, f) The square of wave functions of the VBs at $k=\mathrm{VBM}+(0.03,0,0)$ ($2\pi /\text{Å}$) (e) and $k=\mathrm{VBM}+(0,0,0.03)$ ($2\pi /\text{Å}$) (f). Red and black dots in a unit cell indicate Ti and O atoms in the cross sections, respectively. The isosurface value is 0.002 $e$/${\text{Å}}^3$.

Figure 4

Comparisons of effective mass anisotropy (log scale) in the presence of dopants and defects.

Figure 5

Direction dependence of polaronic transport of both excess electron (a–c) and excess hole (d–f) in anatase ${\mathrm{TiO}}_2$. (a) DOS of the extra electron case ($U=5$ eV). (b) Spin density contour of polaron states (localized on a single Ti atom). (c) Potential energy for polaron hopping along the [100], [001], and [201] directions. (d) DOS of the extra hole case ($U=5$ eV). (b) Spin density contour of polaron states (localized on a single O atom). (f) Potential energy for polaron hopping along the [100] and [001] directions.