Quantum confinement and surface relaxation effects in rutile TiO${}_2$ nanowires

Spin-polarized density functional theory calculations within the generalized gradient approximation have been applied to investigate the size- and shape-dependent atomic and electronic properties of [001]-oriented rutile ${\mathrm{TiO}}_2$ nanowires of rectangular cross section. We find pronounced even-odd oscillation in the formation energy and band structure of the nanowires as a function of the number of ${\mathrm{TiO}}_2$ layers, which are largely connected to the presence or absence of a mirror Ti-O plane along either confinement direction. We demonstrate that the relative stability and the oscillation in the band structure characters of the rutile ${\mathrm{TiO}}_2$ nanowires arise from the interplay between surface relaxation and quantum confinement effects, which depend on the even-odd parity of the number of ${\mathrm{TiO}}_2$ layers and can be tuned separately along each confinement direction.

Figure 1

Atomic structures of the $6\times 8$ (left figure in A), $5\times 8$ (right figure in A), $3\times 7_1$ (left figure in B), and $3\times 7_2$ (right figure in B) rutile ${\mathrm{TiO}}_2$ nanowires oriented along the [001] direction. For the type-1 $3\times 7_{\text{1}}$ nanowire, the mirror Ti-O plane along the first confinement direction terminates on the Ti-$5c$ atoms at the corresponding facet surfaces. For the type-2 $3\times 7_{\text{2}}$ nanowire, the mirror Ti-O plane instead terminates on the surface Ti-$6c$ atoms which are bound to the twofold coordinated bridging O atoms (O-$2c$). The large gray and small red spheres represent Ti and O atoms, respectively. For each nanowire, both the axial view (upper figure) and side view (lower figure) are shown.

Figure 2

The upper figures show the dependence of the axial lattice constant on the perimeter (left) and cross-sectional area (right) with the PBE functional. The horizontal line denotes the corresponding bulk value. The lower figures show the formation energy per ${\mathrm{TiO}}_2$ unit as a function of the perimeter (left) and cross-sectional area (right). In all figures, the filled markers denote a direct band gap at the $\Gamma$ point and empty markers denote an indirect gap.

Figure 3

Dependence of the band gap (upper figures) and the conduction- and valence-band edges (lower figures) of the rectangular rutile ${\mathrm{TiO}}_2$ nanowires on the perimeter (left figures) and cross-sectional area (right figures). The band edges are aligned by lining up the Fermi levels of the different nanowires, which are chosen as the energy reference. In all figures, the filled markers denote a direct band gap at the $\Gamma$ point and empty markers denote an indirect gap. The two horizontal lines in the band-gap plot show the bulk indirect and direct band gaps, respectively. The two horizontal lines in the band-edge plot show the positions of the bulk conduction- and valence-band edges relative to the bulk Fermi level (set at the energy reference).

Figure 4

Dependence of the band gap (upper figures) and the conduction- and valence-band edges (lower figures) of the rectangular rutile ${\mathrm{TiO}}_2$ nanowires on the perimeter (left figures) and cross-sectional area (right figures) computed at the PBE relaxed geometries using the LDA exchange-correlation functional and corresponding pseudopotentials, similar to those shown in Fig. 3.

Figure 5

Density of states projected onto the $3d$ orbitals of Ti and the $2p$ orbitals of O atoms on both the first and second ${\mathrm{TiO}}_2$ layers along the horizontal (a) and vertical (b) confinement directions for $6\times 8$ nanowire. The zero of the DOS energy scale is the vacuum level. We have also shown the total DOS of the nanowire as a common reference in both figures.

Figure 6

Isosurface plots of the local density of states (LDOS) at the valence-band maximum (left figure) and the conduction-band minimum (right figure) for the $6\times 8$ (A) and $5\times 8$ (B) rutile ${\mathrm{TiO}}_2$ nanowires. The large gray and small red spheres represent Ti and O atoms, respectively. For each nanowire, both the top view (upper figure) and side view (lower figure) of the LDOS are shown.

Figure 7

Cross-sectional view of the type-1 $3\times 7_1$ (A) and type-2 $3\times 7_2$ (B) nanowires. For each nanowire, we have shown the isosurface plots of the LDOS at the valence-band edge (left figure) and the conduction-band edge (right figure). The type-1 $3\times 7_{\text{1}}$ nanowire has an indirect gap due to a shift in the valence-band maximum away from the $\Gamma$ point (empty marker in Fig. 3), while the type-2 $3\times 7_{\text{2}}$ nanowire has a direct gap at the $\Gamma$ point (filled marker in Fig. 3).

Figure 8

COOP plot for the $6\times 8$, $5\times 8$, $3\times 7_1$, and $3\times 7_2$ ${\mathrm{TiO}}_2$ nanowires. For each nanowire, a and b show the COOP between the $3d$ orbitals of the surface Ti-$5c$ and subsurface Ti-$6c$ atoms (“$5c$-$6c$”) and the COOP between the $3d$ orbitals of the surface and subsurface Ti-$6c$ atoms (“$6c$-$6c$”) along both horizontal (a) and vertical (b) confinement directions. c and d show the COOP between the $3d$ orbitals of the second layer Ti atoms and the $2p$ orbitals of the O atoms above (“Ti $3d$–O $2p$ $2:1$”) and the COOP between the $3d$ orbitals of the second layer Ti atoms and the $2p$ orbitals of the O atoms underneath (“Ti $3d$–O $2p$ $2:3$”) along both horizontal (c) and vertical (d) confinement directions. For the two $3\times 7$ nanowires, (e) shows the COOP plots between the $2p$ orbitals of the neighbor O layers along both the horizontal (“O $2p$–O $2p$ $m$”) and vertical (“O $2p$–O $2p$ $n$”) confinement directions. The zero of each energy scale is set to the vacuum level.[41]