First-principles real-space study of electronic and optical excitations in rutile ${\mathrm{TiO}}_2$ nanocrystals

We model rutile titanium dioxide nanocrystals (NCs) up to $\sim 1.5$ nm in size to study the effects of quantum confinement on their electronic and optical properties. Ionization potentials (IPs) and electron affinities (EAs) are obtained via the perturbative $GW$ approximation ($G_0{W}_0$) and $\Delta \mathrm{SCF}$ method for NCs up to 24 and 64 ${\mathrm{TiO}}_2$ formula units, respectively. These demanding $GW$ computations are made feasible by using a real-space framework that exploits quantum confinement to reduce the number of empty states needed in $GW$ summations. Time-dependent density functional theory (TDDFT) is used to predict the optical properties of NCs up to 64 ${\mathrm{TiO}}_2$ units. For a NC containing only 2 ${\mathrm{TiO}}_2$ units, the offsets of the IP and the EA from the corresponding bulk limits are of similar magnitude. However, as NC size increases, the EA is found to converge more slowly to the bulk limit than the IP. The EA values computed at the $G_0{W}_0$ and $\Delta \mathrm{SCF}$ levels of theory are found to agree fairly well with each other, while the IPs computed with $\Delta \mathrm{SCF}$ are consistently smaller than those computed with $G_0{W}_0$ by a roughly constant amount. TDDFT optical gaps exhibit weaker size dependence than $GW$ quasiparticle gaps, and result in exciton binding energies on the order of eV. Altering the dimensions of a fixed-size NC can change electronic and optical excitations up to several tenths of an eV. The largest NCs modeled are still quantum confined and do not yet have quasiparticle levels or optical gaps at bulk values. Nevertheless, we find that classical Mie-Gans theory can quite accurately reproduce the line shape of TDDFT absorption spectra, even for (anisotropic) ${\mathrm{TiO}}_2$ NCs of subnanometer size.

Figure 1

The atomic structures and orientations of the $1\times 1\times 1$, $1\times 1\times 4$, $1\times 4\times 1$, and $2\times 4\times 4$ NCs. Oxygens are red, titaniums are light blue, passivating pseudohydrogens with $4e$/3 charge are pink, and passivating pseudohydrogens with $2e$/3 charge are gray.

Figure 2

$G_0{W}_0$ IP (top) and EA (bottom) of the $1\times 1\times 1$ (black circles), $1\times 1\times 2$ (red pluses), $1\times 1\times 3$ (green squares), and $1\times 1\times 4$ (blue triangles) NCs as a function of ${\varepsilon }_N$, the KS-DFT energy of the highest state included in the Green's function summation. Exponential fits to each NC's converging quasiparticle energies are also shown.

Figure 3

The EA of the $1\times 1\times 1$ NC at various levels of convergence in $GW$, for simulations performed in real-space cells with radii ranging from $R=10$ to 20 a.u. The $G_0{W}_0$ EA is plotted relative to the KS-DFT energy of highest unoccupied state included in each calculation, ${\varepsilon }_N$. Results are already converged with respect to radii at $R=12$ a.u., but not at $R=10$ a.u., for which the predicted EA value is $\sim 0$.2 eV lower than the true EA value after achieving convergence with respect to ${\varepsilon }_N$.

Figure 4

Comparison of IPs (above 6 eV) and EAs (below 6 eV) as a function of the number of ${\mathrm{TiO}}_2$ units in the NC, calculated at three levels of theory: $G_0{W}_0$ (green triangles), $\Delta \mathrm{SCF}$ (red circles), and KS-DFT (blue crosses). For values of $N_{{\mathrm{TiO}}_2}$ that correspond to more than one NC, the average value is plotted. Experimental bulk levels of 7.96 eV and 4.9 eV for IP and EA, respectively, are indicated by arrows at the right. Lines are a guide for the eye.

Figure 5

IPs (top) and EAs (bottom) predicted by $G_0{W}_0$ as a function of $N_{{\mathrm{TiO}}_2}$. Quasiparticle energies of the $1\times n_{\parallel }\times 1$ series are indicated by blue squares, the $1\times 1\times n_{\ell }$ NC series by red pluses, and the remaining NCs by black circles. All $G_0{W}_0$ quasiparticle energies are also listed in Table 2.

Figure 6

TDLDA absorption cross sections for ${\mathrm{TiO}}_2$ NCs as a function of energy. The spectra are scaled by $N_{{\mathrm{TiO}}_2}$ for easier comparison of features. Spectra have a Gaussian broadening of 0.1 eV.

Figure 7

TDLDA optical gaps evaluated with an absolute threshold $p=0.02$ (top) and with a scaled threshold $p_{\mathrm{scaled}}=N_{{\mathrm{TiO}}_2}\times {10}^{-4}$ (bottom). The red crosses correspond to NCs with $n_{\ell }=1$, the orange triangles to NCs with $n_{\ell }=2$, the green squares to NCs with $n_{\ell }=3$, and the blue circles to NCs with $n_{\ell }=4$. All TDLDA gaps for $p=0.02$ are also included in Table 2.

Figure 8

Quasiparticle gaps at the $\Delta \mathrm{SCF}$ (red circles) and $G_0{W}_0$ (green triangles) levels, and optical gaps inferred from TDLDA absorption spectra with $p=0.02$ threshold (blue pluses) as a function of $N_{{\mathrm{TiO}}_2}$. At a given size, the values for NCs differing in shape are averaged. Lines are a guide for the eye.

Figure 9

TDLDA versus Mie-Gans optical spectra for the $1\times 1\times 1$ (top) and $2\times 4\times 4$ (bottom) NCs. The depolarization factors used to compute the Mie-Gans spectra for the NCs are given in each figure.