Fingerprints of order and disorder in the electronic and optical properties of crystalline and amorphous TiO${}_2$

We have investigated the structural and electronic properties as well as the linear optical response of amorphous TiO${}_2$ within density functional theory and a numerically efficient density functional based tight-binding approach as well as many-body perturbation theory. The disordered TiO${}_2$ phase is modeled by molecular dynamics. The equivalence to experimentally characterized amorphous phases is demonstrated by atomic structure factors and radial pair-distribution functions. By density functional theory calculations, using both the semilocal Perdew-Burke-Ernzerhof functional and the nonlocal Heyd-Scuseria-Ernzerhof screened hybrid functional, the electronic energy gap is found to be larger than in the crystalline TiO${}_2$ phases rutile and brookite but close to the anatase band gap. The quasiparticle energy gap of amorphous TiO${}_2$ is determined to be $\gtrsim$3.7 eV, while the optical gap is estimated to $\lesssim$3.5 eV. The disorder-induced formation of localized electronic states has been analyzed by the information entropy of the charge density distributions. The frequency-dependent optical constants, calculated from the complex dielectric function, have been determined in independent particle approximation. Besides similar absorption characteristics between the most common crystalline phases and amorphous TiO${}_2$, we find distinct differences in the optical spectra in the energy region between 5 eV and 8 eV. These differences can be assigned to the loss of symmetry in the local atomic structure of the disordered material. While the composition of the crystalline phases rutile, anatase, and brookite is well described by periodic arrangements of distorted TiO${}_6$ octahedra building blocks, the amorphous phase is characterized by partial loss of this octahedral coordination and the disorder-induced formation of under- and over-coordinated Ti ions. This leads to the absence of the characteristic crystal-field splitting of unoccupied Ti${}_{3d}$ states into $e_g$ and $t_{2g}$ like subbands. The optical characteristics of the amorphous phase are interpreted as a superposition of optical transitions that reflect the various local symmetries of the manifold of synthesizable crystalline TiO${}_2$ phases. The linear optical properties, calculated within the independent-particle approximation, are found to be in good agreement with the available experimental data.

Figure 1

Polyhedra representation (2.5 Å Ti-O bond-length cutoff) of the TiO${}_x$ building blocks for the 216-atom rectangular cuboidal unit cells ($a=13.80$ Å, $a=11.93$ Å) of amorphous TiO${}_2$ with a mass density of 4.20 g/cm${}^3$. The structures on the left-hand side are room-temperature (300 K) structures generated by (a) CPMD and (c) DFTB based MD. On the right-hand side the geometry after structural relaxation with VASP is shown [(b) CPMD+VASP, (d) DFTB+VASP].

Figure 2

(a) Reduced static structure factors and (b) reduced atomic RDFs of DFTB, CPMD, and VASP optimized 4.20 g/m${}^3$, 216-atom unit cells. Also displayed are the experimental results (extracted from Ref. 15) of diffraction experiments on sol-gel prepared and sputtered TiO${}_2$ layers as well as TiO${}_2$ powders.

Figure 3

Total density of occupied and unoccupied states calculated from the DFT-PBE eigenvalues and local $l$-decomposed components for $3d$ states of Ti and $2p$ states of O.

Figure 4

Kohn-Sham eigenvalues of the VASP relaxed CPMD structure calculated at the $\Gamma$ point using the PBE and the HSE06 functional. Also shown are the QP energies obtained by G${}_0$W${}_0$ QP corrections on top of the PBE calculation. For details on the numerical convergence of the G${}_0$W${}_0$ results see discussion in the text.

Figure 5

Information entropy based measures of state localization $W^{-1}\left(\nu \right)$ and $R^{-1}\left(\nu \right)$ for the CPMD MD and DFTB MD generated as well as VASP relaxed models of 4.20 g/cm${}^3$ a-TiO${}_2$ (cf. Fig. 1). Based on the definitions in Sec. 2d, a higher value corresponds to stronger localization for both quantities. For the two VASP optimized structures, the IPRs based on the HSE06 electron densities have been included. The DFT-PBE electron densities for marked electronic states are visualized in Fig. 6. Also indicated are the estimated positions of the mobility edges that separate the strongly localized states from the delocalized Bloch-type states. The estimated position of the valence states mobility edge has been chosen as the zero point of the energy scale ($x$ axis).

Figure 6

Isosurfaces of various electronic states in a-TiO${}_2$. The states are labeled corresponding to the energy levels in Fig. 5. The states represent different degrees of localization for occupied and unoccupied states. The subfigures d and l are examples for completely delocalized Bloch-type states.

Figure 7

Local densities of states for the various coordination numbers occurring in the CPMD+VASP model. The total site-projected densities of states have been averaged over all O ions (a) and Ti ions (b) showing the particular coordination. The lower graphs show the differences between the normal coordinated ions (threefold in the case of O ions and sixfold for Ti ions) and the over- and under-coordinated ions.

Figure 8

Real (dashed line) and imaginary (solid lines) parts of the isotropic complex dielectric functions $\varepsilon \left(\omega \right)$ of the four a-TiO${}_2$ models (cf. Fig. 1).

Figure 9

(a) Real part ${\varepsilon }_1\left(\omega \right)$ and imaginary part ${\varepsilon }_2\left(\omega \right)$ of the complex dielectric function $\varepsilon \left(\omega \right)={\varepsilon }_1\left(\omega \right)+i{\varepsilon }_2\left(\omega \right)$, (b) real part $n\left(\omega \right)$ and imaginary part $\kappa \left(\omega \right)$ of the complex refractive index $\tilde{n}=n+i\kappa$, (c) absorption coefficient $\alpha \left(\omega \right)$, and (d) reflectivity $R\left(\omega \right)$ (left axis) and energy loss spectrum $R\left(\omega \right)$ (right axis). The optical spectra are calculated from the DFT-PBE eigenvalues (CPMD+VASP or DFTB structure, 1920 electronic bands, 3$\times$3$\times$3 $k$ points) applying a scissors shift of 1.3 eV to the unoccupied states. For comparison, experimental data for the dielectric function from Ref. 133 and refractive data from Ref. 30 have been included.

Figure 10

Real (dashed line) and imaginary (solid lines) parts of the isotropic complex dielectric function $\varepsilon \left(\omega \right)$ calculated for the CPMD+VASP structure model on different levels of theory.

Figure 11

Comparison of the real part ${\varepsilon }_1\left(\omega \right)$ and imaginary part ${\varepsilon }_2\left(\omega \right)$ of the isotropic dielectric function of the crystalline TiO${}_2$ phases rutile, anatase, and brookite and the disordered phase (CPMD+VASP structure) of TiO${}_2$. Also plotted is the equally weighted average of the three crystalline phases.

Figure 12

Real and imaginary part of the isotropic dielectric functions of (in ascending order) the crystalline TiO${}_2$ phases rutile (ru-TiO${}_2$), anatase (an-TiO${}_2$), brookite (br-TiO${}_2$), TiO${}_2$(B), TiO${}_2$(R), TiO${}_2$(H), TiO${}_2$(II), TiO${}_2$(MI), c-TiO${}_2$, TiO${}_2$(OI), and TiO${}_2$(OII).

Figure 13

Real and imaginary part of the isotropic dielectric function of a-TiO${}_2$ and the equally weighted averages of the eleven crystalline phases listed in Fig. 12.

Figure 14

Real (dashed line) and imaginary (solid lines) parts of the isotropic dielectric function $\varepsilon \left(\omega \right)$ for a-TiO${}_2$ structure models with mass densities of 3.8, 4.0, 4.2, and 4.5 g/cm${}^3$ obtained using 1632 electronic bands, the $\Gamma$ points only, and a 1.3 eV scissors shift in all calculations. The inset on the upper right gives a detailed view on the imaginary part of the dielectric function close to the absorption edge.