In the last decades, new materials with large possibilities of technological applications have appeared as a direct consequence of developments in nanoscience. These materials exhibit novel optic, electronic or magnetic properties among others. Among these systems, the photonic materials can manipulate and control the transport of light. In fact, photonic crystals are considered equivalent to semiconductors for the light.^[1] The term photonic crystals was introduced in 1987 by Yablonovitch and John. ^[2,3] In the last years, there are many expectations on photonic crystals regarding their uses and applications in different fields of knowledge and technology, like medicine, optics, optoelectronics, and telecommunications. An important characteristic of photonic systems is the versatility to be grown in one, two or three dimensions that should permit its integration in optoelectronic devices.

Photonic crystals use the periodic variation of the dielectric function to control the light in a similar way that in crystals happen with the electrons. Generally, photonic crystals are a material or matrix with dielectric constant Є₁, where there are immersed an array of forms made of a second material with dielectric constant Є₂. This picture resembles the atomic organization of a crystal where the space is the first material and the atoms correspond to the second material. So, traditional concepts from electronic band theory are useful in this subject. However, light does not obey the Fermi-Dirac statistics and the second material, "the atoms" can have different shapes: circular, square and triangular, to mention a few examples. Another difference is that there are two different components in the dispersion relation: Transverse Electric and Transverse Magnetic modes. For this reason it is possible to have photonic band gap (PBG) for either both modes or one of them. The PBG is a function of the geometric properties of the photonic crystal and the difference between the dielectric constants. These characteristics are a powerful tool to develop photonic devices for possible technological applications.

In a recent paper was reported the formation of TiO₂ nanostructures as a secondary effect during the grow of TiO₂ nanotubes^[4] for the anodization method. ^[5] These nanostructures that we will call as traces appeared on the Ti surface and can be visible after remove the nanotubes by mechanical stress. The traces resemble an egg box and present photonic properties. TiO₂ Anatase and Rutile phases were identified in the traces. This work sought to bring a connection between the relevant parameters of the PGB with the radii of the circular forms of traces.

We used a representation of the traces as a photonic system where long cylindrical rods of air were immersed in a TiO₂ matrix. In the xy plane the rods were distributed in a hexagonal array such that a 2D system was formed. The dielectric function remained constant in the z direction. This picture was inspired by SEM micrographs from traces, as we can see in Fig 1(a). Numerical experiments were developed in order to explore the effect produced on PBG when radius of traces changed.

FIGURE 1

In particular, we studied the evolution of the position and the width of the first partial gap as a function the radii of traces considering both Anatase and Rutile phases for the M and K symmetry points. The traces packing factor dependence with the radii of TiO₂ traces was determinate. These calculations would enable the acquisition of information to prepare materials with specific characteristics, suitable for practical applications.

This theoretical study sought to identify the evolution of partial gaps of the TE and TM modes as a function of the size of the radius of TiO₂ traces. This artificial material exhibits a photonic behavior and presents a hexagonal array, which has the matrix of Anatase and Rutile phases (black) with air cylinders in the representation of TiO₂ traces (white) Fig. 2(a-d), with radii of cylinders varying from 12 nm to 42 nm. When the radius of traces increases, the packing factor ^_PF , the relation between the occupied area of air cylinders and TiO₂ matrix increases with a quadratic behavior, following the equation P _F = ^_2Πr2 /(), r ≤ 2a , which is for 2D hexagonal lattices. The packing factor is shown in Fig. 2(e). The photonic band structures obtained for Anatase and Rutile phases are shown in Figs. 4 to 7 for radii of 12 nm, 22 nm, 32 nm, and 42 nm, respectively.

Figure 2

The increase of the partial gap of TE mode (TE) is shown for both phases of TIO₂. This region defines the frequencies in which the light cannot propagate in the periodical photonic crystal of TiO₂-air implemented in this study. When the radii of traces are small (r = 12nm. Fig. 2(a)), in comparison to reference radius of 42nm (Fig. 2 (d)), the gap region is not open; the dielectric and air band has a very close frequency value, which inhibits the appearance of gaps (Fig. 3).

Figure 3

Figure 6 shows the maximum value of the photonic band gap region, corresponding to the radius taken as reference. This value is obtained from SEM micrographs of TiO₂ nanotubes. ^[6] For this case, the first band gap for TE modes has an interesting property: the lowest state of second band occurs at M point while the highest state of first band is at K point. This behavior is similar as the indirect electronic gap in semiconductors. For this reason, the effectively gap for light transportation must take in account this characteristic. However, photons obey Bose-Einstein statistics and direct-like transitions can happen. This effective gap disappears for radius lower than 22 nm even that at K or M point is possible to observe a local gap, see Figs. 3 to 6.

Figure 4

Figure 5

Figure 6

The evolution of the central frequency of the local gap, for M and K point, as function of radius in Anatase and Rutile phases are shown in Figs. 7. Local gap at K point is higher than local gap at M point for both modes and phases.

Figure 7

A study of the effect of the change in the radii of TiO2 traces on the properties of gaps for a 2D photonic material compose of Titania (Anatase and Rutile phases) - air hole was conducted. From them, it is possible to determine that the value of the photonic gap increases, according to the radii increase. This behavior is observed both K and M high symmetry points. In the same form, the photonic gap could be tuned with the packing factor which can be considered a new tool more useful in technology applications. Our results predict a threshold of 150 Thz for the maximum value of the central frequency of the first photonic gap when the packing factor reaches the maximum value of 0.906 for a 2D hexagonal lattice.