1. Introduction

Since the proposal of photonic crystal concept by Yablonovich [1] and John [2] in their early works of 1987, PhC has attracted considerable amount of research interests and is becoming one of the most active research fields nowadays. The periodic variation of dielectric media in PhC introduces gaps in the energy band of photons by Bragg diffraction, preventing light of certain energies from propagating in certain directions. Thus, PhC can be utilized to control and manipulate the light flow in a highly controllable way [3]. Till now, numerous novel PhC devices have been proposed and some of them have already been practically fabricated and tested [4–7]. Most of them, however, are based on semiconductor materials such as silicon, GaAs/AlGaAs or InP to take advantages of the existing mature nano-fabrication techniques developed in the LSI industry as well as the large refractive index of these materials.

On the other hand, titanium dioxide (TiO₂) is attracting wide attentions due to its novel applications in photocatalysts, sensors and solar cells. Recently, we started to fabricate micro-nano structures on TiO₂ [8,9] to make use of its several unique characteristics. First of all, TiO₂ has a fairly large refractive index among the transparent metal oxides (>2.5 at anatase phase, and >2.7 at rutile phase), which is large enough to form a photonic band gap in PhC structures. Secondly, the optical absorption loss of TiO₂ is about 10 times lower than that of silicon at the optical communication wavelength of 1.5µm [10]. Thirdly, its thermal expansion coefficient is very small (7~9×10^-6 K^-1 over a wide range from room temperature to 1000°C). This is invaluable for PhC applications because the properties of PhC are essentially sensitive to the geometric size variation which may originate from temperature fluctuations in the environment. Last and most importantly, TiO₂ is transparent over a wide range of wavelength and is especially attractive in its potential of constructing PhC devices for the visible light. Considering all these advantages, TiO₂ should be considered as an important candidate for PhC engineering. If further combining with TiO₂’s other special abilities such as photocatalyst or sensor etc, we believe TiO₂ PhC will find lots of novel applications in fields besides optical integration.

However, till recently there were no well-established techniques for TiO₂ nano-scale fabrication. Though there have been quite a lot of work on the artificial titania opal PhCs formed by self-assembly pattern [11,12], they suffer from the irregularity and it is hard to adjust the local parameters. Later, to fabricate the TiO₂ structure in a more controllable way, several new trials were made [13,14]. But the titania body formed in [13] by sol-gel filling technique is somewhat porous with voids in it and of rather poor homogeneity. On the other hand, the PhC fabricated by [14] is of relatively large period (tens of micrometers), and need to sinter at high temperature (e.g., over 1000°C) to obtain rigid structure. Therefore, recently we developed a liquid-phase deposition method for the template fabricated by deep x-ray lithography [9]. For the first time, we could create tight TiO₂ structure at rather low temperature (60°C) with a sub-micrometer resolution. We have used this method to fabricate both the two-dimensional and three-dimensional TiO₂ PhC structures. As an example, Fig. 1 shows the field-emission scanning electron microscope (FE-SEM) photograph of a reversed 3D Yablonovite structure prepared by our method. The refractive index of the deposited TiO₂ was measured to be close to that of anatase TiO₂, thus proving that the amorphous TiO₂ is tightly packed into the tiny complex 3D structures. In addition, the low temperature (60°C) operation of this liquid-phase deposition method greatly alleviates the requirements on the template materials and avoids the deforming problems of other metal oxide molding methods such as chemical vapor deposition (CVD) or spray-pyrolysis processes. Furthermore, thanks to the deep x-ray lithography, it is feasible to fabricate large-area devices at high speed compared to usual electron beam lithography.

Therefore, it is becoming possible to create a new category of PhCs which are based on TiO₂. To understand their performance and potential as optical band gap engineering materials, it is necessary to clarify to what extent a material with a relatively low refractive index as TiO₂ is capable of forming the optical band gap in PhC. In this paper we compute systematically and summarize the characteristics of 2D and 3D PhC when using TiO₂ as the construction material. All the results are plotted as normalization to the unit cell size for easy scaling to any PhC size. Therefore, they can serve as a guideline for TiO₂ PhC design. The optimization and machining requirement for these structures are also discussed.

 

Fig. 1. The reversed Yablonovite structure of titania fabricated by the liquid phase deposition method using a PMMA template (honeycomb lattice). Diameter of individual rod is 400nm.

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2. The optical band structures of two-dimensional TiO₂ PhC

In this paper, the two-dimensional structure means that there is periodicity only in one plane. To calculate the optical band structures of a PhC, the planewave expansion method is utilized [15,16]. By invoking the periodic condition in PhC, for a specific frequency ω and in a unit cell, we can expand the magnetic field $\overrightarrow{H}$ ($\overrightarrow{r}$) into the planewave form of $\overrightarrow{H}$($\overrightarrow{r}$)=∑_G⃗$\overrightarrow{h}$_{$\overrightarrow{H}$,$\overrightarrow{H}$} exp[i($\overrightarrow{k}$+$\overrightarrow{G}$)·$\overrightarrow{r}$], where $\overrightarrow{k}$ is the Bloch wavevector, $\overrightarrow{G}$ is reciprocal lattice vector in order for $\overrightarrow{H}$($\overrightarrow{r}$) to satisfy the Bloch theorem $\overrightarrow{H}$($\overrightarrow{r}$+$\overrightarrow{R}$)=$\overrightarrow{H}$($\overrightarrow{r}$)e^{i$\overrightarrow{k}$ ·$\overrightarrow{R}$} for any lattice translation vector $\overrightarrow{R}$ , and the unknown vector amplitude $\overrightarrow{h}$ _{$\overrightarrow{k}$ ,$\overrightarrow{G}$} of the planewave exp[i($\overrightarrow{k}$ +$\overrightarrow{G}$ )·$\overrightarrow{r}$ ] is chosen to be perpendicular to $\overrightarrow{k}$ +$\overrightarrow{G}$ to satisfy the transverse condition ∇·$\overrightarrow{H}$ ($\overrightarrow{r}$ )=0. $\overrightarrow{h}$ _{$\overrightarrow{k}$ ,$\overrightarrow{G}$} is further expressed in its two polarization components which are also perpendicular to the propagation direction. Substituting the above expansion as well as the expansion of the dielectric function ${\epsilon }^{-1}(\overrightarrow{r})={\displaystyle \sum _{\overrightarrow{G}}{\kappa }_{\overrightarrow{G}}\exp (i\overrightarrow{G}\cdot \overrightarrow{r})}$ into the source-free Maxwell’s equation for $\overrightarrow{H}\left(\nabla \times \left(\frac{1}{\epsilon }\nabla \times \overrightarrow{H}\right)={\left(\frac{\omega }{c}\right)}^2\overrightarrow{H}\right)$, we can obtain a system of linear matrix equations on the expansion coefficients. Solving this eigenvalue problem numerically, we finally get the sequence of discrete eigenfrequencies ω_n for each Bloch wavevector $\overrightarrow{k}$ and ω_n($\overrightarrow{k}$) is the “band structure” of PhC. The real implementation of the numerical code, however, needs to address much more issues such as accuracy and convergence speed, and they are well summarized in [16].

In all of our following computations, we set the refractive index of TiO₂ as ε√=2.52, as measured in the experiments [9]. We assume the TiO₂ material is isotropic since the liquid-phase deposition method is used to fabricate the structure.

2.1 Two-dimensional TiO₂ PhC with infinite height

We first consider the case of an ideal two-dimensional PhC, that is, the length of the non-periodic direction is infinite. For in-plane propagation, the field modes can be divided into two groups: TM mode or TE mode, according to whether their electric field or magnetic field is parallel to the non-periodic dimension. Figure 2 shows an example of the computed band structures for a triangular-lattice PhC where air holes are perforated on TiO₂. The horizontal axis is the wavevectors along the irreducible Brillouin zone edges and the vertical axis is the normalized frequency defined as $\omega \frac{a}{2\pi c}$ or a/λ (c is the velocity of light, a is the distance between the neighboring holes or referred as lattice constant, and λ is the optical wavelength). In Fig. 2, the TE mode shows a large band gap within which no light can be transmitted in this polarization, while TM mode does not have such gap. The band gap width depends on the filling ratio of the dielectric material TiO₂. To find the optimal parameter, it will be convenient to plot the optical band gap width against the radius of the air holes (Fig. 3).

 

Fig. 2. An example of the band structure for a 2D PhC of triangular lattice air holes in TiO2 background. The hole radius is r=0.4a.

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Figure 3(a) shows the optical band gap map for the air hole-type PhC. The horizontal axis is the hole radius normalized by lattice constant (r/a), indicating the filling ratio of air. The shaded areas are the optical band gaps: red for the TE modes and blue for the TM modes. To evaluate the gap width more definitely, Fig. 3(b) also plots the ratio of gap width to midgap frequency (Δω/ω) versus hole radius for the lowest TE mode gap. At r/a=0.41, we can obtain a band gap width as wide as 33%. Although it is narrower than that obtainable in semiconductor due to the relatively lower ε of TiO₂ comparing with ε≈12 of semiconductors, it is wide enough for most PhC applications. Other higher order band gaps are much thinner here and not suitable for any practical usage. In addition, the complete band gap (i.e., overlapped band gap for both TE and TM modes) in semiconductor PhC (see appendix of [3]) does not show up here. On the other hand, Figs. 3(c) and 3(d) show the case for TiO₂ rod-type PhC. Here we have several wide TM mode band gaps stacked up at higher frequencies and the lowest gap width is as wide as that of air hole-type PhC. For both kinds of PhCs, the widest low gaps occur at relatively low filling ratio of TiO₂, i.e., r/a are 0.41 and 0.21, respectively. When applying to optical communication wavelength λ=1.55µm, i.e., centering the midgap frequency of lowest gap to this wavelength, the thinnest TiO₂ wall thickness and air hole diameter for air hole-type PhC are estimated to be 122nm and 559nm, respectively. While for TiO₂ rod-type PhC, the TiO₂ rod diameter and rod spacing distance are 273nm and 378nm, respectively. Both should be of no problem in practical fabrication. Even for the visible light wavelength (for example, argon laser wavelength λ=488nm), the TiO₂ rod-type PhC should still be feasible regarding the fabrication sizes. Thanks to its relatively lower refractive index, TiO₂ PhC gains some relaxation on the machining size requirement compared to the PhC of very high refractive index materials.

 

Fig. 3. The optical band gap maps for two-dimensional TiO₂ PhC of triangular lattice. (a) and (b) are for air holes in TiO₂ background. (c) and (d) are for TiO₂ rods in air.

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Furthermore, Fig. 4 summarizes the band gap maps for two other types of lattice, the square lattice and honeycomb lattice. The air hole-type PhC of square lattice (Fig. 4(a)) shows sparse and small band gaps. The TE mode gap almost disappears here. As the air holes become large enough to approach each other, there appear the TM mode gaps because in this case the PhC is in fact converting to the TiO₂ rod-type PhC similar to Fig. 4(b), that is, small TiO₂ area surrounded by large area of air. The TiO₂ rod-type PhC shows some similar characteristics as the triangular lattice Fig. 3(b), but the gap widths are narrower.

Figures 4(c) and 4(d) give the band gap maps for honeycomb lattice. These PhCs in fact can be roughly approximated as the mixture of two reversed types of PhCs of triangular lattice. For example, in Fig. 4(c) PhC of small air holes is overlapped by a PhC of larger TiO₂ “rods” in a larger triangular lattice, while in Fig. 4(d) the situation is reversed. Therefore, we see that the TE mode gaps are intertwined with the TM mode gaps in this type of lattice. In some areas, we even obtain narrow complete band gaps for both polarizations. In Fig. 4(c), the approximated TiO₂ “rods” are of larger lattice constant, thus its characteristic TM mode gaps occur at lower frequency than the TE mode gaps, and they are concentrated at larger r/a. When r/a approaches 0.5, i.e., the air holes begin to connect to each other, the PhC in fact becomes a triangular lattice PhC of TiO₂ rods and TM mode gaps dominate. Similar explanation applies to Fig. 4(d).

By comparing Figs. 3 and 4, we may say that triangular lattice is the most reasonable choice for TiO₂ based PhC because of its wider band gaps and moderate fabrication difficulty. For other types of lattices, for example, the air hole structures (Figs. 4(a) and 4(c)) achieve their wide band gaps near r/a=0.5 which means a thin wall between holes and is difficult to build. The TiO₂ rod structures in square and honeycomb lattices (Figs. 4(b) and 4(d)) show inferior gap sizes than triangular lattice case and thus should be of interest only in some special usages.

 

Fig. 4. The optical band gap maps for two-dimensional TiO₂ PhC with square lattice ((a) and (b)) and honeycomb lattice ((c) and (d)). The TiO₂ dielectric material is indicated in black in the insets.

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2.2 Two-dimensional TiO₂ PhC slab

When building optical devices with PhC, the slab structure of finite thickness is a more practical solution not only because it is far more easy to fabricate, but also because it can add confinement in the third dimension of 2D PhC by the total internal reflection principle [17, 18], thus avoiding the optical loss due to scattering and leakage in the non-periodic direction. Based on the conclusions of last section, we choose triangular lattice to construct the TiO₂ PhC slab, and employ the supercell technique in combination with the planewave expansion method to analyze the band structures of PhC slab.

First, we consider the air-bridge type slab structure. Figure 5(a) shows the computation model and Fig. 5(b) is the computed band curves for even (TE-like) mode when r/a=0.4. In Fig. 5(b), the horizontal axis indicates the wavevector in the slab plane, and the orange thick line indicates the light line of air. Those modes below it are the guided modes in the TiO2 slab. Obviously, there exists a band gap between the guided modes and we still have the PhC effect in slab plane propagation. This band gap width depends not only on the filling ratio of air (i.e., r/a) as in the last section, but also depends on the slab thickness now. Figure 6 shows the slab thickness dependence of the gap size for r/a=0.4. At the thickness of h/a=0.8 (aspect ratio of 1:2), we have a maximal ratio of gap to midgap frequency near 30%. When the slab is thinner, the modes are less trapped into the slab and all band curves are close to the light line, thus a narrower band gap. On the other hand, if the slab is too thick, there will be more guided modes cramming into the slab. The result is that the gap between modes also becomes narrower. For the odd (TM-like) modes, there is no band gap existing.

 

Fig. 5. The computational model of the air-bridge type PhC slab (Fig. (a)), and the computed TE-like band curves for r/a=0.4, h/a=0.6 slab (Fig. (b)).

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Fig. 6. The dependence of even-band gap width on the slab thickness for even mode at r/a=0.4.

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Fig. 7. The calculated TE transmission spectrum of the PhC slab for r/a=0.4 and h/a=0.6, by 3D FDTD.

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It is worth noting that those frequencies which lie in the gap and have no guided modes in the PhC slab may be coupled into air at the neighborhood of light line. Thus, the confinement of slab structure is not perfect in the direction perpendicular to the slab plane. When designing waveguide or cavities with the PhC slab, the vertical leakage problems need to be carefully handled to ensure good confinement.

To verify the above results of band computation, a virtual experiment by the 3D finite differential time domain (FDTD) simulation is carried out to investigate the light propagation property in the PhC slab of Fig. 5. Figure 7 shows the transmission spectrum for a 10-period long PhC slab along the Γ→K direction with the TE polarization. A very deep band gap (more than 30dB) was observed and its bandwidth and position agree very well with that predicated in Fig. 5(b) by planewave expansion method.

From the point of view of device engineering, a PhC slab built on a substrate is preferable because of the easier fabrication and more robust structure than air-bridge type structure. Also the potential heat amass problem in air-bridge type PhC can be avoided since the substrate plays the role of heat sink. Here we consider the case of TiO₂ PhC slab on a SiO₂ substrate. If the space above the slab is air, the asymmetry above and below the slab will break the condition for pure even or odd modes and thus no strict band gap exists [17,18]. Therefore, we propose a “sandwich-structure” and cover the PhC slab with either SiO₂ or a transparent PMMA material of similar refractive index value as SiO₂ (as shown in Fig. 8(a)) to maintain the symmetry. This structure can also simplify the fabrication process because there is no need to remove the template material in PhC holes. However, now the much lower light line of SiO₂ has to be taken as the boundary of guided modes instead of the air light line of Fig. 5(b). Figure 8(b) shows the band gap width versus slab thickness for different hole radii. We see that the band gap width is much narrower than that of air-bridge structure because of the dimmer contrast in dielectric media. Nevertheless, at r/a=0.375 and h/a=1.125 the gap can still be as wide as 12% and is enough for a PBG application. The optimal slab thickness is larger than that of air-bridge type slab because we need a thicker TiO₂ layer to attract guiding modes against the heavier background. Similarly, the hole radius needs to be reduced somewhat to increase the average refractive index of the slab. This improves the weak confinement of light in the slab. The optimal radius is a result of tradeoff between the horizontal band gap width and the vertical confinement effect.

 

Fig. 8. (a) The proposed “sandwich-stucture” of TiO₂ PhC slab on SiO₂ substrate. (b) Its evenmode bandgap width against the slab thickness for different air-hole radii.

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3. The optical band structures for three-dimensional PhC (Yablonovite and its reverse)

As mentioned in the last section, the PhC slab cannot provide a true complete band gap in all spatial directions. To obtain such a strict band gap, one needs to apply the three-dimensional periodic structures. Yablonovite structure, which was first proposed by Yablonovitch in 1991, is one of the earliest 3D structures which demonstrates complete band gap [19]. Since it is suitable to be fabricated by our TiO₂ deposition method, we will analyze its band characteristics when using TiO₂ as the building material.

Figure 9 shows the computer graphics of Yablonovite structure and the unit cell which is used to compute its band structure. The corresponding first Brillouin zone is illustrated in Fig. 9(c), where the irreducible Brillouin zone is shown in shade. The reason why we choose such a rather big unit cell is to conveniently investigate the effect of drilling angle (θ) of Yablonovite, as will be discussed later. θ is defined as the angle between the drilling direction and the mask-plane normal. When the angle θ varies, the Brillouin zone vertexes in the reciprocal space do not change, thus eliminating the need to calculate new sets of wavevector coordinates for each angle. Compared with other smaller unit cells, the band curves are folded back in the Brillouin zone. But it does not affect the band gap width of interest.

 

Fig. 9. (a) The Yablonovite structure, (b) the unit cell for computation, and (c) the corresponding first Brillouin Zone.

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Figure 10(a) shows the calculated band curves along the edges of the irreducible Brilloun zone when the air hole radius is 0.325a (here a is the distance between neighboring holes on the mask plane) and the drilling angle is set as the standard diamond angle of 35.26°. For simplicity, the shape of air holes is taken as cylinders. Obviously there exists a distinct band gap in all spatial directions, thus a complete band gap. If we look at the band curves closely, we see that the upper boundary of the gap is mainly limited by A and L wavevector points. This reflects the fact that the Yablonovite is essentially an anisotropic structure, i.e., the optical property along Γ→A direction is quite different from that in the mask plane directions, and Γ→L direction is the direction where dielectric media is most concentrated. Thus there are large variations of band curve at these points.

 

Fig. 10. (a) The band structures of TiO₂ Yablonovite with drilling angle of 35.26° and air hole radius of 0.325a. (b) the band gap map for different air hole radii.

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To investigate the dependence of this gap width on the filling ratio of TiO₂, Fig. 10(b) summarizes the band gap map of the TiO₂ Yablonovite structure for different air hole radii. At r/a=0.31, the gap becomes widest and the ratio of gap width to midgap frequency is 7.2%. When applied to wavelength of λ=1.55µm, the required air hole diameter and spacing are roughly 435nm and 267nm, respectively. These are feasible sizes in practical fabrication.

In order to examine the precision requirement for the drilling angles when fabricating Yablonovite, we compute the angle dependence of the band gap width for several different hole radii (Fig. 11). It is seen that the diamond angle of 35.26° is almost the optimal angle except that there is a tiny shift for the larger hole radii. To explain the reason of angle dependence of the band gap, Fig. 12 shows the band curves for two different drilling angles of 32.5° and 38°. When the angle becomes smaller, the lattice distance in the direction perpendicular to the mask plane (Γ→A direction) is expanded, thus the frequency of bands at A-point are pulled down and results in the fast drop of band gap. On the other hand, when the drilling angle increases beyond 35.26°, the lattice along Γ→A direction is compressed, thus the upper bands around A-points are lifted and become flatter. However, the anisotropism at L-point is not removed and degrades with larger angles. Therefore, the overall band gap also shows a drop at larger angles. According to Fig. 12, the drilling angle error should be kept within 1° so as to maintain a good band gap. Notably, the angle error below 35.26° should be avoided.

 

Fig. 11. The dependence of band gap width of TiO₂ Yablonovite on drilling angle for different hole radii.

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Fig. 12. The band structures of TiO₂ Yablonovite with drilling angle of (a) 31.26°, (b) 39.26°.

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In addition, because we can now fabricate the reversed Yablonovite structure by TiO₂ [9], we also compute the band structure for this type of 3D PhC (Fig. 13(a)). Figure 13(b) shows the band curves for r/a=0.19 and Fig. 14(a) summarizes its band gap versus different rod radii. The results show clearly that there exists also a narrow complete band gap area. The maximum gap occurs at r/a=0.19 and its width is 2.43%. To our knowledge, this is the first result showing the existence of a complete band gap in the reversed Yablonovite structure. Although it is very narrow for the TiO₂ material, if a high refractive index material such as a semiconductor is used, the gap width is calculated to be as wide as 12% (assuming ε=12).

Similar to the case of normal Yablonovite structure, the band gap width depends on the drilling angle (Fig. 14(b)) and it reaches the peak at about the standard angle of 35.26°. Another characteristic is that the band gap closes very fast when the drilling angle is smaller than 35.26°, while for the angles larger than 35.26°, the degradation speed is fairly slow.

 

Fig. 13. The band structure of reversed Yablonovite in TiO₂ with drilling angle=35.26°. (a) the structure image. (b) the band structure for r/a=0.19.

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Fig. 14. (a) The band gap map for different air hole radii. (b)The dependence of band gap width of TiO₂ reversed Yablonovite on the drilling angle (r/a=0.19).

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It should be noted that all the above computations assume cylindrical shape for the holes or rods in Yablonovite. In practical fabrication, when using a mask with circular holes, the generated holes or rods are of the shape of ellipsoid, and the ellipticity varies with the drilling angle. Figure 15 shows the results for the practical Yablonovite when strict ellipsoid model is introduced. The gap width of Fig. 10(b) is plotted here in blue as a reference, and the optimal drilling angle is shown in the grey dotted line. We observe an improved gap size of 9.1% for ellipsoid case at a larger optimal hole radius of 0.355a. The optimal drilling angle is also a little bit smaller than 35.26° (the step variation in angles is due to the numerical setting).

 

Fig. 15. The band gap size of TiO₂ Yablonovite with ellipsoidal holes. The gray line shows the corresponding optimal drilling angles.

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4. Summary

We systematically compute the band structures for the new type of PhC made of TiO₂ material. The results show that two-dimensional very long PhC can have fairly large optical band gap despite the relatively lower refractive index of TiO₂. The triangular lattice is preferable both from its large band gap at low frequency and from the loose requirement for fabrication resolution. The air-bridge type PhC slab of TiO₂ can confine the light effectively and there exists an optimal slab thickness. For the slab on SiO₂ structure, the proposed sandwich structure can achieve a band gap width of 12% by reducing the air-hole radius and increasing slab thickness for better confinement. The three-dimensional Yablonovite structure is capable of constructing a complete band gap of which the width is 9.1% for TiO₂. For the first time, we show that the reversed Yablonovite structure also exhibits a complete band gap, although it is smaller than that of normal Yablonovite. The dependence of band gap size on filling ratio and drilling angle are also summarized to provide guidance for practical fabrication.

As a compensation to the smaller band gap, the size of TiO₂ PhC structure is relatively larger than that of semiconductor PhC thanks to its lower refractive index. This loosens the requirement on fabrication resolution and makes it easier to couple light to optical fiber. All these band structure characteristics, along with other unique properties of TiO₂ material such as low absorption at wide wavelength range and extremely low thermal expansion, show that TiO₂ is an important candidate for the future optical PhC devices.