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A two-dimensional photonic crystal with a large full band gap has been designed, fabricated, and characterized. The photonic crystal design was based on a calculation using inverse iteration with multigrid acceleration. The fabrication of the photonic crystal on silicon was realized by the processes of electron-beam lithography and inductively coupled plasma reactive ion etching. It was found that the hexagonal array of circular columns and rods has an optimal full photonic band gap. In addition, we show that a larger extraction of light from our designed photonic crystal can be obtained when compared with the frequently used photonic crystals reported previously. Our designed PC structure therefore should be very useful for creating highly efficient optoelectronic devices.
©2005 Optical Society of America
1. Introduction
Photonic crystal (PC) is a lattice of dielectric media with a periodically modulated refractive indices [1]. PC with a complete full band gap can be used to strongly localize electromagnetic waves to specific areas, to inhibit or enhance spontaneous emission, and to guide propagation of electromagnetic waves along certain directions at restricted frequencies. It also allows a detailed manipulation of photonic defect states [2,3]. For two-dimensional PCs, if we decompose an electromagnetic wave into the E and H polarizations, PCs usually only forbid the omni-directional propagation in one of polarizations in a frequency region. A complete two-dimensional full photonic band gap (PBG) exists if the band gaps for both E and H polarizations are present and overlap with each other, which can be obtained only in specially designed periodic lattices. Since there exists many prominent phenomena emerging from PCs, much of research efforts has been devoted to the study of the coupling between PCs and optoelectronic devices. For example, because most of the light emitted from a semiconductor light emitted diode (LED) is lost to guided modes within the high dielectric material, it results in a low extraction efficiency of only about 5% for the emitted light out of the top surface. Therefore, the study of the light extraction using two-dimensional PCs in LED devices has attracted a great deal of attention [5–7].
In the present paper, we demonstrate an optimal design for two-dimensional PCs with a hexagonal array of circular columns connected to its nearest neighbors by slender rectangular rods [3]. The reason for choosing this geometry is that the existence of TM (E polarization) band gap favors in a lattice of isolated high-ε regions, and that for TE (H polarization) band gap favors in a connected lattice [3,4]. Our designed PC has a large full band gap and light extraction when compared with that of the PCs with triangular and graphite lattices. The fabrication of PCs on silicon was realized by the processes of electron-beam lithography and inductively coupled plasma reactive ion etching (ICP-RIE). The reflectance and transmittance spectra were performed to confirm our theoretical prediction. Therefore, it is expected that our designed PC should be very useful for creating highly efficient optoelectronic devices.
2. Calculation and fabrication
For the design of two-dimensional PCs, two fast and accurate methods of inverse iteration with multigrid acceleration have been developed to compute band structures of PCs of general shapes [3]. We found theoretically that two-dimensional PCs of silicon air as shown in Fig. 1 can have an optimal full band gap of gap-midgap ratio Δω/ωmid = 0.242, which is larger than ever reported in the literature. The crystals consist of a hexagonal array of circular columns, each connected to its nearest neighbors by slender rectangular rods. The reason for choosing this geometry is that the existence of band gaps for E polarization favors in a lattice of isolated high-e region, and that for H polarization favors in a connected lattice [4]. Based on our calculation, the largest gap-midgap ratio for silicon air (ε/ε0 = 13) corresponds to r/a = 0.155 and d/a = 0.035, where r is the radius of columns, d is the width of rods, and a is the lattice constant of the hexagonal lattice [3]. Figure 2 shows a map of band gaps for the hexagonal lattice defined in Fig. 1 by varying radius r/a of circular columns. It is worth noting that the calculation shown here was based on ideal 2D PCs with infinite height rods, which is quite different from the finite aspect ratio used in experiments. However, it has been shown that the aspect ratio of PBG maximum is at 2.3 [8]. We do not need to demonstrate a very high aspect ratio to obtain the guided modes in 2D PCs. Thus, the result of Fig. 2 can be used as long as the ratio is around 2.3.
Fig. 2. A map of band gaps for hexagonal lattice in Fig. 1 by varying radius r/a of circle columns with fixed d/a = 0.035 [3].
To fabricate our designed structure, in the first step, a periodical hexagonal array of circular columns connected to its nearest neighbors by slender rectangular rods of openings was prepared in the ZEP-520 (the positive electron-beam resists produced by Japan ZEON Co. Ltd.) layer on a silicon substrate using electron-beam lithography (ELS-7500EX). In the second step, silicon dioxide molecules were deposited on the ZEP-520 by electron-beam evaporation. The deposited silicon dioxide reached the bottom of the openings to form circular columns and slender walls with the sizes determined by the sizes of openings. In the third step, the ZEP-520 layer was removed by rinsing in ZMDAC (the specific remover of ZEP-520) liquid and the hexagonal lattice of silicon dioxide on the silicon substrate was prepared. In the fourth step, the sample was etched in SF6, O2, and Ar gases using ICP-RIE. Finally, the hexagonal array of silicon circular columns connected to its nearest neighbors by slender rectangular rods was realized. A typical PC sample is shown in Figs. 3(a) and 3(b) for the side view and top view, respectively. The area of the hexagonal array of PCs is about 100μm × 100μm, and its height is about 1μm. The aspect ratio of our designs is about 2, which is approximately equal to the value for the aspect ratio of PBG maximum.
Fig. 3. Sample of hexagonal array of circular columns and rods with lattice constant a = 2.5μm, radius r = 0.328μm, width d = 0.114μm, and height~1μm: (a) side view at a tilt angle of 45°, (b) top view.
3. Experimental measurements
The PCs with various radii of circular columns and width of rods as our proposed geometry have been fabricated as shown in Table 1. In order to characterize their optical properties, the reflectance and transmittance spectra were measured by the Nicolet Continuμm Infrared
4. Results and discussion
Typical reflectance and transmittance spectra are shown in Figs. 4 (a) and 4(b), respectively. The Fourier transform-infrared (FT-IR) spectra cover a range between 4000 cm-1 (2.5μm) and 750 cm-1 (~13.3μm). In Fig. 4, the peak at around 2360 cm-1 is due to the absorption of CO2 molecules, and the other peaks at 1500 cm-1 and 4000 cm-1 are due to H2O molecules. The largest broad band structure at around 2500 cm-1 is a result of the PBG. Our measurement system can not make a distinction between the TM and TE. The measured PBG shown in Fig. 4 corresponds to the calculated full photonic band gap in Fig. 2. There are two possible ways to explain the improvement of light reflectance due to PCs [9]. First, because multiple scattering of photons by lattices of periodically varying refractive indices in the PCs acts to form PBGs in which lateral propagation of the Bloch guided modes is prohibited, light can radiate outward only. Second, the refractive index periodically creates a cut-off frequency for guided modes. Guided modes are folded by the PCs at the Brillouin zone boundaries, allowing phase match to the radiation modes that lie above this cut-off frequency. The guided modes become leaky resonances of the PCs and scatter the light in the active region [5,6,10].
Fig. 4. (a) Reflectance spectra of hexagonal array of photonic crystals with structure shown in Fig. 3. (b) Transmittance spectra of hexagonal array of photonic crystals with structure shown in Fig. 3. The arrows indicate the position of photonic band gap (PBG).
According to the previous reports [5–7,9,10], we can define the full width at half maximum (FWHM) as the value of PBG. The obtained PBGs and gap-midgap ratios are shown in Table 1 for our design PCs. We can see that when the radius of columns and the width of rods decrease, the maximal peak is blue-shifted. This tendency is in good agreement with the result one should expect for the property of PCs. The measured PBGs display a slight high-energy shift compared to the calculated PBG. This discrepancy may be due to the finite size of the hexagonal array in the real samples, while the calculated PCs are infinite hexagonal array [11]. The measured gap-midgap ratio Δω/ωmid can be as large as 0.414, which is the largest value ever reported in the literature [3]. The measured gap-midgap ratios of Δω/ωmid show about 1.5 times larger than that obtained by the calculation for the optimal design. The main reason attributed to the fact that the FWHM may broaden due to the collected infrared light is scattered by the roughness of the PCs. Therefore, the distinction between the photonic allowed and stop bands in the measurement is not as clear as in the calculation.
In comparison, we have also fabricated and characterized the triangular lattice of silicon pillars with the lattice constant of 2μm and the radius of 0.5μm (r/a = 0.25) as shown in Fig. 5(a). The triangular lattice is a typical structure used in most of the published papers to enhance the light extraction from LEDs [5–7]. The gap map of theoretical calculation of triangular structure had been reported in Meade et al [4]. The map of the calculated band gaps is shown in Fig. 5(b) [4]. The corresponding reflectance and transmittance spectra are shown in Fig. 6. The peak of the reflectance spectrum at around 3000 cm-1 is defined as the PBG compared with our theoretical calculation. The measured PBG corresponds to the calculated TM-PBG in Fig. 5(b). The FTIR spectra show a poor confinement of light, in which the reflectance at the midgap wavelength is five times less than that of our proposed hexagonal structure as shown in Fig. 4. In addition, the magnitude of the gap-midgap ratio Δω/ωmid is reduced by a factor of 2.
Fig. 5. (a) Top view of a triangular lattice with lattice constant a = 2μm, radius r = 0.5μm, and height~1μm. The ratio r/a = 0.25. (b) A map of band gaps for triangular lattice by varying radius r/a [4].
Fig. 6. Reflectance and transmittance spectra of a triangular lattice of silicon pillars with lattice constant a = 2μm and radius r = 0.5μm The arrows indicate the position of photonic band gap (PBG).
Another interesting two dimensional PC is the graphite structure proposed by D. Cassagne [12–14]. It has been shown to have a wider gap and can be easily fabricated, when compared with the triangular lattice [12–14]. We have fabricated the graphite structure of silicon pillars with the lattice constant of 2.5μm and the radius of 0.4μm as shown in Fig. 7(a). In the same way, the result of calculation is shown in Fig. 7(b) [4]. The corresponding FTIR spectra are shown in Fig. 8. The highest peak of the reflectance spectrum at about 2000 cm-1 is defined as the PBG for the graphite structure. The measured PBG corresponds to the calculated TM-PBG in Fig. 7(b). As compared with Fig. 4, the light confinement at midgap wavelength and gap-midgap ratio are not as good as our designed hexagonal structure. For an easy comparison, all the data shown above have been collected in Table 1. In the sixth column, we also have listed the dimensionless values of aΔω/2πc. The FWHM value Δω is normalized relative to the lattice constant a, which can serve as a good indication for the magnitude of the band width of PCs.
Fig. 7. (a)Top view of a graphite lattice with lattice constant a = 2.5μm, radius r = 0.4μm, and height~1μm. The ratio r/a = 0.16. (b) A map of band gaps for graphite lattice, also called honeycomb lattice, by varying radius r/a [4].
Fig. 8. Reflectance and transmittance spectra of a graphite lattice of silicon pillars with lattice constant a = 2.5μm and radius r = 0.4μm. The arrows indicate the position of photonic band gap (PBG).
Table 1. Radius (defined as r), and width (defined as d), and their gap-midgap ratios for the hexagonal array of silicon PCs. The lattice constant (defined as a) of the silicon PCs is 2.5 μm. Δω is the FWHM of PBG. aΔω/2πc is a dimensionless value.
5. Conclusion
In conclusion, from our theoretical and experimental results, a two dimensional photonic crystal with a large full band gap can be realized by the hexagonal array of circular columns and rods. Its reflectance at midgap wavelength is larger than that of the triangular lattice by a factor of five and the gap-midgap ratio Δω/ωmid also increases by two times. These results indicate that our designed PC has a much better light confinement than all the reported two dimensional PC structures. Due to the intensive research on light extraction and wave guide using PCs recently, the geometry of the connected hexagonal array of PCs as our designed structure should be very useful for improving the efficiency of optoelectronic devices, such as LEDs and lasers.
Acknowledgments
We like to thank P.T. Chou and Ming-Wei Tsai for the process of ICP-RIE. This work was supported by the Education of Ministry and National Science Council of the Republic of China.
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