Open Access
Add to BibSonomy
Abstract
We materialized the isotropic Dirac-cone dispersion relation in the mid-infrared range by fabricating photonic crystal slabs of the C4v symmetry in SOI (silicon-on-insulator) wafers by electron beam lithography. The dispersion relation was examined by the angle-resolved reflection spectra with our home-made high-resolution apparatus, which showed a good agreement with the dispersion relation and the reflection spectra calculated by the finite element method. The reflection spectra also agreed with the selection rules derived from the spatial symmetry of the Dirac-cone modes, which proved to be a powerful tool for the mode assignment.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The dispersion curve of the electromagnetic field is generally an even function with respect to the wave number, $\textbf {k}$, for periodic systems with time-reversal symmetry. So, odd terms are absent and the dispersion relation is quadratic in $\textbf {k}$ around the origin (the $\Gamma$ point) of the first Brillouin zone. Then, in particular, the group velocity is vanishing on the $\Gamma$ point, so the eigenmode is a standing wave. This basic feature has been known for a long time and used in various applications like distributed feedback lasers.
However, there is an important exception. When two dispersion curves cross each other on the $\Gamma$ point and have the same slope with opposite signs, the two dispersion curves as a whole satisfy the requirment from the time-reversal symmetry and still they can be linear in $\textbf {k}$. So, they are not standing waves but travelling waves.
The linear dispersion relation on the $\Gamma$ point of the first Brillouin zone in periodic structures was found in microwave frequencies [1–3], which is known as the CRLH (composite right-/left- handed) transmission line. Later, the theory was extended to more general cases and higher dimensions first by the tight-binding approximation [4–6] and later by the $\textbf {k} \cdot \textbf {p}$ perturbation theory [7–9]. For highly symmetric structures, we can materialize the isotropic linear dispersion, which is called the Dirac cone (Fig. 1) after similar energy spectra of massless particles. For example, we can materialize the Dirac cone with an auxiliary flat dispersion surface in the two-dimensional square lattice of the $C_{4v}$ symmetry [5,8,9] and doubly degenerate Dirac cones (double Dirac cone) in the triangular lattice of the $C_{6v}$ symmetry [6,8,9].
Fig. 1. (a) Dirac cone with an auxiliary flat dispersion surface (dotted lines), and (b) double Dirac cone on the $\Gamma$ point ($\textbf {k} = 0$) of the two-dimensional Brillouin zone materialized by accidental degeneracy of two modes with particular combinations of their spatial symmetries.
Because the essential point of Dirac-cone formation is the accidental degeneracy of two eigenmodes with particular combinations of their spatial symmetries, we can expect the formation of the Dirac-cone dispersion in other wave systems such as phonons [7] and electrons [10] as well. Because the effective refractive index is vanishing at the Dirac point frequency, $\omega _{D}$ (see Fig. 1), we can expect curious phenomena and their applications such as tunneling through sharp bends and cloaking [11]. We can also control the propagation direction of the Dirac-cone modes by the polarization of the incident wave [12]. When the constituent eigenmodes of the Dirac cone have a finite lifetime due to diffraction loss, etc., the linear dispersion is distorted and the group velocity calculated from the slope of the dispersion curve exceeds the speed of light in the vicinity of exceptional points [13,14]. This fact may sound contradictory to special relativity. But actually it is not because the dispersion is highly frequency dependent so that the velocity of wave packets is not well defined. The amount of the distortion is governed by the quality factor of the Dirac-cone modes. The distortion is not significant when the quality factor is sufficiently large, larger than 400 for example, as far as its experimental detection is concerned, which will be discussed later in this paper.
In this study, we designed the Dirac-cone dispersion relation by tuning the structural parameters of photonic crystal slabs and materialized it by the EB (electron beam) lithography of SOI (silicon-on-insulator) wafers. We confirmed the Dirac-cone dispersion by the angle-resolved reflection spectroscopy with our home-made high-resolution apparatus.
This paper is organized as follows. In Section 2, we briefly describe the condition for the formation of Dirac cones in the square lattice of the $C_{4v}$ symmetry by the $\textbf {k} \cdot \textbf {p}$ perturbation theory. We also derive the selection rules for reflelction peaks, which will be used for the peak assignment of the angle-resolved reflection spectra. In Section 3, we show a sample design to materialize the Dirac cone by the FEM (finite element method) calculation. The sample fabrication by the EB lithography is described in Section 4. In Section 5, we explain about our home-made high-resolution apparatus to measure the angle-resolved reflection spectra. The measured spectra showed a good agreement with the calculated dispersion relation and the selection rules. A brief summary is given in Section 6.
2. Theory
We can materialize the Dirac cone with an auxiliary flat dispersion surface in the square lattice of the $C_{4v}$ symmetry by the accidental degeneracy of an $E$-symmetric mode with an $A_{1}$-, $A_{2}$-, $B_{1}$-, or $B_{2}$-symmetric mode [5,8,9]. We should note that the symmetries of the electric field and the magnetic field are generally different because the former is a genuine vector whereas the latter is an axial vector [15]. Here we denote the symmetry of the magnetic field. As we will show in Section 3, the Dirac cone was materialized by the $E$ and $B_{2}$ modes in our experimental study, so we derive the selection rule of the angle-resolved reflection spectra for this case. Other cases can be dealt with in the same manner.
In [8], we proved by the $\textbf {k} \cdot \textbf {p}$ perturbation theory that the first-order solutions of the Dirac-cone dispersion for the combination of the $E$ and $B_{2}$ modes were obtained by diagonalizing the following matrix:
To derive the selection rule, we need the spatial symmetry of these two solutions. It is easy to show that their magnetic fields are given as follows.
Fig. 2. Configuration of the incident plane wave for the angle-resolved reflection measurement. $\theta$ and $\phi$ denote the tilt angle from the normal ($z$) direction and the azimuthal angle from the $x$ axis, respectively. The inset shows the three symmetric points of the first Brillouin zone.
Table 1. Selection rules for reflection peaks.
3. Sample design
To confirm the formation of Dirac cones and their basic features, we fabricated photonic crystal slabs in SOI wafers by EB lithography. Figure 3 shows their design. The specimens consist of a regular square array of circular air cylinders fabricated in the top Si layer. The thickness of the top silicon layer and that of the SiO$_{2}$ layer were assumed to be 400 nm and 3000 nm, respectively, according to available SOI wafers. The typical depth of the air cylinders was 210 nm and their radius was changed from 440 nm to 620 nm to materialize the accidental degeneracy of the $E$ and $B_{2}$ modes. Their dispersion relation was calculated by FEM with a commercial software, COMSOL. The Bloch-type boundary condition was imposed on the unit cell of the photonic crystal slab in the lateral directions and the perfectly-matched-layer absorbing boundary condition was imposed in the vertical direction. The refractive indices of Si and SiO$_{2}$ were assumed to be 3.427 [17] and 1.440 [18].
Fig. 3. Illustration of the specimen structure. Two-dimensional photonic crystal slabs were fabricated in the top silicon layer of SOI wafers, which consisted of a regular-square array of cylindrical air holes. Thickness of the top silicon layer, 400 nm; thickness of the SiO$_2$ layer, 3000 nm; lattice constant $a$, 2270 nm; air-hole radius $R$, 440 - 620 nm; typical air-hole depth $d$, 210 nm.
Figure 4 shows their dispersion relation. Figures 4(a) and 4(b) show the non-degenerate and degenerate cases, respectively. In the frequency range of our interest, there are $A_{1}$-, $B_{2}$-, and $E$-symmetric modes on the $\Gamma$ point. By tuning the radius and the depth of the air cylinders, we obtained the structural parameters for the accidental degeneracy that materialized a Dirac cone with an auxiliary flat band at $\omega _{D} = 2610$ cm$^{-1}$, which consisted of the $E$ and $B_{2}$ modes as shown in Fig. 4(b).
Fig. 4. Dispersion relation of photonic crystal slabs fabricated in SOI wafers. The radius and depth of air cylinders were assumed to be (a) $R=440$ nm , $d=210$ nm and (b) $R=526$ nm, $d=216$ nm. The vertical axis is the frequency of the electromagnetic eigenmodes and the horizontal axis is the wave vector in the first Brillouin zone. The dispersion relation is plotted in the $\Gamma$-to-M and $\Gamma$-to-X directions. M/10, for example, implies that the horizontal axis is magnified by 10 times.
The $E$ and $B_{2}$ modes are non-degenerate in Fig. 4(a), so their dispersion curves are all horizontal and quadratic on the $\Gamma$ point. On the other hand, two lines (Dirac cone) are linear and the third one (flat band) is horizontal and quadratic in the vicinity of the $\Gamma$ point in Fig. 4(b). Actually, the flat band is very flat in the $\Gamma$-to-M direction whereas it is flat just around the $\Gamma$ point and soon shows a quadratic deviation in the $\Gamma$-to-X direction. On the other hand, the Dirac cone in the $\Gamma$-to-X direction shows a linear behavior whereas its lower branch soon deviates from the linear relation in the $\Gamma$-to-M direction. In any case, their linear and flat features are correct in the vicinity of the $\Gamma$ point as predicted by the $\textbf {k}\cdot \textbf {p}$ perturbation theory.
The eigenmodes of photonic crystal slabs near the $\Gamma$ point are generally located above the light line [15], so their eigenfrequencies are not real but complex values due to the diffraction loss. In Fig. 4, their real part is plotted. As we already mentioned, the Dirac-cone dispersion relation is distorted by the diffraction loss and its slope is divergent at the exceptional points located close to the $\Gamma$ point. However, the distortion is not significant when the diffraction loss is small, or in other words, the quality factor of the eigenmodes is large. In fact, we do not observe an appreciable distortion in Fig. 4(b), in which the quality factor of the $E$ mode is 400.
In addition to the dispersion relation, we calculated the angle-resolved reflection spectra by FEM. Figure 5 shows the spectra calculated for the structural parameters of Fig. 4(b) that materialize the Dirac cone with a flat band. The incident plane wave was tilted from the normal direction to the (1,0) ($\Gamma$-to-X) direction ($\phi =0^{\circ }$). Figures 5(a) and 5(b) show the reflection spectra for the s and p polarizations, respectively. In each figure, six reflection spectra with different incident angles by $0.2^{\circ }$ steps are drawn with an equal spacing.
Fig. 5. Angle-resolved reflection spectra calculated for an incident light tilted from the normal direction to the (1,0) ($\Gamma$-to-X) direction. The incident light was assumed to be polarized (a) perpendicular and (b) parallel to the incidence plane. In each figure, the upper and lower limits of the reflection spectrum for $\theta = 0^{\circ }$ are 1 and 0, respectively. Other spectra are drawn in the same scale and shifted by unity from each other in the vertical direction. The following specimen parameters were assumed to materialize the Dirac cone with a flat band: $a=2270$ nm, $d=216$ nm, $R=526$ nm.
The selection rule in Table 1 shows that the flat band is active to the s polarization whereas the Dirac cone is active to the p polarization. As predicted, we observed a nearly $\theta$-independent peak for the s polarization in addition to another small peak originating from an $A_{1}$ mode. On the other hand, there are two $\theta$-dependent peaks for the p polarization that diverge linearly from each other with increasing $\theta$, which agrees with the features of the Dirac cone. The selection rule for the $A_{1}$ mode is essentially the same as for the Dirac cone materialized by the $E$ and $A_{1}$ modes, so it is only active to the s polarization, which agrees with Fig. 5. In addition, the $A_{1}$ mode is inactive to an incident wave from the normal direction ($\theta = 0^{\circ }$) due to the symmetry mismatching, since the plane wave coming from the normal direction has the $E$ symmetry [15,19]. So, the reflection peak of the $A_{1}$ mode becomes small and disappears at $\theta = 0^{\circ }$. All these features imply that the selection rules derived by the $\textbf {k} \cdot \textbf {p}$ perturbation theory are correct. In addition, the agreement between the dispersion relation and the reflection spectra shows the accuracy of our FEM calculations.
4. Sample fabrication
According to the specimen design described in the last section, we fabricated SOI photonic crystal slabs by EB lithography, which consisted of a regular square array of cylindrical air holes made in the 400 nm-thick top silicon layer of the SOI wafer (SOITEC). We performed the EB patterning using an EB exposure apparatus, Elionix ELS-700 with a resist, Zenon ZEP520A. We etched the Si layer by ICP-RIE (inductively coupled plasma reactive ion etching) with a mixture of Cl$_{2}$ and Ar as an etchant.
Figure 6 shows their photo and SEM images. The typical depth of the circular air cylinders was 210 nm, whereas the lattice constant of the photonic crystal was 2270 nm. We can see regular, flat and smooth surface in the etched regions. By making allowance for the possible errors in both the calculation and fabrication, we fabricated seven specimens with different air-hole radii ranging from 440 to 620 nm on the same wafer. The lateral area of each specimen was 3 mm by 3 mm.
Fig. 6. (a) Photo and (b), (c) SEM images of photonic crystal slabs fabricated in the top silicon layer of an SOI wafer. Seven specimens with different air-hole radii were fabricated on the same wafer.
5. Angle-resolved reflection
We examined the dispersion relation by measuring the angle-resolved reflection spectra of the specimens. To obtain a sufficiently high angle resolution, we fabricated an optical system in the sample chamber of a commercial FT-IR spectrometer (JASCO 6800) as shown in Fig. 7. Its main components are flat mirrors, a polarizer, a rotating sample stage, and a concave lens to compensate the focused input IR beam from the light source of the FT-IR. This home-made apparatus enabled us to measure the reflection spectrum around the $\Gamma$ point with an unfocused and polarized incident beam at small and variable incident angles. The angle resolution, which was limited by the beam divergence, was estimated to be $0.3^{\circ }$.
Fig. 7. Our home-made optics for a high-resolution angle-resolved reflection spectroscopy fabricated in the sample chamber of an FT-IR spectrometer (JASCO 6800).
Now, let us proceed to the confirmation of the Dirac cone. Figure 8 shows the reflection spectra of a specimen for which the eigenfrequencies of the $E$ and $B_{2}$ modes are somewhat different from each other as shown in Fig. 4(a). The reflection spectra were measured for two different incident orientations and different polarizations. In each panel, 27 reflection spectra with different incident angles around the normal incidence by $0.292^{\circ }$ steps are drawn with an equal spacing. Our interesting frequency range is around 2600 cm$^{-1}$ where we find three modes and can assign the mode symmetry according to the spatial symmetry of each mode.
Fig. 8. Angle-resolved reflection spectra for an incident beam tilted to (a) the $\Gamma$-to-$X$ direction ($\phi =0^{\circ }$) and (b) the $\Gamma$-to-$M$ direction ($\phi =45^{\circ }$) . The upper and lower panels are for the s and p polarizations, respectively. 27 spectra were measured for different incident angles by $0.292^{\circ }$ steps for each panel. In each panel, the upper and lower limits of the lowest reflection spectrum are 1 and 0, respectively. Other spectra are drawn in the same scale and shifted by 0.1 from each other in the vertical direction. Sample parameters: $a= 2270$ nm, $R=440$ nm, $d= 210$ nm.
We also find spectral structures around 2070 cm$^{-1}$, 2360 cm$^{-1}$, and 2920 cm$^{-1}$. The first one was attributed to another $E$ mode in Fig. 8 and to an $E$ mode and a $B_{1}$ mode in Fig. 9, which was confirmed by the numerical results of the dispersion relation and the selection rules. The second one is a low reflection region caused by a strong absorption by CO$_2$ in the ambient air. The third one was attributed to an $E$ mode, an $A_{2}$ mode, and a $B_{2}$ mode, which was also confirmed by the numerical results of the dispersion relation and the selection rule.
Fig. 9. Angle-resolved reflection spectra for the specimen closest to the Dirac-cone dispersion relation. The incident beam was tilted to (a) the $\Gamma$-to-$X$ direction ($\phi =0^{\circ }$) and (b) the $\Gamma$-to-$M$ direction ($\phi =45^{\circ }$). The upper and lower panels are for the s and p polarizations, respectively. DC, FB, and $A_{1}$ denote the reflection peaks of the Dirac cone, the flat band, and the $A_{1}$-symmetric mode. 27 spectra were measured for different incident angles by $0.292^{\circ }$ steps for each panel. In each panel, the upper and lower limits of the lowest reflection spectrum are 1 and 0, respectively. Other spectra are drawn in the same scale and shifted by 0.1 from each other in the vertical direction. Sample parameters: $a= 2270$ nm, $R= 530$ nm, $d= 210$ nm.
Let us go back to the frequency range around 2600 cm$^{-1}$. When we look closely at the spectra of the $E$ band, its peak position takes a minimum frequency around the middle of the 27 spectra, while the peaks of the $A_{1}$ and $B_{2}$ bands take maximum frequencies there. In addition, the peak location is symmetric about the middle spectrum. Because the dispersion curves are symmetric about the $\Gamma$ point, these features imply that the incidence angle $\theta \approx 0^{\circ }$ for the middle spectrum and the minimum and maximum frequencies correspond to the eigenfrequencies on the $\Gamma$ point. The peaks due to the $E$ and $B_{2}$ modes approach each other, but do not coincide around the normal incidence. We should also note that the $A_{1}$ and $B_{2}$ peaks disappear at the normal incidence, since they are inactive to the incident light from the normal direction due to symmetry mismatching.
Figure 9 shows the spectra measured for a specimen whose dispersion was closest to the Dirac-cone dispersion, which have several peculiar features. (1) Two linear dispersions due to the Dirac cone cross each other around 2630 cm$^{-1}$. (2) In addition to the Dirac cone, there is a flat band (quadratic dispersion surface) as predicted by the $\textbf {k} \cdot \textbf {p}$ perturbation theory. (3) All modes including a nearby $A_{1}$-symmetric mode show their own selection rules with respect to the tilt direction and polarization of the incident beam. In addition, the $A_{1}$-symmetric mode is inactive to an incident beam from the normal direction, so that its reflection peak disappears at an incident angle of $0^{\circ }$. These features also agree well with the dispersion relation calculated by FEM, which is shown in Fig. 4(b).
6. Conclusion
We successfully fabricated photonic crystal slabs by EB lithography of SOI wafers that materialized the Dirac-cone dispersion relation in the mid infrared range, which was confirmed by the angle-resolved reflection spectra measured by our home-made optics that attained a high angle resolution of $0.3^{\circ }$. The observed spectra agreed quite well with the dispersion relation calculated by FEM and the selection rules due to the spatial symmetry of the eigenmodes. The Dirac-cone dispersion thus materialized may find applications in photonic micro circuits such as transmission through sharp bends and cloaking.
Funding
Innovative Science and Technology Initiative for Security by Acquisition, Technology & Logistics Agency (ATLA), Japan.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microw. Wirel. Compon. Lett. 14(2), 68–70 (2004). [CrossRef]
2. C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microw. Theory Tech. 52(3), 980–992 (2004). [CrossRef]
3. A. Sanada, C. Caloz, and T. Itoh, “Planar distributed structures with negative refractive index,” IEEE Trans. Microw. Theory Tech. 52(4), 1252–1263 (2004). [CrossRef]
4. K. Sakoda and H.-F. Zhou, “Role of structural electromagnetic resonances in a steerable left-handed antenna,” Opt. Express 18(26), 27371–27386 (2010). [CrossRef]
5. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012). [CrossRef]
6. K. Sakoda, “Double Dirac cones in triangular-lattice metamaterials,” Opt. Express 20(9), 9925–9939 (2012). [CrossRef]
7. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012). [CrossRef]
8. K. Sakoda, “Proof of the universality of mode symmetries in creating photonic Dirac cones,” Opt. Express 20(22), 25181–25194 (2012). [CrossRef]
9. K. Sakoda, “Photonic Dirac cones and relevant physics,” in Electromagnetic Metamaterials, K. Sakoda, ed. (Springer-Verlag, 2019), Chap. 16.
10. Y. Yao and K. Sakoda, “Dirac cones in periodically modulated quantum wells,” J. Phys. Soc. Jpn. 85(6), 065002 (2016). [CrossRef]
11. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef]
12. K. Sakoda, “Polarization-dependent continuous change in the propagation direction of Dirac-cone modes in photonic crystal slabs,” Phys. Rev. A 90(1), 013835 (2014). [CrossRef]
13. K. Sakoda and H. Takeda, “Dirac cones in photonic crystal slabs,” in Program and Abstract Book of 10th International Symposium on Modern Optics and its Applications (2015), pp. 68–69.
14. B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer, A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić, “Spawning rings of exceptional points out of Dirac cones,” Nature 525(7569), 354–358 (2015). [CrossRef]
15. K. Sakoda, Optical Properties of Photonic Crystals, 2nd Ed. (Springer-Verlag, 2004).
16. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1990).
17. D. Chandler-Horowitz and P. M. Amirtharaj, “High-accuracy, midinfrared (450 cm−1 ≤ ω ≤ 4000 cm−1) refractive index values of silicon,” J. Appl. Phys. 97(12), 123526 (2005). [CrossRef]
18. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965). [CrossRef]
19. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63(12), 125107 (2001). [CrossRef]
