We investigated semi-disordered photonic crystals (PCs), digital alloys, and made thorough comparisons with their counterparts, random alloys. A set of diamond lattice PC digital alloys operating in a microwave regime were prepared by alternately stacking two kinds of sub-PC systems composed of alumina and silica spheres of the same size. Measured transmission spectra as well as calculated band structures revealed that when the digital alloy period is short, band-gaps of the digital alloys are practically the same as those of the random alloys. This study indicates that the concept of digital alloys holds for photons in PCs as well.
© 2011 OSA
Because of the spatially periodic variation in the dielectric constant profile, a photonic crystal (PC) may possess a photonic band-gap (PBG) in which the propagation of electromagnetic waves is prohibited [1,2]. In fact, the PBG is a photonic analogy to the electronic band-gap in a semiconductor. It is now a proven fact that close physical analogies do exist between photonic and electronic crystals, and the list of the analogies continues to expand. Among others, heterostructures turned out to be an extremely useful concept in photonics as well, producing an entirely new type of optical cavity with extraordinarily high Q-factors [3,4]. Photon localizations in disordered PCs are scientifically exciting subjects [5,6], which were inspired by electron localizations where complex electronic scatterings by impurities and defects result in localized states .
It was found that electronic band-gaps can be actively controlled in the format of semiconductor alloys (or mixed semiconductors), which has enabled semiconductor heterostructures and therefore modern optoelectronic devices with an extremely high impact, such as room-temperature continuous-wave laser diodes [8,9] and high-speed transistors . The authors’ group has been investigating their photonic counterpart, PC alloys, and verified using various PC platforms that the PBG of the alloy PCs can also be tuned by adjusting the mixing composition ratio [11–13], similar to band-gap engineering in semiconductor alloys . In addition, we found that the virtual crystal approximation theory , originally developed to explain the band-gap properties of semiconductor alloys, is equally successful for the PC alloys, and therefore provides additional strong evidence that analogies exist between photonics and electronics.
Despite many advantages and fruitful outcomes, however, alloy crystals (which are random both typically and conventionally) can sometimes be difficult to grow due to the immiscibility of constituent atoms. The immiscibility, however, can be overcome by alternately stacking pure crystals in a short period. Such artificial alloy crystals have been called digital alloys (DAs), as opposed to random alloys (RAs). For example, a AlAsSb/GaAsSb single-quantum-well structure grown in the DA format showed a dramatic improvement in surface morphology and crystalline quality, resulting in strong room-temperature photoluminescence, while the band-gap property as an alloy was preserved . It is worth noting that, in terms of structure, a DA can be regarded as a special type of superlattice with a relatively short lattice period. In fact, a few research groups have studied PC superlattices [17–20]. For example, Rengarajan et al.  prepared three-dimensional (3D) PC superlattices by alternately stacking two kinds of colloidal PCs, which differ from the DAs in that the periods of their superlattices are much longer than the lattice constant of the PC (Λ >> a). In particular, they reported that their structure exhibited extra bands due to the super-periodicity.
No systematic studies, however, have been performed on the PC DAs, probably due to the difficulty in preparing short-period superlattice structures in a reproducible manner. Here, we have realized PC DAs and examined whether the concept of DAs still holds for PCs as well. To circumvent the difficulty in sample preparation, we employed PC DAs operating in the microwave frequency range. We investigated experimentally how the resultant PBGs behave for various DA compositions and periods. For a theoretical confirmation, the band structures of the PC DAs were calculated by a 3D plane-wave-expansion (PWE) method and compared with experimental results.
2. Sample preparation
We employed the diamond lattice PC structure, as it is known to have a full PBG over a relatively wide frequency range . For the ease of sample preparation, we built our PCs on the millimeter scale so that their PBGs could develop at microwave frequencies (f ~12-15 GHz). Our PC alloys consisted of silica (SiO2) and alumina (Al2O3) spheres of the same diameter, ϕ = 5 mm. The dielectric constants of silica and alumina are 5.0 and 9.0 at the microwave frequencies, respectively . Because the diamond structure has a face-centered cubic (fcc) lattice with two basis atoms at the sites of (0,0,0) and (¼,¼,¼) within the conventional cubic cell, the diamond lattice can be constructed by vertically stacking identical unit plates in a special sequence, each unit plate containing the two basis atoms arranged in a two-dimensional (2D) hexagonal lattice. The direction in which the unit plates are stacked then corresponds to the  direction of the conventional cubic cell.
The forming material of the unit plates was a highly porous sponge with a dielectric constant of approximately 1.05. A 2D hexagonal lattice array of holes (5 mm in diameter) was punched through each plate. Each hole was then filled with a pair of either silica or alumina spheres, and each sphere pair represented the two basis atoms of the diamond lattice. The sponge plate, which is 5 mm thick, was intentionally designed to be thinner than the full length of the sphere pair (10 mm); this ensures that the spheres in one unit plate make comfortable physical contact with the spheres in the adjacent unit plates when the plates are stacked by gravity, which allows all of the spheres to settle at the right positions of the close-packed diamond lattice structure. To improve the mechanical robustness, each pair of spheres was glued with a tiny amount of epoxy before they were inserted into the holes. Then, the unit plates were stacked in the sequence of (…ABCABC…) to complete the PC in the diamond lattice . In particular, DAs were constructed by stacking the unit plates in groups: multiple layers of the unit plates of either silica or alumina spheres were stacked alternately in predetermined numbers.
We denote our PC DAs as [(Al2O3)l(SiO2)m]n, where l and m indicate the numbers of alumina and silica unit plates contained within one superlattice period, while n stands for the number of repetitions of the period. Figure 1 illustrates an example of how to construct the PC DA of l = 2 and m = 1. The yellow and red spheres in Fig. 1(a) represent the alumina and silica spheres, respectively. Figure 1(b) shows photographic images of the unit plates and also a completed DA structure. The DA structure shown in Fig. 1(b) has 12 repetitions (n = 12) of the period; thus, it can be expressed as [(Al2O3)2(SiO2)1]12. Note that, in terms of the conventional RA expression of (Al2O3)x(SiO2)1- x, the equivalent RA composition ratio of this particular DA structure becomes x = l/(l+m) = 2/3.
3. Measurements and analyses
3.1 Band structure calculations
We calculated the band structures of photonic DAs and their counterparts, RAs. This allowed us to make thorough comparisons not only between experimental and theoretical results but also between DAs and RAs. For this purpose, we employed the 3D PWE method that has been proven reliable in calculating PC band structures. The periodic nature of PCs leads to the Bloch form of electromagnetic waves in the PWE calculation. In the PWE method, a unit cell of the PC should be determined first. In the case of RAs, however, no unit cell can be clearly defined because the component photonic atoms are randomly distributed; thus, no strict periodicity exists. In our previous study on RAs (previously called mixed PCs), which were composed of two kinds of photonic atoms, we modeled the RAs as a homogeneous PC consisting of a single photonic element but with an effective refractive index . The effective refractive index was assumed to follow Vegard’s law , which has been used successfully to estimate band-gaps, lattice constants, and other properties of mixed semiconductors. In the case of RAs in which alumina and silica spheres of the same size are randomly distributed, the effective dielectric constant can be written as
In contrast, the unit cell for the PWE calculation is obvious for DAs; as illustrated in Fig. 1, a DA is formed by repeating a basic unit structure composed of multiple unit plates, which can be naturally chosen as the unit cell for PWE calculation.
Figure 2(a) shows the unit cell of the diamond lattice PC. a 1 and a 2 are the primitive lattice vectors of the 2D hexagonal lattice, which forms the basic building block of the unit plate, while a 3 is the unit vector along the  direction (perpendicular to the plane defined by a 1 and a 2). In case of the DA, the magnitude of a 3 depends on the period in the stacking sequence of (…ABCABC…). Shown in Fig. 2(b) is the first Brillouin zone of the diamond lattice PC, where point A is the zone boundary in the  direction. The actual diamond structure fabricated for experiments was found to be elongated slightly in the direction of a 3, which is due to the swelling of the matrix material of sponge. Although its thickness was carefully determined, the sponge plates swelled after the air holes were filled with dielectric spheres. In fact, direct measurement revealed that the vertical distance was 1.18 times longer than what it should be. This multiplication factor was then taken into account when we calculated photonic band structures.
3.2 Digital alloys versus random alloys
We used a network analyzer combined with a pair of horn antennas to measure transmission spectra of the PC DAs for various composition ratios and periods. The transmission measurements were made along the  direction, which coincided with the direction in which the unit plates were stacked. For comparison, we also prepared RAs and measured their transmission spectra as well. It is natural to expect that, when the period of a DA is small (comparable to or less than the photon wavelength), it should behave as an RA with the corresponding composition ratio. In fact, we have confirmed this conjecture experimentally, as we describe below.
Figure 3 displays the transmission spectra of DAs measured for a few representative compositions, where the transmission dips provide the information on the PBGs. Note here that the samples possess the shortest superlattice periods for the given composition ratios of x. First, it is clear from the experimental spectra that the PBG of the RA system monotonically shifts as x changes. Overall, the photonic properties of the RAs are in excellent agreement with the PWE calculation results (based on the virtual crystal approximation), which are indicated by the shaded regions. This observation is basically a reconfirmation of what we have already demonstrated in our previous study . However, the transmission spectra for the DAs tell us that the PBGs of the DAs are practically the same as the PBGs of the RAs, regardless of composition ratios. This is a direct indication that photonic DAs are indistinguishable from the RA counterparts in terms of their PBG characteristics, as long as the period of the DA superlattice is kept small. This is consistent with what is known for semiconductor DAs: when the period remains short, semiconductor DAs exhibit properties identical to the properties of RAs [24–26].
3.3 Superlattice properties of digital alloys
As shown already in Fig. 3, the main band-gaps of the short-period PC DAs are identical to those of the RAs of the same composition. However, close examination reveals that the transmission spectra of the DAs differ from those of the RAs, especially in the high frequency region. Apart from the gradual decrease beyond 16 GHz (which occurs commonly for both DAs and RAs), additional dips are seen for the DAs―see the regions indicated by curly brackets in Fig. 3 (b) and (d). While the gradual decrease is due to size fluctuations in constituent spheres (for alumina spheres, ϕ = 5.0 ± 0.2 mm) and therefore increases in the scattering cross-section , we attribute the additional dips to the super-periodicity of the DAs (i.e., a mini-gap) . Note that those extra dips do not appear in the transmission spectra of the RAs. Although the corresponding extra dip seems to be absent for [(Al2O3)1(SiO2)1]12―Fig. 3(c), the PWE calculations revealed that the corresponding mini-gap is to be formed in the frequency range of 18-19 GHz, which happens to fall in a low transmission region so that the corresponding dip is not as distinctly noticeable as in other cases―Fig. 3(b) and 3(d).
To further investigate the existence of the additional mini-gaps, we performed band structure calculations for the DAs in which the actual superlattice period was chosen as the unit cell in the PWE supercell calculation. The magnitude of a 3, which should depend on the period of DA, was therefore varied, while the Brillouin zone in the reciprocal space was changed accordingly. As already shown in Fig. 2(b), A represents the wavevector for the first Brillouin zone boundary in the  direction with a magnitude of π/a 3. Figure 4 shows the band structures calculated for the PC DAs of the same composition ratio of x = 2/3 but of different superlattice periods. Shown from the left are the resultant band structures for the DAs: [(Al2O3)2(SiO2)1], [(Al2O3)4(SiO2)2], and [(Al2O3)6(SiO2)3]. Note that the structural repetition number, n, is omitted here because the PWE calculation assumes an infinitely periodic structure. Additionally, the unit cell of the corresponding DA is shown in each figure. For clarity, PBG regions (not only the main PBG but also mini-gaps) are shaded. The vertical axes are the normalized frequency, a/λ, where a is the lattice constant of the conventional cubic cell of the diamond structure. Because of the variation in the unit cell, the wavevector A that represents the zone boundary along the  direction differs accordingly. We expressed them as A, A´, and A˝ for [(Al2O3)2(SiO2)1], [(Al2O3)4(SiO2)2], and [(Al2O3)6(SiO2)3], respectively. As can be seen in Fig. 4, additional mini-bands and gaps are developed due to the longer periods superimposed on the DAs. As the superlattice period increases (from A to A´ to A˝), the additional mini-bands and gaps appear more densely. It is worth noting that the main PBG remains the same, regardless of the superlattice periods.
Figure 5 shows the transmission spectra measured for a few representative superlattice periods and compositions of the PC DAs. For comparison, calculated band-gaps are shaded in red. The overall correspondence between the experimental and theoretical band-gaps is reasonably good, including the fact that the spacing between adjacent dips decreases as the DA period increases for a given alloy composition ratio. This observation indicates that the DAs indeed induce mini-gaps (in addition to the main PBG), which proves that the DAs bear the intrinsic nature of the superlattice. Slight discrepancies in spectral positions of the mini-gaps are believed to be caused by the finite size of the real DAs, as opposed to the infinitely periodic structure assumed in simulations.
3.4 Digital alloy with a long superlattice period
Intuitively, it is rather apparent that when the superlattice period of a DA becomes much larger (than the photon wavelength), the structure cannot behave as an alloy any longer but should act as a combined structure of two independent PCs. If that happens, then we can anticipate that two PBGs are simply added together, which results in a simple extended PBG and not a new PBG of the corresponding alloy. This can be understood in the same context as the previous research on the PC superlattices with long periods [17,20]. To investigate this aspect, we prepared DAs of x = ½ with various superlattice periods: [(Al2O3)1(SiO2)1]24, [(Al2O3)3(SiO2)3]8, [(Al2O3)4(SiO2)4]6, [(Al2O3)6(SiO2)6]4, and [(Al2O3)12(SiO2)12]2. Note that the total number of unit plates is fixed to 48 for all of the DA structures.
Figure 6 displays transmission spectra measured for the DA structures (and also those of the pure alumina and silica PCs for comparison). As the superlattice period of DA is increased, the transmission spectrum evolves gradually. When the period is relatively short (l+m = 2, 6, 8), a completely new PBG is formed somewhere between the PBGs of the pure PCs, while its bandwidth is comparable to those of the pure PCs. When the superlattice period is large (l+m = 12, 24), however, the PBG of DA becomes very large, its bandwidth extended to those of the two pure PCs. Although limited to the present PC system, the validity condition for DA can be formulated as l + m ≈10 (or Λ/a ≈6). These observations agree with our intuitive predictions. Therefore, we can say that a DA with a long period should not serve as an alloy or mixed crystal.
The validity of the concept of digital alloys in photonics has been explored. Photonic crystals operating in the microwave frequency range were employed for precise structure fabrication and high accuracy measurements. Three-dimensional digital alloys composed of silica and alumina spheres of the same size were prepared in a diamond lattice structure. When the digital alloy period is short, the properties of the main photonic band-gap are almost the same as those of random alloys. Because an extra periodicity is inevitably superimposed on the digital alloy structure, however, additional mini-gaps are developed. The number of mini-gaps increases in proportion to the period of the digital alloy. A three-dimensional plane-wave-expansion method correctly reproduced the photonic band-gaps and mini-gaps of the digital alloys. Both the experimental and the theoretical results support the existence of the one-to-one correspondence between photonic and electronic digital alloys.
This study was supported by the Mid-career Researcher Program funded by the National Research Foundation (2010-0014470) and by the World-Class University (WCU) Project funded by the Ministry of Education, Science & Technology of Korea (R31-10032).
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