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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19255–19264
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Photonic crystal digital alloys and their band structure properties

Jeongkug Lee, Dong-Uk Kim, and Heonsu Jeon  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19255-19264 (2011)
http://dx.doi.org/10.1364/OE.19.019255


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Abstract

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

1. Introduction

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

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]

,2

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]

]. 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

3. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]

,4

4. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006). [CrossRef]

]. Photon localizations in disordered PCs are scientifically exciting subjects [5

5. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007). [CrossRef] [PubMed]

,6

6. C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4(10), 794–798 (2008). [CrossRef]

], which were inspired by electron localizations where complex electronic scatterings by impurities and defects result in localized states [7

7. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]

].

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

8. Z. I. Alferov, V. M. Andreev, D. Z. Garbuzov, Y. V. Zhilyaev, E. P. Morozov, E. L. Portnoi, and V. G. Trofim, “Investigation of the influence of the AlAs–GaAs heterostructure parameters on the laser threshold current and the realization of continuous emission at room temperature,” Sov. Phys. Semicond. 4, 1573–1575 (1971).

,9

9. I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970). [CrossRef]

] and high-speed transistors [10

10. H. Kroemer, “Theory of a wide-gap emitter for transistors,” Proc. IRE 45, 1535–1537 (1957).

]. 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

11. H. J. Kim, Y.-G. Roh, and H. Jeon, “Photonic bandgap engineering in mixed colloidal photonic crystals,” Jpn. J. Appl. Phys. 44(40), L1259–L1262 (2005). [CrossRef]

13

13. S. Kim, S. Yoon, H. Seok, J. Lee, and H. Jeon, “Band-edge lasers based on randomly mixed photonic crystals,” Opt. Express 18(8), 7685–7692 (2010). [CrossRef] [PubMed]

], similar to band-gap engineering in semiconductor alloys [14

14. F. Capasso, “Band-gap engineering: from physics and materials to new semiconductor devices,” Science 235(4785), 172–176 (1987). [CrossRef] [PubMed]

]. In addition, we found that the virtual crystal approximation theory [15

15. L. Nordheim, “The electron theory of metals,” Ann. Phys Lpz. 9(5), 607–640 (1931). [CrossRef]

], 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 [16

16. Y.-H. Zhang and D. H. Chow, “Improved crystalline quality of AlAsxSb1-x grown on InAs by modulated molecular-beam epitaxy,” Appl. Phys. Lett. 65(25), 3239–3241 (1994). [CrossRef]

]. 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

17. P. Jiang, G. N. Ostojic, R. Narat, D. M. Mittleman, and V. L. Colvin, “The fabrication and bandgap engineering of photonic multilayers,” Adv. Mater. (Deerfield Beach Fla.) 13(6), 389–393 (2001). [CrossRef]

20

20. R. Rengarajan, P. Jiang, D. C. Larrabee, V. L. Colvin, and D. M. Mittleman, “Colloidal photonic superlattices,” Phys. Rev. B 64(20), 205103 (2001). [CrossRef]

]. For example, Rengarajan et al. [20

20. R. Rengarajan, P. Jiang, D. C. Larrabee, V. L. Colvin, and D. M. Mittleman, “Colloidal photonic superlattices,” Phys. Rev. B 64(20), 205103 (2001). [CrossRef]

] 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 [21

21. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990). [CrossRef] [PubMed]

]. 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 [12

12. H. J. Kim, D.-U. Kim, Y.-G. Roh, J. Yu, H. Jeon, and Q. H. Park, “Photonic crystal alloys: a new twist in controlling photonic band structure properties,” Opt. Express 16(9), 6579–6585 (2008). [CrossRef] [PubMed]

]. 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 [111] 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 [22

22. C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1974).

]. 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
Fig. 1 (a) Schematic of the photonic crystal digital alloy structure constructed in the diamond lattice. The superlattice period of the example structure corresponds to 3 unit plates, composed of 2 alumina (yellow) plates and 1 silica (red) plate, which results in the alumina composition ratio of x = 2/3. (b) Actual photo images of the individual unit plates and the final digital alloy structure, the schematic of which is illustrated in (a). The stacking direction of the unit plates corresponds to [111] in the conventional cubic cell. The inset illustrates a schematically drawn side-view of the unit plate.
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

εeff(x)=xεalumina+(1x)εsilica
(1)

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)
Fig. 2 (a) Unit cell of the diamond structure used in the 3D PWE calculations. a 1 and a 2 are the primitive vectors of the 2D hexagonal lattice structure on the unit plate, whereas a 3 is along the stacking direction of the unit plates. (b) The 1st Brillouin zone of the structure illustrated in (a). (A) is the wavevector at the zone boundary in the [111] direction.
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 [111] 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 [111] 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 [111] 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.

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 [27

27. İ. İ. Tarhan II and G. H. Watson, “Photonic band structure of fcc colloidal crystals,” Phys. Rev. Lett. 76(2), 315–318 (1996). [CrossRef] [PubMed]

], we attribute the additional dips to the super-periodicity of the DAs (i.e., a mini-gap) [28

28. L. Esaki and R. Tsu, “Superlattice and negative differential conductivity in semiconductors,” IBM J. Res. Develop. 14(1), 61–65 (1970). [CrossRef]

]. 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]12Fig. 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).

Figure 5
Fig. 5 Transmission spectra for PC digital alloys with different composition ratios: (a) x = 1/3, (b) x = 1/2, and (c) x = 2/3. Panels in each column represent different superlattice periods for the given composition. The PBGs calculated for the given digital alloy structures using the 3D PWE method are shaded in red.
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

Figure 6
Fig. 6 Transmission spectra measured for the PC digital alloys with a fixed composition ratio of x = l/(l + m) = 1/2, but of different superlattice periods: (a) l = m = 1, (b) l = m = 3, (c) l = m = 4, (d) l = m = 6, and (e) l = m = 12. Transmission spectra for the pure PCs (x = 0 and 1) are also presented in (f) for comparison. Eye-guides for the PBGs of x = 0 (red), x = 1 (blue), and x = 1/2 (purple) are shaded.
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.

4. Conclusions

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.

Acknowledgments

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).

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [CrossRef] [PubMed]

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987). [CrossRef] [PubMed]

3.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005). [CrossRef]

4.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006). [CrossRef]

5.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007). [CrossRef] [PubMed]

6.

C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys. 4(10), 794–798 (2008). [CrossRef]

7.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109(5), 1492–1505 (1958). [CrossRef]

8.

Z. I. Alferov, V. M. Andreev, D. Z. Garbuzov, Y. V. Zhilyaev, E. P. Morozov, E. L. Portnoi, and V. G. Trofim, “Investigation of the influence of the AlAs–GaAs heterostructure parameters on the laser threshold current and the realization of continuous emission at room temperature,” Sov. Phys. Semicond. 4, 1573–1575 (1971).

9.

I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17(3), 109–111 (1970). [CrossRef]

10.

H. Kroemer, “Theory of a wide-gap emitter for transistors,” Proc. IRE 45, 1535–1537 (1957).

11.

H. J. Kim, Y.-G. Roh, and H. Jeon, “Photonic bandgap engineering in mixed colloidal photonic crystals,” Jpn. J. Appl. Phys. 44(40), L1259–L1262 (2005). [CrossRef]

12.

H. J. Kim, D.-U. Kim, Y.-G. Roh, J. Yu, H. Jeon, and Q. H. Park, “Photonic crystal alloys: a new twist in controlling photonic band structure properties,” Opt. Express 16(9), 6579–6585 (2008). [CrossRef] [PubMed]

13.

S. Kim, S. Yoon, H. Seok, J. Lee, and H. Jeon, “Band-edge lasers based on randomly mixed photonic crystals,” Opt. Express 18(8), 7685–7692 (2010). [CrossRef] [PubMed]

14.

F. Capasso, “Band-gap engineering: from physics and materials to new semiconductor devices,” Science 235(4785), 172–176 (1987). [CrossRef] [PubMed]

15.

L. Nordheim, “The electron theory of metals,” Ann. Phys Lpz. 9(5), 607–640 (1931). [CrossRef]

16.

Y.-H. Zhang and D. H. Chow, “Improved crystalline quality of AlAsxSb1-x grown on InAs by modulated molecular-beam epitaxy,” Appl. Phys. Lett. 65(25), 3239–3241 (1994). [CrossRef]

17.

P. Jiang, G. N. Ostojic, R. Narat, D. M. Mittleman, and V. L. Colvin, “The fabrication and bandgap engineering of photonic multilayers,” Adv. Mater. (Deerfield Beach Fla.) 13(6), 389–393 (2001). [CrossRef]

18.

K. Baert, K. Song, R. A. L. Vallée, M. Van der Auweraer, and K. Clays, “Spectral narrowing of emission in self-assembled colloidal photonic superlattices,” J. Appl. Phys. 100(12), 123112 (2006). [CrossRef]

19.

N. C. Panoiu, R. M. Osgood Jr, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23(3), 506–513 (2006). [CrossRef]

20.

R. Rengarajan, P. Jiang, D. C. Larrabee, V. L. Colvin, and D. M. Mittleman, “Colloidal photonic superlattices,” Phys. Rev. B 64(20), 205103 (2001). [CrossRef]

21.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990). [CrossRef] [PubMed]

22.

C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1974).

23.

L. Vegard, “Die Konstitution der Mischkristalle und die Raumfüllung der Atome,” Z. Phys. 5(1), 17–26 (1921). [CrossRef]

24.

R. Kaspi and G. P. Donati, “Digital alloy growth in mixed As/Sb heterostructures,” J. Cryst. Growth 251(1-4), 515–520 (2003). [CrossRef]

25.

M. A. Khan, J. N. Kuznia, D. T. Olson, T. George, and W. T. Pike, “GaN/AlN digital alloy short-period superlattices by switched atomic layer metalorganic chemical vapor deposition,” Appl. Phys. Lett. 63(25), 3470–3472 (1993). [CrossRef]

26.

J. D. Song, D. C. Heo, I. K. Han, J. M. Kim, Y. T. Lee, and S.-H. Park, “Parametric study on optical properties of digital-alloy In(Ga1-zAlz)As/InP grown by molecular-beam epitaxy,” Appl. Phys. Lett. 84(6), 873–875 (2004). [CrossRef]

27.

İ. İ. Tarhan II and G. H. Watson, “Photonic band structure of fcc colloidal crystals,” Phys. Rev. Lett. 76(2), 315–318 (1996). [CrossRef] [PubMed]

28.

L. Esaki and R. Tsu, “Superlattice and negative differential conductivity in semiconductors,” IBM J. Res. Develop. 14(1), 61–65 (1970). [CrossRef]

OCIS Codes
(350.4010) Other areas of optics : Microwaves
(160.5293) Materials : Photonic bandgap materials
(160.5298) Materials : Photonic crystals
(160.2710) Materials : Inhomogeneous optical media

ToC Category:
Photonic Crystals

History
Original Manuscript: June 14, 2011
Revised Manuscript: August 4, 2011
Manuscript Accepted: September 11, 2011
Published: September 19, 2011

Citation
Jeongkug Lee, Dong-Uk Kim, and Heonsu Jeon, "Photonic crystal digital alloys and their band structure properties," Opt. Express 19, 19255-19264 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19255


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References

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987). [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58(23), 2486–2489 (1987). [CrossRef] [PubMed]
  3. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater.4(3), 207–210 (2005). [CrossRef]
  4. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys.78(2), 455–481 (2006). [CrossRef]
  5. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446(7131), 52–55 (2007). [CrossRef] [PubMed]
  6. C. Conti and A. Fratalocchi, “Dynamic light diffusion, three-dimensional Anderson localization and lasing in inverted opals,” Nat. Phys.4(10), 794–798 (2008). [CrossRef]
  7. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109(5), 1492–1505 (1958). [CrossRef]
  8. Z. I. Alferov, V. M. Andreev, D. Z. Garbuzov, Y. V. Zhilyaev, E. P. Morozov, E. L. Portnoi, and V. G. Trofim, “Investigation of the influence of the AlAs–GaAs heterostructure parameters on the laser threshold current and the realization of continuous emission at room temperature,” Sov. Phys. Semicond.4, 1573–1575 (1971).
  9. I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett.17(3), 109–111 (1970). [CrossRef]
  10. H. Kroemer, “Theory of a wide-gap emitter for transistors,” Proc. IRE 45, 1535–1537 (1957).
  11. H. J. Kim, Y.-G. Roh, and H. Jeon, “Photonic bandgap engineering in mixed colloidal photonic crystals,” Jpn. J. Appl. Phys.44(40), L1259–L1262 (2005). [CrossRef]
  12. H. J. Kim, D.-U. Kim, Y.-G. Roh, J. Yu, H. Jeon, and Q. H. Park, “Photonic crystal alloys: a new twist in controlling photonic band structure properties,” Opt. Express16(9), 6579–6585 (2008). [CrossRef] [PubMed]
  13. S. Kim, S. Yoon, H. Seok, J. Lee, and H. Jeon, “Band-edge lasers based on randomly mixed photonic crystals,” Opt. Express18(8), 7685–7692 (2010). [CrossRef] [PubMed]
  14. F. Capasso, “Band-gap engineering: from physics and materials to new semiconductor devices,” Science235(4785), 172–176 (1987). [CrossRef] [PubMed]
  15. L. Nordheim, “The electron theory of metals,” Ann. Phys Lpz.9(5), 607–640 (1931). [CrossRef]
  16. Y.-H. Zhang and D. H. Chow, “Improved crystalline quality of AlAsxSb1-x grown on InAs by modulated molecular-beam epitaxy,” Appl. Phys. Lett.65(25), 3239–3241 (1994). [CrossRef]
  17. P. Jiang, G. N. Ostojic, R. Narat, D. M. Mittleman, and V. L. Colvin, “The fabrication and bandgap engineering of photonic multilayers,” Adv. Mater. (Deerfield Beach Fla.)13(6), 389–393 (2001). [CrossRef]
  18. K. Baert, K. Song, R. A. L. Vallée, M. Van der Auweraer, and K. Clays, “Spectral narrowing of emission in self-assembled colloidal photonic superlattices,” J. Appl. Phys.100(12), 123112 (2006). [CrossRef]
  19. N. C. Panoiu, R. M. Osgood, S. Zhang, and S. R. J. Brueck, “Zero-n bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B23(3), 506–513 (2006). [CrossRef]
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