## Excitonic polaritons in Fibonacci quasicrystals

Optics Express, Vol. 16, Issue 20, pp. 15382-15387 (2008)

http://dx.doi.org/10.1364/OE.16.015382

Acrobat PDF (190 KB)

### Abstract

The fabrication and characterization of light-emitting one-dimensional photonic quasicrystals based on excitonic resonances is reported. The structures consist of high-quality GaAs/AlGaAs quantum wells grown by molecular-beam epitaxy with wavelength-scale spacings satisfying a Fibonacci sequence. The polaritonic (resonant light-matter coupling) effects and light emission originate from the quantum well excitonic resonances. Measured reflectivity spectra as a function of detuning between emission and Bragg wavelength are in good agreement with excitonic polariton theory. Photoluminescence experiments show that active photonic quasicrystals, unlike photonic crystals, can be good light emitters: While their long-range order results in a stopband similar to that of photonic crystals, the lack of periodicity results in strong emission.

© 2008 Optical Society of America

1. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. **53**, 1951–1953 (1984). [CrossRef]

3. A. Ledermann et al, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Mater. **5**, 942–945 (2006). [CrossRef] [PubMed]

6. M. A. Kaliteevski et al, “Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals,” J. Phys.: Condens. Matter **13**, 10459–10470 (2001). [CrossRef]

7. R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, and P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. **55**, 1768–1770 (1985). [CrossRef] [PubMed]

8. F. Laruelle and B. Etienne, “Fibonacci invariant and electronic properties of GaAs/Ga_{1-x}Al_{x}As quasiperiodic superlattices,” Phys. Rev. B **37**, 4816–4819 (1988). [CrossRef]

9. W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. **72**, 633–636 (1994). [CrossRef] [PubMed]

15. M. A. Kaliteevski, V. V. Nikolaev, R. A. Abram, and S. Brand, “Bandgap structures of optical Fibonacci lattices after light diffraction,” Opt. Spectrosc. **91**, 109–118 (2001). [CrossRef]

_{x}/SiO

_{2}) have been fabricated with enhanced emission compared with homogeneous SiNx dielectrics [16]. Quasicrystal lasers have also been reported [17

17. K. Nozaki and T. Baba, “Quasiperiodic photonic crystal microcavity lasers,” Appl. Phys. Lett. **84**, 4875–4877 (2004). [CrossRef]

18. S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, and Y. H. Lee, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. **86**, 031101 (2005). [CrossRef]

17. K. Nozaki and T. Baba, “Quasiperiodic photonic crystal microcavity lasers,” Appl. Phys. Lett. **84**, 4875–4877 (2004). [CrossRef]

13. L. Dal Negro et al., “Photon band gap properties and omnidirectional reflectance in Si/SiO_{2} Thue—Morse quasicrystals,” Appl. Phys. Lett. **84**, 5186–5188 (2004). [CrossRef]

19. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Reports **376**, 225–337 (2003). [CrossRef]

19. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Reports **376**, 225–337 (2003). [CrossRef]

3. A. Ledermann et al, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Mater. **5**, 942–945 (2006). [CrossRef] [PubMed]

*quasicrystalline*structures to be discussed below, we first briefly review the well-known yet appealing physics of a

*crystalline*arrangement of

*N*optically active QWs, periodically spaced by half the exciton wavelength λ in the material (typically,

*N*>10 is sufficient), so that the period of the structure

*d*equals the Bragg value

*d*

_{Bragg}≡λ/2=λ

_{0}/2

*n*, where λ

_{0}is the vacuum wavelength, and

*n*is the effective refractive index of the materials between the QW centers. In this periodic case, the coupling of the

*N*QWs via the electromagnetic field lifts the

*N*-fold degeneracy in a particular manner: While (

*N*-

*1*) subradiant modes experience only very little coupling to the field, most of the oscillator strength is concentrated in a single superradiant mode [26-30

30. J. P. Prineas et al, “Exciton-polariton eigenmodes in light-coupled In_{0.04}Ga_{0.96}As/GaAs semiconductor multiple-quantum-well periodic structures,” Phys. Rev. B **61**, 13863–13872 (2000). [CrossRef]

*N*. This increase makes competing nonradiative dephasing channels (e.g., disorder or phonon damping) irrelevant for large enough

*N*. At first sight, this enhanced light-matter coupling appears to be ideal for an efficient light-emitting structure. Yet, it is not. For the exact Bragg condition and for emission normal to the layers of the Bragg stack, the corresponding standing wave pattern has nodes at the QW positions in real space – obviously leading to inefficient emission in this direction. Correspondingly, in the frequency domain, the emission wavelength lies within the photonic crystal stopband. We will see that these unfavorable conditions no longer exist in 1D quasicrystalline structures where the Bragg condition can be effectively maintained, yet not all of the QWs are located at field-node positions.

_{j}contains QWs with two different separations A and B between the centers of the wells. The ratio of the optical pathlengths of B to A equals the golden mean (√5+1)/2; the Fibonacci recursion is illustrated in Fig. 1. We chose the well established GaAs/AlGaAs material as a model system. The Bragg condition for the Fibonacci structure of the smallest possible thickness can be presented in the same form as for the periodic one,

*d*̄=

*d*

_{Bragg}, where

*d*̄=(5-√5)A/2 stands for the mean period of the lattice, and

*d*

_{Bragg}=λ/2 [31

31. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B **77**, 113306 (2008). [CrossRef]

*ηNΓ*, where

_{0}*η*is close to unity and

*Γ*is the radiative decay rate of a single QW. Frequencies of the other (

_{0}*N*-

*1*) modes have much smaller but nonzero imaginary parts. These weakly radiating modes contribute to the fine structure of the reflection spectrum in the vicinity of the exciton resonance frequency as shown below.

_{0.3}Ga

_{0.7}As barriers. The target lengths

*L*and

_{A}*L*for A and B of a Fibonacci chain sample were then reached by growing Al

_{B}_{0.04}Ga

_{0.96}As superlattices. The physical thicknesses of A and B are roughly 82 and 134 nm for the Bragg condition; the ratio of their optical thicknesses is the golden mean value. The optical thickness of A is

*d*

_{QW}*n*

*+ (*

_{QW}*L*-

_{A}*d*)nbarrier, where

_{QW}*d*is the thickness of the QW; here the QW background index

_{QW}*n*is almost the same as

_{QW}*n*

*. The samples were not anti-reflection coated; for the Bragg condition, the optical thickness from the vacuum interface to the center of the first QW is λ/2.*

_{barrier}*N*=21 (sample FIB12) and 54 (FIB13) as well as a periodically spaced sample with

*N*=21 (FIB14). Figure 2 compares the reflectivity for the two samples with

*N*=21. A reflectivity maximum with broad linewidth occurs at the Bragg condition for both samples for both the heavy-hole (HH) and the light-hole (LH) resonances. This maximum is just the well known photonic stopband, which is in agreement with previous work on active 1D photonic crystals using excitonic resonances in QWs of other materials [32

32. V. P. Kochereshko et al, “Giant exciton resonance reflectance in Bragg MQW structures,” Superlatt. Microstruct. Vol. **15**, No. 4, 471–473 (1994). [CrossRef]

30. J. P. Prineas et al, “Exciton-polariton eigenmodes in light-coupled In_{0.04}Ga_{0.96}As/GaAs semiconductor multiple-quantum-well periodic structures,” Phys. Rev. B **61**, 13863–13872 (2000). [CrossRef]

31. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B **77**, 113306 (2008). [CrossRef]

*N*=54 QW sample, as shown in Fig. 4.

30. J. P. Prineas et al, “Exciton-polariton eigenmodes in light-coupled In_{0.04}Ga_{0.96}As/GaAs semiconductor multiple-quantum-well periodic structures,” Phys. Rev. B **61**, 13863–13872 (2000). [CrossRef]

33. L. I. Deych, M. V. Erementchouk, A. A. Lisyansky, E. L. Ivchenko, and M. M. Voronov, “Exciton luminescence in one-dimensional resonant photonic crystals: A phenomenological approach,” Phys. Rev. B **76**, 075350 (2007). [CrossRef]

31. A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B **77**, 113306 (2008). [CrossRef]

*single*QW to experiment (Fig. 5(a) & (b)) using a Lorentzian form: χ(ω)=Γ

_{0HH}/(

*ω*

_{HH}-

*ω*- iΓ

_{HH})+Γ

_{0LH}/(

*ω*

_{LH}-

*ω*- iΓ

_{LH}), where

*ω*

_{HH}is the radial frequency of the resonance, Γ

_{0HH}is the radiative HWHM linewidth, and Γ

_{HH}is the nonradiative HWHM of the heavy hole, etc. Next, this susceptibility is used for calculating the optical spectra of the Fibonacci structure FIB13 (

*N*=54) in Fig. 5(d). Given the complexity of the optical spectra and the simplicity of the assumed susceptibility, the agreement has to be considered as very good. In particular, this comparison gives us additional confidence that the discussed fine structure in the optical spectra is actually due to the 1D Fibonacci photonic quasicrystal. That the field overlap with QWs is much better for the Fibonacci spacings than for periodic is shown in Fig. 6; this explains the much stronger PL in the Fibonacci case.

## Acknowledgments

## References and links

1. | D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. |

2. | C. Janot, |

3. | A. Ledermann et al, “Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths,” Nat. Mater. |

4. | W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosehdral quasicrystals,” Nature |

5. | B. Freedman et al, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature |

6. | M. A. Kaliteevski et al, “Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals,” J. Phys.: Condens. Matter |

7. | R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, and P. K. Bhattacharya, “Quasiperiodic GaAs-AlAs heterostructures,” Phys. Rev. Lett. |

8. | F. Laruelle and B. Etienne, “Fibonacci invariant and electronic properties of GaAs/Ga |

9. | W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. |

10. | N. Liu, “Propagation of light waves in Thue-Morse dielectric multilayers,” Phys. Rev. B |

11. | R. W. Peng et al, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett. |

12. | L. Dal Negro et al, “Light transport through the band-edge states of Fibonacci quasicrystals,” Phys. Rev. Lett. |

13. | L. Dal Negro et al., “Photon band gap properties and omnidirectional reflectance in Si/SiO |

14. | T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, “Photonic dispersion relation in a one-dimensional quasicrystal,” Phys. Rev. B |

15. | M. A. Kaliteevski, V. V. Nikolaev, R. A. Abram, and S. Brand, “Bandgap structures of optical Fibonacci lattices after light diffraction,” Opt. Spectrosc. |

16. | L. Dal Negro et al., “Spectrally enhanced light emission from aperiodic photonic structures,” Appl. Phys. Lett. |

17. | K. Nozaki and T. Baba, “Quasiperiodic photonic crystal microcavity lasers,” Appl. Phys. Lett. |

18. | S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, and Y. H. Lee, “Photonic quasicrystal single-cell cavity mode,” Appl. Phys. Lett. |

19. | E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Reports |

20. | M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B |

21. | S. V. Gaponenko, S. V. Zhukovsky, A. V. Lavrinenko, and K. S. Sandomirskii, “Propagation of waves in layered structures viewed as number recognition,” Optics Commun. |

22. | M. S. Vasconcelos, E. L. Albuquerque, and A. M. Mariz, “Optical localization in quasi-periodic multilayers,” J. Phys. Cond. Mat. |

23. | E. M. Nascimento, F. A. B. F. de Moura, and M. L. Lyra, “Scaling laws for the transmission of random binary dielectric multilayered structures,” Phys. Rev. B |

24. | M. S. Vasconcelos and E. L. Albuquerque, “Plasmon-polariton fractal spectra in quasiperiodic multilayers,” Phys. Rev. B |

25. | E. L. Albuquerque and M. G. Cottam, |

26. | E. L. Ivchenko, “Excitonic polaritons in periodic quantum-well structures,” Sov. Phys. Solid State |

27. | E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, “Bragg reflection of light from quantum-well structures,” Phys. Sol. Stat. |

28. | E. L. Ivchenko, |

29. | M. Hübner et al, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. |

30. | J. P. Prineas et al, “Exciton-polariton eigenmodes in light-coupled In |

31. | A. N. Poddubny, L. Pilozzi, M. M. Voronov, and E. L. Ivchenko, “Resonant Fibonacci quantum well structures in one dimension,” Phys. Rev. B |

32. | V. P. Kochereshko et al, “Giant exciton resonance reflectance in Bragg MQW structures,” Superlatt. Microstruct. Vol. |

33. | L. I. Deych, M. V. Erementchouk, A. A. Lisyansky, E. L. Ivchenko, and M. M. Voronov, “Exciton luminescence in one-dimensional resonant photonic crystals: A phenomenological approach,” Phys. Rev. B |

**OCIS Codes**

(230.5590) Optical devices : Quantum-well, -wire and -dot devices

(160.5293) Materials : Photonic bandgap materials

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 11, 2008

Revised Manuscript: July 30, 2008

Manuscript Accepted: August 31, 2008

Published: September 15, 2008

**Citation**

J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, A. N. Poddubny, E. L. Ivchenko, M. Wegener, and H. M. Gibbs, "Excitonic polaritons in Fibonacci quasicrystals," Opt. Express **16**, 15382-15387 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15382

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### References

- D. Shechtman, I. Blech, D. Gratias, J. W. Cahn, "Metallic phase with long-range orientational order and no translational symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984). [CrossRef]
- C. Janot, Quasicrystals (Clarendon Press, Oxford, 1994).
- A. Ledermann et al, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nat. Mater. 5, 942-945 (2006). [CrossRef] [PubMed]
- W. Man, M. Megens, P. J. Steinhardt, P. M. Chaikin, "Experimental measurement of the photonic properties of icosehdral quasicrystals," Nature 436, 993-996 (2005). [CrossRef] [PubMed]
- B. Freedman et al, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006). [CrossRef] [PubMed]
- M. A. Kaliteevski et al, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter 13, 10459-10470 (2001). [CrossRef]
- R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P. K. Bhattacharya, "Quasiperiodic GaAs-AlAs heterostructures," Phys. Rev. Lett. 55, 1768-1770 (1985). [CrossRef] [PubMed]
- F. Laruelle and B. Etienne, "Fibonacci invariant and electronic properties of GaAs/Ga1-xAlxAs quasiperiodic superlattices," Phys. Rev. B 37, 4816-4819 (1988). [CrossRef]
- W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, "Localization of light waves in Fibonacci dielectric multilayers," Phys. Rev. Lett. 72, 633-636 (1994). [CrossRef] [PubMed]
- N. Liu, "Propagation of light waves in Thue-Morse dielectric multilayers," Phys. Rev. B 55, 3543-3547 (1997). [CrossRef]
- R. W. Peng et al, "Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers," Appl. Phys. Lett. 80, 3063-3065 (2002). [CrossRef]
- L. Dal Negro et al, "Light transport through the band-edge states of Fibonacci quasicrystals," Phys. Rev. Lett. 90, 055501 (2003). [CrossRef]
- L. Dal Negro et al., "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186-5188 (2004). [CrossRef]
- T. Hattori, N. Tsurumachi, S. Kawato, H. Nakatsuka, "Photonic dispersion relation in a one-dimensional quasicrystal," Phys. Rev. B 50, 4220-4223 (1994). [CrossRef]
- M. A. Kaliteevski, V. V. Nikolaev, R. A. Abram, S. Brand, "Bandgap structures of optical Fibonacci lattices after light diffraction," Opt. Spectrosc. 91, 109-118 (2001). [CrossRef]
- L. Dal Negro et al., "Spectrally enhanced light emission from aperiodic photonic structures," Appl. Phys. Lett. 86, 261905 (2005).
- K. Nozaki and T. Baba, "Quasiperiodic photonic crystal microcavity lasers," Appl. Phys. Lett. 84, 4875-4877 (2004). [CrossRef]
- S.-K. Kim, J.-H. Lee, S.-H. Kim, I.-K. Hwang, Y. H. Lee, "Photonic quasicrystal single-cell cavity mode," Appl. Phys. Lett. 86, 031101 (2005). [CrossRef]
- E. L. Albuquerque and M. G. Cottam, "Theory of elementary excitations in quasiperiodic structures," Phys. Reports 376, 225-337 (2003). [CrossRef]
- M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, E. L. Albuquerque, "Photonic band gaps in quasiperiodic photonic crystals with negative refractive index," Phys. Rev. B 76, 165117 (2007). [CrossRef]
- S. V. Gaponenko, S. V. Zhukovsky, A. V. Lavrinenko, K. S. Sandomirskii, "Propagation of waves in layered structures viewed as number recognition," Optics Commun. 205, 49-57 (2002). [CrossRef]
- M. S. Vasconcelos, E. L. Albuquerque, A. M. Mariz, "Optical localization in quasi-periodic multilayers," J. Phys. Cond. Mat. 10, 5839-5849 (1998). [CrossRef]
- E. M. Nascimento, F. A. B. F. de Moura, M. L. Lyra, "Scaling laws for the transmission of random binary dielectric multilayered structures," Phys. Rev. B 76,115120 (2007). [CrossRef]
- M. S. Vasconcelos and E. L. Albuquerque, "Plasmon-polariton fractal spectra in quasiperiodic multilayers," Phys. Rev. B 57, 2826-2833 (1998). [CrossRef]
- E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, Amsterdam, The Netherlands, 2004).
- E. L. Ivchenko, "Excitonic polaritons in periodic quantum-well structures," Sov. Phys. Solid State 33, 1344-1349 (1991).
- E. L. Ivchenko, A. I. Nesvizhskii, S. Jorda, "Bragg reflection of light from quantum-well structures," Phys. Sol. Stat. 36, 1156-1161 (1994).
- E. L. Ivchenko, Optical Spectroscopy of Semiconductor Nanostructures (Alpha Science International, Harrow, U. K., 2005).
- M. Hübner et al, "Optical lattices achieved by excitons in periodic quantum well structures," Phys. Rev. Lett. 83, 2841-2844 (1999). [CrossRef]
- J. P. Prineas et al, "Exciton-polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structures," Phys. Rev. B 61, 13863-13872 (2000). [CrossRef]
- A. N. Poddubny, L. Pilozzi, M. M. Voronov, E. L. Ivchenko, "Resonant Fibonacci quantum well structures in one dimension," Phys. Rev. B 77, 113306 (2008). [CrossRef]
- V. P. Kochereshko et al, "Giant exciton resonance reflectance in Bragg MQW structures," Superlatt. Microstruct. Vol. 15, No. 4, 471-473 (1994). [CrossRef]
- L. I. Deych, M. V. Erementchouk, A. A. Lisyansky, E. L. Ivchenko, M. M. Voronov, "Exciton luminescence in one-dimensional resonant photonic crystals: A phenomenological approach," Phys. Rev. B 76, 075350 (2007). [CrossRef]

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