## Photonic band gaps analysis of Thue-Morse multilayers made of porous silicon

Optics Express, Vol. 14, Issue 13, pp. 6264-6272 (2006)

http://dx.doi.org/10.1364/OE.14.006264

Acrobat PDF (219 KB)

### Abstract

Dielectric aperiodic Thue-Morse structures up to 128 layers have been fabricated by using porous silicon technology. The photonic band gap properties of Thue-Morse multilayers have been theoretically investigated by means of the transfer matrix method and the integrated density of states. The theoretical approach has been compared and discussed with the reflectivity measurements at variable angles for both the transverse electric and transverse magnetic polarizations of light. The photonic band gap regions, wide 70 nm and 90 nm, included between 0 and 30°, have been observed for the sixth and seventh orders, respectively.

© 2006 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited spontaneous emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059 (1987). [CrossRef] [PubMed]

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

3. S. M. Weiss, M. Haurylau, and P. M. Fauchet, “Tunable photonic bandgap structures for optical interconnects,” Opt. Mater. **27**, 740 (2005). [CrossRef]

*n*) and high (

_{A}*n*) refractive index, whose thicknesses satisfy the Bragg condition:

_{B}*n*. The Fourier spectrum (FS) of an infinite PhC shows a δ-function peak at those wavevectors corresponding to the reciprocal lattice vectors. On the other hand, an infinite random structure has a nearly homogeneous FS [4

_{A}d_{A}+n_{B}d_{B}=λ0/24. C. M. Soukoulis and E. N. Economou, “Localization in one-dimensional lattices in the presence of incommensurate potentials,” Phys. Rev. Lett. **48**, 1043 (1982). [CrossRef]

4. C. M. Soukoulis and E. N. Economou, “Localization in one-dimensional lattices in the presence of incommensurate potentials,” Phys. Rev. Lett. **48**, 1043 (1982). [CrossRef]

5. A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystal,” Appl. Phys. Lett. **82**, 3227 (2003). [CrossRef]

6. V. Agarwal and J. A. del Rio, “Tailoring the photonic band gap of a porous silicon dielectric mirror,” Appl. Phys. Lett. **82**, 1512 (2003). [CrossRef]

7. N. Liu, “Propagation of light waves in Thue-Morse dielectric multilayers,” Phys. Rev. B **55**, 3543 (1997). [CrossRef]

8. J. M. Luck, “Cantor spectra and scaling of gap widths in deterministic aperiodic systems,” Phys. Rev. B **39**, 5834 (1989). [CrossRef]

_{2}and TiO

_{2}/SiO

_{2}as high/low refractive index materials [9

9. M. Dulea, M. Severin, and R. Riklund, “Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,” Phys. Rev. B **42**, 3680 (1990). [CrossRef]

11. F. Qui, R. W. Peng, X. Q. Huang, X. F. Hu, Mu Wang, A. Hu, S. S. Jiang, and D. Feng, “Omnidirectional reflection of electromagnetic waves on Thue-Morse dielectric multilayers,” Europhys. Lett. **68**, 658–663 (2004). [CrossRef]

*S*sequences by exploiting the porous silicon (PSi) technology. Due to the high quality optical response exhibited by the multilayered structures, the PSi is a very attractive material which can be fabricated by a fast and quite simple process. The PSi is, in fact, obtained by the electrochemical etching of the crystalline silicon in a hydrofluoridric acid (HF) based solution. The porosity and the thickness of a single layer are linear functions of the current density and the anodization time for a fixed doping level of the silicon wafer and HF concentration. The refractive index of the PSi film depends on its porosity, and can be calculated in the frame of several effective medium approximations, like the Bruggemann or Maxwell-Garnett models [12]. Fibonacci QC of up to 233 porous silicon layers has been fabricated in order to study the light propagation in aperiodic structures [13

_{1}–S_{7}13. L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, M. Righini, L. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci Quasicrystals,” Phys. Rev. Lett. **90**, 55501 (2003). [CrossRef]

*et al*. have recently exploited the oxidation effect in T-M multilayers up to more than one thousand layers [14

14. V. Agarwal, J. A. Soto-Urueta, D. Becera, and M. E. Mora-Ramos, “Light propagation in polytype Thue-Morse structures made of porous silicon,” Photonics Nanostruct. Fundam. **3**, 155 (2005). [CrossRef]

## 2. Theory

*n*(

_{A}*n*) and thickness

_{B}*d*(

_{A}*d*). Applying the substitution rules A→AB and B→BA [7

_{B}7. N. Liu, “Propagation of light waves in Thue-Morse dielectric multilayers,” Phys. Rev. B **55**, 3543 (1997). [CrossRef]

*S*

_{0}=A,

*S*

_{1}=AB,

*S*

_{2}=ABBA,

*S*

_{3}=ABBABAAB,

*S*

_{4}=ABBABAABBAABABBA, and so on. The layers number of

*S*is

_{N}*2*, where

^{N}*N*is the T-M order. A non-recursive expression of the T-M sequence is given by assigning the numerical values 1 and 0 to the A and B symbols, respectively:

*S*,

_{5}*S*and

_{7}*S*strings with 32, 128 and 4096 terms, respectively. The Fourier power spectrums (FPS) are plotted in Fig. 1 as function of the percentage sampling frequency

_{12}*q*. The FPS of an aperiodic structure is composed by a finite sequence of peaks which changes, differently from that of a periodic structure, as the system length

*d*, increases. In particular, the intensity and the distribution of the peaks changes, concentrating the peaks around the two central resonances which correspond to the band gaps. From Fig. 1(c) is well evident that the self-similarity of the T-M geometry also emerges in the fractal behavior of its FPS.

_{N}=2^{N}**χ**(

*z*) represents the two-component wave function:

**M**(

*z*) is described by following relation:

*ϕ(z)*is the phase modulation,

*θ*is the incident angle,

*k*

_{0}is the wave vector in the vacuum,

*n(z)*is the refractive index profile. If an

*N*-th order T-M sequence starts at

*z*

_{0}and ends at

*z*, where

_{N}*z*is the stratification direction, we can write:

*is the transfer matrix that can be deduced from the following recursion relation:*

_{N}*is the complement of Γ*

_{N}*obtained by interchanging*

_{N}*A*and

*B*in Γ

*.*

_{N}*et al*. [16

16. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107 (1996). [CrossRef]

*ρ=dk/dω*, can be written as function of real and imaginary part of

*t*:

_{N}*ω*. To investigate the photonic band gap properties is advantageous to introduce the integrated density of states (IDOS) defined as:

*ω*, and can be readily determined by numerical calculations. Being the modulation phase (

*ϕ*, the value 2

_{A}+ϕ_{B})/π=ω/ω_{0}*ω*

_{0}represent a phase shift of 2

*π*.

*S*, with a resonance wavelength

_{7}*λ*of 700 nm, as function of the wavelength, and its transmittance,

_{0}*T*=|

_{N}*t*|

_{N}^{2}. From the plot of transmittance is interesting to note the same hierarchy peaks of the

*S*FPS in Fig. 1(b). The plateau zones of IDOS represent the band gaps of the T-M structure. The larger band gap is obtained for (

_{7}*ϕ*+

_{A}*ϕ*)=0.66π(≈2/3π,

_{B}*θ*

_{2}) corresponding with stronger frequency of FPS in Fig. 1(b). Jiang

*et al*. identified this gap as a fractal one, which origin is due to the complex correlation depending on the system length [17

17. X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. **86**, 201110 (2005). [CrossRef]

*ϕ*+

_{A}*ϕ*)=0.57π (

_{B}*θ*) and 0.83π (

_{1}*θ*). Furthermore the strong slopes in IDOS near the PBGs are connected with the narrow transmittance peaks in the

_{3}*S*structure, as we can note from the Fig. 2(b).

_{7}## 3. Experiment

^{+}-silicon, <100> oriented, 0.01 Ω cm resistivity, 400 µm thick was used as the substrate in the T-M structures fabrication. The samples were fabricated in dark light at room temperature using a solution of 30% volumetric fraction of aqueous HF (50% wt) and 70% of Ethanol. Before anodisation, we have removed the thin film of native oxide from the silicon wafer by rapid rinsing in a diluted HF solution. Thicknesses and porosities have been estimated by variable angle spectroscopic ellipsometry (VASE) (J. A. Wollam Company) measurements on single PSi layers.

*p*=81 %), with an average refractive index

_{A}*n*≅1.3 and a thickness

_{A}*d*≅135 nm, were obtained applying an etching current density of 150 mA/cm

_{A}^{2}for 0.88 s. Low porosity layers (

*p*=56 %), with

_{B}*n*≅1.96 and a thickness of

_{B}*d*≅90 nm, were obtained with a current density of 5 mA/cm

_{B}^{2}for 0.53 s. The thickness

*d*of each layer was designed to satisfy the Bragg condition

_{i}*n*where

_{i}d_{i}=λ_{0}/4*n*is the average refractive index and

_{i}*λ*=700 nm. A schematic of the PSi T-M sequences realized is reported in Fig. 3: while the number of the layers increases, like 2

_{0}*where*

^{N}*N*is the Thue-Morse order, the thickness of the devices is given by the simple relationship

*d*=2

_{N}*d*for

_{N-1}*N*>1. The realized samples

*S*have thicknesses spanning the range between 0.135 µm and 14.4 µm.

_{0}–S_{7}## 4. Results and discussion

*S*[Fig. 4(a)],

_{3}*S*[Fig. 4(b)],

_{4}*S*[Fig. 4(c)],

_{5}*S*[Fig. 5(a)], and

_{6}*S*[Fig. 5(b)] T-M structures. The good control in the fabrication process of the devices is demonstrated by the agreement between the measured and calculated spectra. The not perfect matching can be ascribed to the non-uniformities of thicknesses and porosities of layers along the etching direction. The spectrum of the

_{7}*S*is characterized by two band gaps separated by a large transmission peak at 1000 nm. On increasing the order of T-M sequence, the PBG splits and very narrow transmission peaks appear (FWHM about 6 nm). The band gap structure, predicted by the numerical calculations, is clearly recognizable at higher the T-M orders. In particular, from the reflectivity spectrum of

_{3}*S*in Fig. 5(a) we can distinguish quite clearly the three gaps

_{6}*θ*, and

_{3}, θ_{2}*θ*that are centered at the wavelengths of 830, 1050 and 1250 nm, respectively. These values of PBGs are in good agreement with the theoretical ones of Fig. 2. On the other hand the

_{1}*S*reflectivity spectrum shows a light blue-shift, probably due to a mismatch of deeper layer thickness.

_{7}*S*and

_{6}*S*for both the TE (solid curve) and the TM (dashed curve) polarization at different incident angles up to 45° since they show the larger PBG,

_{7}*θ*, of all. The results are reported in Figs. 6 and 7. The grey area highlights a PBG region of 70 nm, centered at 1100 nm which exists in the incident angle range between 0 and 30° in case of

_{2}*S*. An even more extend PBG respect to the previous sequence, of about 90 nm, centered at 950 nm in the angular range between 0°–30° can be observed for

_{6}*S*.

_{7}## 5. Conclusions

*S*and

_{6}*S*T-M structures, respectively. Even if it is not possible to obtain an omidirectional PBG with a T-M structures as in the case of Bragg mirrors [5

_{7}5. A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystal,” Appl. Phys. Lett. **82**, 3227 (2003). [CrossRef]

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | S. M. Weiss, M. Haurylau, and P. M. Fauchet, “Tunable photonic bandgap structures for optical interconnects,” Opt. Mater. |

4. | C. M. Soukoulis and E. N. Economou, “Localization in one-dimensional lattices in the presence of incommensurate potentials,” Phys. Rev. Lett. |

5. | A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystal,” Appl. Phys. Lett. |

6. | V. Agarwal and J. A. del Rio, “Tailoring the photonic band gap of a porous silicon dielectric mirror,” Appl. Phys. Lett. |

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

8. | J. M. Luck, “Cantor spectra and scaling of gap widths in deterministic aperiodic systems,” Phys. Rev. B |

9. | M. Dulea, M. Severin, and R. Riklund, “Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,” Phys. Rev. B |

10. | L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc, and J. Haavisto, “Photon band gap properties and omnidirectional reflectance in Si/SiO |

11. | F. Qui, R. W. Peng, X. Q. Huang, X. F. Hu, Mu Wang, A. Hu, S. S. Jiang, and D. Feng, “Omnidirectional reflection of electromagnetic waves on Thue-Morse dielectric multilayers,” Europhys. Lett. |

12. | L. Canham, |

13. | L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, M. Righini, L. Colocci, and D. Wiersma, “Light transport through the band-edge states of Fibonacci Quasicrystals,” Phys. Rev. Lett. |

14. | V. Agarwal, J. A. Soto-Urueta, D. Becera, and M. E. Mora-Ramos, “Light propagation in polytype Thue-Morse structures made of porous silicon,” Photonics Nanostruct. Fundam. |

15. | M. Born and E. Wolf, |

16. | J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E |

17. | X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, “Photonic band gaps and localization in the Thue-Morse structures,” Appl. Phys. Lett. |

**OCIS Codes**

(130.0250) Integrated optics : Optoelectronics

(160.4760) Materials : Optical properties

(230.4170) Optical devices : Multilayers

**ToC Category:**

Optical Devices

**History**

Original Manuscript: April 14, 2006

Revised Manuscript: June 13, 2006

Manuscript Accepted: June 14, 2006

Published: June 26, 2006

**Citation**

Luigi Moretti, Ilaria Rea, Lucia Rotiroti, Ivo Rendina, Giancarlo Abbate, Antigone Marino, and Luca De Stefano, "Photonic band gaps analysis of Thue-Morse multilayers made of porous silicon," Opt. Express **14**, 6264-6272 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-13-6264

Sort: Year | Journal | Reset

### References

- E. Yablonovitch, "Inhibited spontaneous emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987). [CrossRef] [PubMed]
- S. M. Weiss, M. Haurylau, P. M. Fauchet, "Tunable photonic bandgap structures for optical interconnects," Opt. Mater. 27, 740 (2005). [CrossRef]
- C. M. Soukoulis and E. N. Economou, "Localization in one-dimensional lattices in the presence of incommensurate potentials," Phys. Rev. Lett. 48, 1043 (1982). [CrossRef]
- A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, "All-silicon omnidirectional mirrors based on onedimensional photonic crystal," Appl. Phys. Lett. 82, 3227 (2003). [CrossRef]
- V. Agarwal and J. A. del Rio, "Tailoring the photonic band gap of a porous silicon dielectric mirror," Appl. Phys. Lett. 82, 1512 (2003). [CrossRef]
- N. Liu, "Propagation of light waves in Thue-Morse dielectric multilayers," Phys. Rev. B 55, 3543 (1997). [CrossRef]
- J. M. Luck, "Cantor spectra and scaling of gap widths in deterministic aperiodic systems," Phys. Rev. B 39, 5834 (1989). [CrossRef]
- M. Dulea, M. Severin and R. Riklund, "Transmission of light through deterministic aperiodic non-Fibonaccian multilayers," Phys. Rev. B 42,3680 (1990). [CrossRef]
- L. Dal Negro, M. Stolfi, Y. Yi, J. Michel, X. Duan, L. C. Kimerling, J. LeBlanc and J. Haavisto, "Photon band gap properties and omnidirectional reflectance in Si/SiO2 Thue-Morse quasicrystals," Appl. Phys. Lett. 84, 5186 (2004). [CrossRef]
- F. Qui, R. W. Peng, X. Q. Huang, X. F. Hu, Mu Wang, A. Hu, S. S. Jiang and D. Feng, "Omnidirectional reflection of electromagnetic waves on Thue-Morse dielectric multilayers," Europhys. Lett. 68, 658-663 (2004). [CrossRef]
- L. Canham, Properties of Porous Silicon, (London IEE-INSPEC, 1997).
- L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, M. Righini, L. Colocci, and D. Wiersma, "Light transport through the band-edge states of Fibonacci Quasicrystals," Phys. Rev. Lett. 90, 55501 (2003). [CrossRef]
- V. Agarwal, J. A. Soto-Urueta, D. Becera, M. E. Mora-Ramos, "Light propagation in polytype Thue-Morse structures made of porous silicon," Photonics Nanostruct. Fundam. 3, 155 (2005). [CrossRef]
- M. Born, E. Wolf, Principles of Optics, (Cambridge University Press, New York, 2001).
- J. M. Bendickson, J. P. Dowling, M. Scalora, "Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures," Phys. Rev. E 53, 4107 (1996). [CrossRef]
- X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, "Photonic band gaps and localization in the Thue-Morse structures," Appl. Phys. Lett. 86, 201110 (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.