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A mid-infrared tunable filter in a semiconductor–dielectric photonic crystal containing doped semiconductor defect
Solid State Communications 151 (2011) 1677–1680
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A mid-infrared tunable filter in a semiconductor–dielectric photonic crystal containing doped semiconductor defect Hui-Chuan Hung a , Chien-Jang Wu b,∗ , Shoou-Jinn Chang a a
Institute of Microelectronics and Department of Electrical Engineering, Center for Micro/Nano Science and Technology, and Advance Optoelectronic Technology Center, National Cheng Kung University, Tainan 701, Taiwan b
Institute of Electro-Optical Science and Technology, National Taiwan Normal University, Taipei 116, Taiwan
article
info
Article history: Received 20 July 2011 Accepted 2 August 2011 by Y.E. Lozovik Available online 7 August 2011 Keywords: A. Semiconductors A. Photonic crystals D. Wave properties
abstract In this work, we theoretically analyze tunable filtering properties in a semiconductor–dielectric photonic crystal (SDPC) containing doped semiconductor defect in the mid-infrared frequency region. We consider two possible configurations of filter structures, the symmetric and asymmetric ones. With a defect of the doped n-type semiconductor, n-Si, the resonant transmission peak can be tuned by varying the doping concentration, that is, the peak wavelength will be shifted to the position of lower wavelength for both structures. Additionally, by increasing the defect thickness, it is also possible to have a filter with multiple resonant peaks, leading to a multichannel filter. The results provide another type of tunable filter in the defective SDPC that could be of technical use for semiconductor applications in optical electronics. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction With their inherent physical and optical properties as well as important photonic applications, the research on photonic crystals (PCs) have been attractive in the optical and physical communities [1–6]. PCs are periodic in structure and, conventionally, made of dielectric materials. In these all-dielectric PCs, there exist some photonic band gaps (PBGs) in which the propagation of electromagnetic waves is forbidden when the frequencies of waves fall inside the PBGs. These PBGs are also called the Bragg gaps because they originate from the Bragg scattering in the periodic structure. For a simple one-dimensional (1D) PC, the size of the PBG can be widened by increasing the index contrast in the constituent layers. In addition, the gap size is also dependent on the angle of incidence for both the transverse electric (TE) and transverse magnetic (TM) waves [7]. The properties of PBGs in 1D PCs have been proven to play an important role in some promising applications [8–11]. One of the familiar applications in PBGs is to engineer it for realizing a narrowband transmission filter. This can be achieved by adding a defect layer into the PC such that the structurally periodic feature is broken. In a 1D PC, there will be two possible filter structures, i.e., (H /L)Np D(L/H )Np and (H /L)Np D(H /L)Np where H and L are the high- and low-index layers, Np is the number of periods for the periodic bilayers, and D is the defect layer. The
∗
Corresponding author. Tel.: +886 2 77346724; fax: +886 2 86631954. E-mail address: [email protected] (C.-J. Wu).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.08.005
first structure (H /L)Np D(L/H )Np is referred to as a symmetric filter and the second filter (H /L)Np D(H /L)Np is asymmetric. With the addition of defect layer, resonant transmission peaks (also called the defect modes) can be generated within the PBG [7,12,13]. From the application point of view, filters with tunable feature are of more promising in optical electronics and communications. There are several methods that can be used to make the filters tunable. For instance, with the use of electric-field-dependent defect, filters can be tuned by the externally applied electric field, which is called E-tuning [14–17]. Filters can also be tuned by the operating temperature T (called T -tuning) if the defect layer permittivity is a function of temperature. Filters with defect materials like superconductors, liquid crystals or semiconductors belong to this type [18–20]. In this work, focusing on the host SDPC of (Si/SiO2 )Np , we shall consider another type of filter tuning based on the use of an extrinsic semiconductor as a defect layer, that is, the defect is a doped semiconductor of n-type silicon (n-Si). Because the tuning agent is the doping concentration N, the filter is a type of N-tuning. We will show that the position of resonant peak can be shifted by changing N. For the purpose of comparison, both symmetric and asymmetric filters will be used to investigate the tunable behaviors. The effect of defect thickness on the number of transmission peaks will also be illustrated. This SDPC tunable filter is designed to be operated in the mid-infrared frequency region. The analysis in this work could be of technical use in the semiconductor optoelectronics.
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2. Basic equations In this work, the optical filtering properties in a defective SDPC will be investigated through the transmittance response calculated by making use of the transfer matrix method (TMM) [21]. In what follows, we take the temporal part as exp(jωt ) for all optical fields. According to TMM, in a one-dimensional multilayer system, the transmittance T is related to the transmission coefficient t by 1
T = |t |2 =
|M11 |2
,
(1)
where M11 is one of the elements of the total system matrix M. In symmetric filter, Air/(H /L)Np D(L/H )Np /Air = Air/(Si/SiO2 )Np / n-Si/(SiO2 /Si)Np /Air, M can be expressed as
M =
M11 M21
M12 M22
Fig. 1. Calculated wavelength-dependent transmittance for the PC of (Si/SiO2 )Np , in which the thicknesses of Si and SiO2 , dH = dL = 0.3 µm, and Np = 10 are used.
1 −1 N p 1 = DA−1 DH PH D− (DD PD D− H DL PL DL D ) 1 −1 N p × (DL PL D− L DH PH DH ) DA .
(2)
In asymmetric filter, Air/(H /L)Np D(H /L)Np /Air SiO2 )Np /n-Si/(Si/SiO2 )Np /Air, M is given by 1 −1 M = DA−1 DH PH D− H DL PL DL
Np
1 DD PD D− D
= Air/(Si/
1 −1 N p × DH PH D− DA . H DL PL DL
(3)
In Eqs. (2) and (3), the subscripts H , L denotes the high- and low-index layers of Si and SiO2 , D is for the defect layer of doped semiconductor n-Si, and A represents the air, respectively. In addition, the propagation matrix in layer i, Pi (i = H , L, and D), is expressed as exp (jki di ) 0
Pi =
0
exp (−jki di )
,
(4)
√ where di is the thickness and ki = ni ω/c = εi ω/c is the wave number in layer i, where c is the speed√ of light in vacuum. Here, the refractive index is taken to be ni = εi because, in this work, all materials are non-magnetic and thus the relative permeability µi = 1. In addition, Dq (q = A, H , L, and D) is the dynamical matrix in medium q which is written as
Dq =
1 nq
1 −n q
,
(5)
where q = A is for air with nA = 1. The dynamical matrix in Eq. (5) is only for the normal incidence, which is the case we are interested in. With the use of n-Si as a defect layer, we now describe the permittivity for such a doped semiconductor. In the infrared region, the relative permittivity of n-Si is complex-valued and can be simply described by the plasma model, namely [22]
εD (ω) = ε∞ 1 −
2 ωpe
ω2 − jωγe
−
2 ωph
ω2 − jωγh
,
(6)
where ε∞ is the high-frequency limit of the relative permittivity, γe and γh are the damping frequencies for electrons and holes, respectively, and ωpe , ωph are the electron and hole plasma frequencies given by
ωpe,h =
ne,h e2 me,h ε0
.
(7)
Here, me,h are the effective masses of electron and hole, ne,h are the electron concentration in the conduction band and the hole concentration in the valence band, respectively. The carrier concentrations can be determined from the conservation of charge, with the result
Fig. 2. The relative permittivity of n-Si versus wavelength for different doping impurity concentrations of N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 22.4 × 1018 cm−3 in the case of zero damping frequencies, γe = γh = 0.
ne,h =
n2i +
N2 4
±
N 2
,
(8)
where ni is the intrinsic electron concentration of Si, N is the doping donor impurity concentration, and the sign ‘‘+’’ represents the electron concentration whereas the hole concentration is taken for the sign ‘‘−’’. It can be seen from Eqs. (6)–(8) that the permittivity of n-Si is strongly dependent on the doping concentration N. As a result, an N-tuning feature in the optical response can be obtainable when this extrinsic semiconductor is incorporated in the system. 3. Numerical results and discussion Let us first demonstrate the mid-infrared band gap structure for host defective PC of (H /L)Np = (Si/SiO2 )Np embedded in air. The calculated wavelength-dependent transmittance is shown in Fig. 1, in which nH = 3.3 (Si), nL = 1.46 (SiO2 ), dH = dL = 0.3 µm, and Np = 10 are taken. It can be seen that there exists a PBG in the mid-infrared region with the left and right band edges at λL = 2.43 µm and λR = 3.69 µm, respectively. The width of PBG is ∆ = λR − λL = 1.26 µm and the center wavelength is λc = 3.06 µm. This PBG is known as a Bragg gap which can be widened as the index contrast nH /nL increases. Our goal is to engineer this PBG for designing a tunable transmission filter by capitalizing on n-Si as a defect layer. Before we present the filtering properties in the defective PC, it is worth understanding the relative permittivity of n-Si given in Eq. (6). In Fig. 2, we plot the mid-infrared permittivity at different donor impurity concentrations of N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 21.3×1018 cm−3 , respectively. Here, without loss of generality, the damping frequencies, γe = γh = 0, are taken. In addition, the material parameters used in this calculation are ε∞ = 1, ni = 1.02 × 1010 cm−3 , me = 0.26m0 , mh = 0.69m0 , where m0 = 9.1 × 10−31 Kg is the mass of free electron. It can be seen that, except at a low concentration of N = 1 × 1018 cm−3 , the permittivity is a decreasing function of wavelength at a fixed concentration. At a
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Fig. 4. The dependence of peak wavelength on the doping concentration at a fixed defect thickness of dD = 0.8 µm. The peak wavelength decreases as the concentration increases.
Fig. 3. Calculated transmittance vs. wavelength for symmetric filter at N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 20 × 1018 cm−3 . Resonant transmission peak can be found inside the PBG of the host PC in Fig. 1.
fixed wavelength, the permittivity will decrease when the doping concentration increases. In addition, at N = 1 × 1018 , 5 × 1018 , and 10 × 1018 cm−3 , the overall permittivities in the region of interest are positive with values being small than one. At N = 21.3 × 1018 cm−3 , the permittivity will be zero at 3.69 µm, the right band edge of the PBG in Fig. 1. The permittivity at this doping concentration can be negative for wavelengths higher than this band edge. The negative-permittivity n-Si behaves like a metallic material in that region. We now present the filtering properties in a filter which is made by engineering the PBG in Fig. 1. We first consider the symmetric filter of Air/(Si/SiO2 )Np /n-Si/(SiO2 /Si)Np /Air, in which the previous parameters in Si and SiO2 are used and the thickness of defect (n-Si) is taken to be dD = 0.8 µm. The calculated transmittance is shown in Fig. 3. Here, four distinct doping concentrations, N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 20 × 1018 cm−3 , are taken for our simulation. It is seen that there is a single resonant peak within the PBG. The peak wavelengths for these four cases are 3.078, 3.002, 2.916, and 2.770 µm, respectively. At a low concentration of 1 × 1018 cm−3 , the resonant peak (3.078 µm) is near the gap center (λc = 3.06 µm). The peak is then shifted to the short wavelength as the concentration increases. The shifting behavior comes from the decrease in the permittivity at the higher concentration as illustrated in Fig. 2. In addition, we also find that the shift trend is only appreciable when the concentration is higher. The peak will, in fact, be moved to coincide with the left band edge of PBG at N = 77 × 1018 cm−3 , leading to the disappearance of the resonant peak. The dependence of peak wavelength on the doping concentration is illustrated in Fig. 4. The results in Figs. 3 and 4 suggest that there is an upper limit in the doping concentration in order to engineer the band gap to the design of a tunable filter. Now, focusing on a fixed doping concentration, we would like to investigate the effect of defect thickness on the filtering properties in the symmetric filter. In Fig. 5, we plot the calculated transmittance spectra at a fixed N = 1 × 1018 cm−3 for three different thicknesses of the defect layer, i.e., dD = 5, 10, and 20 µm. Comparing with first panel of Fig. 3 (where dD = 0.8 µm and only one resonant peak appears), we see that the number of peaks has been significantly increased as dD increases. There are three, four, and seven resonant peaks inside the PBG when dD = 5, 10, and 20 µm, respectively. The three peak wavelengths at dD = 5 µm are found to be 2.51, 2.90 and 3.57 µm, respectively.
Fig. 5. Calculated transmittance vs. wavelength at a fixed N = 1 × 1018 cm−3 for three different thicknesses of defect layer. The number of resonant transmission peaks increases as defect thickness increases.
The four peak wavelengths are 2.50, 2.72, 3.03 and 3.45 µm for dD = 10 µm. And seven peaks are at 2.46, 2.62, 2.76, 2.94, 3.14, 3.37, and 3.63 µm when dD = 20 µm. Comparing with the filter with a single channel, the appearance of multiple transmission peaks can work as a multichannel filter, which can greatly enhance the spectral efficiency in the use of PBG. In addition, a multichannel filter is of particular use in signal processing such as the wavelength-selective (or frequency-selective) filter, the wavelength-division multiplexer (WDM) system. Filters with multiple transmission peaks have attracted much attention in recent years [23]. In general, the multiple filtering phenomenon can be obtained by using the photonic quantum well (PQW) as a defect in the host PC, which was first reported by Qiao et al. [10]. The results in Fig. 5 reveal that a multichannel transmission filter can also be obtained directly by
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its filtering properties have been analyzed. We have investigated the filtering properties in both the symmetric and asymmetric structures. With a strongly concentration-dependent permittivity of n-Si, a lower value of permittivity is attained at a higher concentration, which, in turn, causes the resonant transmission peaks to be moved to the positions of lower wavelength. We have also demonstrated how to increase the number of resonant peaks by widening the thickness of the defect layer. Increasing the number of transmission peaks can be used to function as a multichannel filter which is of technical use in the optical signal processing. Acknowledgments C.-J. Wu acknowledges the financial support from the National Science Council of the Republic of China (Taiwan) under Contract No. NSC-97-2112-M-003-013-MY3. Fig. 6. Calculated transmittance vs. wavelength at N = 1×1018 , 5×1018 , 10×1018 , and 20 × 1018 cm−3 for the asymmetric filter. Only one resonant transmission peak exists inside the PBG.
controlling the thickness of the defect layer instead of using the idea of using PQW. In the above discussion, we have paid our attention to the symmetric filter. We now investigate the asymmetric one, Air/ (Si/SiO2 )Np /n-Si/(Si/SiO2 )Np /Air. With all the same previous material parameters, the calculated transmittance at N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 20 × 1018 cm−3 are plotted in Fig. 6. It can be seen that there is also a single peak within the PBG in the asymmetric filter. The peak wavelengths at these four conditions are equal to 2.74, 2.70, 2.65 and 2.57 µm, respectively. The shifting behavior is the same as that in the symmetric one, i.e., the peak is shifted to the left when the doping concentration increases. The difference between asymmetric and symmetric filters is that, at N = 1 × 1018 cm−3 , the peak is not located near the PBG center, but to the very left of the gap center. Usually, the PBG center wavelength is a reference wavelength when we design a filter based on the use of the PBG. Thus, for a typical filter design, it is preferable to use the symmetric structure because the peak can initially be designed in the vicinity of the gap center. 4. Conclusion Based on the use of SDPC with a doped semiconductor defect, an N-tuning filter operated in the mid-infrared has been designed and
References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [4] S. Fan, P.R. Villeneuve, J.D. Joanopoulos, H.A. Haus, Phys. Rev. Lett. 80 (1998) 960. [5] S. Noda, A. Chutinan, M. Imada, Nature 407 (2000) 608. [6] P. Lodahl, A. Floris van Driel, I.S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelberg, W.L. Vos, Nature 430 (2004) 654. [7] S.J. Orfanidis, Electromagnetic waves and antennas, Rutger University, 2008 (Chapter 7) www.ece.rutgers.edu/~orfanidi/ewa. [8] Y.H. Chen, J. Opt. Soc. Amer. B 26 (2009) 854. [9] Q. Qin, H. Liu, S.N. Zhu, C.S. Yuan, Y.Y. Zhu, N.B. Ming, Appl. Phys. Lett. 82 (2003) 4654. [10] F. Qiao, C. Zhang, J. Wan, Appl. Phys. Lett. 77 (2000) 3698. [11] Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joannopoulos, L.E. Thomas, Science 282 (1998) 1679. [12] D.R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S.L. McCall, P.M. Platzman, J. Opt. Soc. Amer. B 10 (1993) 314. [13] C.-J. Wu, Z.-H. Wang, Prog. Electromagn. Res. 103 (2010) 169. [14] Y.K. Ha, Y.C. Yang, J.E. Kim, H.Y. Park, Appl. Phys. Lett. 79 (2001) 15. [15] Y.Q. Lu, J.J. Zheng, Appl. Phys. Lett. 74 (1999) 123. [16] Q. Zhu, Y. Zhang, Optik 120 (2009) 195–198. [17] C.-J. Wu, J.-J. Liao, T.-W. Chang, J. Electromagn. Waves Appl. 24 (2010) 531. [18] I.L. Lyubchanskii, N.N. Dadoenkova, A.E. Zabolotin, Y.P. Lee, Th. Rasing, J. Opt. A: Pure Appl. Opt. 11 (2009) 114014. [19] S.C. Howells, L.A. Schlie, Appl. Phys. Lett. 69 (1996) 550. [20] P. Halevi, F. Ramos-Mendieta, Phys. Rev. Lett. 85 (2000) 1875. [21] P. Yeh, Optical Waves in Layered Media, John Wiley & Sons, Singapore, 1991. [22] E. Galindo-Linares, P. Halevi, A.S. Sanchez, Solid State Commun. 142 (2007) 67. [23] W.-H. Lin, C.-J. Wu, T.-J. Yang, S.-J. Chang, Opt. Express 18 (2010) 27155.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
A mid-infrared tunable filter in a semiconductor–dielectric photonic crystal containing doped semiconductor defect Hui-Chuan Hung a , Chien-Jang Wu b,∗ , Shoou-Jinn Chang a a
Institute of Microelectronics and Department of Electrical Engineering, Center for Micro/Nano Science and Technology, and Advance Optoelectronic Technology Center, National Cheng Kung University, Tainan 701, Taiwan b
Institute of Electro-Optical Science and Technology, National Taiwan Normal University, Taipei 116, Taiwan
article
info
Article history: Received 20 July 2011 Accepted 2 August 2011 by Y.E. Lozovik Available online 7 August 2011 Keywords: A. Semiconductors A. Photonic crystals D. Wave properties
abstract In this work, we theoretically analyze tunable filtering properties in a semiconductor–dielectric photonic crystal (SDPC) containing doped semiconductor defect in the mid-infrared frequency region. We consider two possible configurations of filter structures, the symmetric and asymmetric ones. With a defect of the doped n-type semiconductor, n-Si, the resonant transmission peak can be tuned by varying the doping concentration, that is, the peak wavelength will be shifted to the position of lower wavelength for both structures. Additionally, by increasing the defect thickness, it is also possible to have a filter with multiple resonant peaks, leading to a multichannel filter. The results provide another type of tunable filter in the defective SDPC that could be of technical use for semiconductor applications in optical electronics. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction With their inherent physical and optical properties as well as important photonic applications, the research on photonic crystals (PCs) have been attractive in the optical and physical communities [1–6]. PCs are periodic in structure and, conventionally, made of dielectric materials. In these all-dielectric PCs, there exist some photonic band gaps (PBGs) in which the propagation of electromagnetic waves is forbidden when the frequencies of waves fall inside the PBGs. These PBGs are also called the Bragg gaps because they originate from the Bragg scattering in the periodic structure. For a simple one-dimensional (1D) PC, the size of the PBG can be widened by increasing the index contrast in the constituent layers. In addition, the gap size is also dependent on the angle of incidence for both the transverse electric (TE) and transverse magnetic (TM) waves [7]. The properties of PBGs in 1D PCs have been proven to play an important role in some promising applications [8–11]. One of the familiar applications in PBGs is to engineer it for realizing a narrowband transmission filter. This can be achieved by adding a defect layer into the PC such that the structurally periodic feature is broken. In a 1D PC, there will be two possible filter structures, i.e., (H /L)Np D(L/H )Np and (H /L)Np D(H /L)Np where H and L are the high- and low-index layers, Np is the number of periods for the periodic bilayers, and D is the defect layer. The
∗
Corresponding author. Tel.: +886 2 77346724; fax: +886 2 86631954. E-mail address: [email protected] (C.-J. Wu).
0038-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2011.08.005
first structure (H /L)Np D(L/H )Np is referred to as a symmetric filter and the second filter (H /L)Np D(H /L)Np is asymmetric. With the addition of defect layer, resonant transmission peaks (also called the defect modes) can be generated within the PBG [7,12,13]. From the application point of view, filters with tunable feature are of more promising in optical electronics and communications. There are several methods that can be used to make the filters tunable. For instance, with the use of electric-field-dependent defect, filters can be tuned by the externally applied electric field, which is called E-tuning [14–17]. Filters can also be tuned by the operating temperature T (called T -tuning) if the defect layer permittivity is a function of temperature. Filters with defect materials like superconductors, liquid crystals or semiconductors belong to this type [18–20]. In this work, focusing on the host SDPC of (Si/SiO2 )Np , we shall consider another type of filter tuning based on the use of an extrinsic semiconductor as a defect layer, that is, the defect is a doped semiconductor of n-type silicon (n-Si). Because the tuning agent is the doping concentration N, the filter is a type of N-tuning. We will show that the position of resonant peak can be shifted by changing N. For the purpose of comparison, both symmetric and asymmetric filters will be used to investigate the tunable behaviors. The effect of defect thickness on the number of transmission peaks will also be illustrated. This SDPC tunable filter is designed to be operated in the mid-infrared frequency region. The analysis in this work could be of technical use in the semiconductor optoelectronics.
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2. Basic equations In this work, the optical filtering properties in a defective SDPC will be investigated through the transmittance response calculated by making use of the transfer matrix method (TMM) [21]. In what follows, we take the temporal part as exp(jωt ) for all optical fields. According to TMM, in a one-dimensional multilayer system, the transmittance T is related to the transmission coefficient t by 1
T = |t |2 =
|M11 |2
,
(1)
where M11 is one of the elements of the total system matrix M. In symmetric filter, Air/(H /L)Np D(L/H )Np /Air = Air/(Si/SiO2 )Np / n-Si/(SiO2 /Si)Np /Air, M can be expressed as
M =
M11 M21
M12 M22
Fig. 1. Calculated wavelength-dependent transmittance for the PC of (Si/SiO2 )Np , in which the thicknesses of Si and SiO2 , dH = dL = 0.3 µm, and Np = 10 are used.
1 −1 N p 1 = DA−1 DH PH D− (DD PD D− H DL PL DL D ) 1 −1 N p × (DL PL D− L DH PH DH ) DA .
(2)
In asymmetric filter, Air/(H /L)Np D(H /L)Np /Air SiO2 )Np /n-Si/(Si/SiO2 )Np /Air, M is given by 1 −1 M = DA−1 DH PH D− H DL PL DL
Np
1 DD PD D− D
= Air/(Si/
1 −1 N p × DH PH D− DA . H DL PL DL
(3)
In Eqs. (2) and (3), the subscripts H , L denotes the high- and low-index layers of Si and SiO2 , D is for the defect layer of doped semiconductor n-Si, and A represents the air, respectively. In addition, the propagation matrix in layer i, Pi (i = H , L, and D), is expressed as exp (jki di ) 0
Pi =
0
exp (−jki di )
,
(4)
√ where di is the thickness and ki = ni ω/c = εi ω/c is the wave number in layer i, where c is the speed√ of light in vacuum. Here, the refractive index is taken to be ni = εi because, in this work, all materials are non-magnetic and thus the relative permeability µi = 1. In addition, Dq (q = A, H , L, and D) is the dynamical matrix in medium q which is written as
Dq =
1 nq
1 −n q
,
(5)
where q = A is for air with nA = 1. The dynamical matrix in Eq. (5) is only for the normal incidence, which is the case we are interested in. With the use of n-Si as a defect layer, we now describe the permittivity for such a doped semiconductor. In the infrared region, the relative permittivity of n-Si is complex-valued and can be simply described by the plasma model, namely [22]
εD (ω) = ε∞ 1 −
2 ωpe
ω2 − jωγe
−
2 ωph
ω2 − jωγh
,
(6)
where ε∞ is the high-frequency limit of the relative permittivity, γe and γh are the damping frequencies for electrons and holes, respectively, and ωpe , ωph are the electron and hole plasma frequencies given by
ωpe,h =
ne,h e2 me,h ε0
.
(7)
Here, me,h are the effective masses of electron and hole, ne,h are the electron concentration in the conduction band and the hole concentration in the valence band, respectively. The carrier concentrations can be determined from the conservation of charge, with the result
Fig. 2. The relative permittivity of n-Si versus wavelength for different doping impurity concentrations of N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 22.4 × 1018 cm−3 in the case of zero damping frequencies, γe = γh = 0.
ne,h =
n2i +
N2 4
±
N 2
,
(8)
where ni is the intrinsic electron concentration of Si, N is the doping donor impurity concentration, and the sign ‘‘+’’ represents the electron concentration whereas the hole concentration is taken for the sign ‘‘−’’. It can be seen from Eqs. (6)–(8) that the permittivity of n-Si is strongly dependent on the doping concentration N. As a result, an N-tuning feature in the optical response can be obtainable when this extrinsic semiconductor is incorporated in the system. 3. Numerical results and discussion Let us first demonstrate the mid-infrared band gap structure for host defective PC of (H /L)Np = (Si/SiO2 )Np embedded in air. The calculated wavelength-dependent transmittance is shown in Fig. 1, in which nH = 3.3 (Si), nL = 1.46 (SiO2 ), dH = dL = 0.3 µm, and Np = 10 are taken. It can be seen that there exists a PBG in the mid-infrared region with the left and right band edges at λL = 2.43 µm and λR = 3.69 µm, respectively. The width of PBG is ∆ = λR − λL = 1.26 µm and the center wavelength is λc = 3.06 µm. This PBG is known as a Bragg gap which can be widened as the index contrast nH /nL increases. Our goal is to engineer this PBG for designing a tunable transmission filter by capitalizing on n-Si as a defect layer. Before we present the filtering properties in the defective PC, it is worth understanding the relative permittivity of n-Si given in Eq. (6). In Fig. 2, we plot the mid-infrared permittivity at different donor impurity concentrations of N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 21.3×1018 cm−3 , respectively. Here, without loss of generality, the damping frequencies, γe = γh = 0, are taken. In addition, the material parameters used in this calculation are ε∞ = 1, ni = 1.02 × 1010 cm−3 , me = 0.26m0 , mh = 0.69m0 , where m0 = 9.1 × 10−31 Kg is the mass of free electron. It can be seen that, except at a low concentration of N = 1 × 1018 cm−3 , the permittivity is a decreasing function of wavelength at a fixed concentration. At a
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Fig. 4. The dependence of peak wavelength on the doping concentration at a fixed defect thickness of dD = 0.8 µm. The peak wavelength decreases as the concentration increases.
Fig. 3. Calculated transmittance vs. wavelength for symmetric filter at N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 20 × 1018 cm−3 . Resonant transmission peak can be found inside the PBG of the host PC in Fig. 1.
fixed wavelength, the permittivity will decrease when the doping concentration increases. In addition, at N = 1 × 1018 , 5 × 1018 , and 10 × 1018 cm−3 , the overall permittivities in the region of interest are positive with values being small than one. At N = 21.3 × 1018 cm−3 , the permittivity will be zero at 3.69 µm, the right band edge of the PBG in Fig. 1. The permittivity at this doping concentration can be negative for wavelengths higher than this band edge. The negative-permittivity n-Si behaves like a metallic material in that region. We now present the filtering properties in a filter which is made by engineering the PBG in Fig. 1. We first consider the symmetric filter of Air/(Si/SiO2 )Np /n-Si/(SiO2 /Si)Np /Air, in which the previous parameters in Si and SiO2 are used and the thickness of defect (n-Si) is taken to be dD = 0.8 µm. The calculated transmittance is shown in Fig. 3. Here, four distinct doping concentrations, N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 20 × 1018 cm−3 , are taken for our simulation. It is seen that there is a single resonant peak within the PBG. The peak wavelengths for these four cases are 3.078, 3.002, 2.916, and 2.770 µm, respectively. At a low concentration of 1 × 1018 cm−3 , the resonant peak (3.078 µm) is near the gap center (λc = 3.06 µm). The peak is then shifted to the short wavelength as the concentration increases. The shifting behavior comes from the decrease in the permittivity at the higher concentration as illustrated in Fig. 2. In addition, we also find that the shift trend is only appreciable when the concentration is higher. The peak will, in fact, be moved to coincide with the left band edge of PBG at N = 77 × 1018 cm−3 , leading to the disappearance of the resonant peak. The dependence of peak wavelength on the doping concentration is illustrated in Fig. 4. The results in Figs. 3 and 4 suggest that there is an upper limit in the doping concentration in order to engineer the band gap to the design of a tunable filter. Now, focusing on a fixed doping concentration, we would like to investigate the effect of defect thickness on the filtering properties in the symmetric filter. In Fig. 5, we plot the calculated transmittance spectra at a fixed N = 1 × 1018 cm−3 for three different thicknesses of the defect layer, i.e., dD = 5, 10, and 20 µm. Comparing with first panel of Fig. 3 (where dD = 0.8 µm and only one resonant peak appears), we see that the number of peaks has been significantly increased as dD increases. There are three, four, and seven resonant peaks inside the PBG when dD = 5, 10, and 20 µm, respectively. The three peak wavelengths at dD = 5 µm are found to be 2.51, 2.90 and 3.57 µm, respectively.
Fig. 5. Calculated transmittance vs. wavelength at a fixed N = 1 × 1018 cm−3 for three different thicknesses of defect layer. The number of resonant transmission peaks increases as defect thickness increases.
The four peak wavelengths are 2.50, 2.72, 3.03 and 3.45 µm for dD = 10 µm. And seven peaks are at 2.46, 2.62, 2.76, 2.94, 3.14, 3.37, and 3.63 µm when dD = 20 µm. Comparing with the filter with a single channel, the appearance of multiple transmission peaks can work as a multichannel filter, which can greatly enhance the spectral efficiency in the use of PBG. In addition, a multichannel filter is of particular use in signal processing such as the wavelength-selective (or frequency-selective) filter, the wavelength-division multiplexer (WDM) system. Filters with multiple transmission peaks have attracted much attention in recent years [23]. In general, the multiple filtering phenomenon can be obtained by using the photonic quantum well (PQW) as a defect in the host PC, which was first reported by Qiao et al. [10]. The results in Fig. 5 reveal that a multichannel transmission filter can also be obtained directly by
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its filtering properties have been analyzed. We have investigated the filtering properties in both the symmetric and asymmetric structures. With a strongly concentration-dependent permittivity of n-Si, a lower value of permittivity is attained at a higher concentration, which, in turn, causes the resonant transmission peaks to be moved to the positions of lower wavelength. We have also demonstrated how to increase the number of resonant peaks by widening the thickness of the defect layer. Increasing the number of transmission peaks can be used to function as a multichannel filter which is of technical use in the optical signal processing. Acknowledgments C.-J. Wu acknowledges the financial support from the National Science Council of the Republic of China (Taiwan) under Contract No. NSC-97-2112-M-003-013-MY3. Fig. 6. Calculated transmittance vs. wavelength at N = 1×1018 , 5×1018 , 10×1018 , and 20 × 1018 cm−3 for the asymmetric filter. Only one resonant transmission peak exists inside the PBG.
controlling the thickness of the defect layer instead of using the idea of using PQW. In the above discussion, we have paid our attention to the symmetric filter. We now investigate the asymmetric one, Air/ (Si/SiO2 )Np /n-Si/(Si/SiO2 )Np /Air. With all the same previous material parameters, the calculated transmittance at N = 1 × 1018 , 5 × 1018 , 10 × 1018 , and 20 × 1018 cm−3 are plotted in Fig. 6. It can be seen that there is also a single peak within the PBG in the asymmetric filter. The peak wavelengths at these four conditions are equal to 2.74, 2.70, 2.65 and 2.57 µm, respectively. The shifting behavior is the same as that in the symmetric one, i.e., the peak is shifted to the left when the doping concentration increases. The difference between asymmetric and symmetric filters is that, at N = 1 × 1018 cm−3 , the peak is not located near the PBG center, but to the very left of the gap center. Usually, the PBG center wavelength is a reference wavelength when we design a filter based on the use of the PBG. Thus, for a typical filter design, it is preferable to use the symmetric structure because the peak can initially be designed in the vicinity of the gap center. 4. Conclusion Based on the use of SDPC with a doped semiconductor defect, an N-tuning filter operated in the mid-infrared has been designed and
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