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Near-infrared tunable narrow filter properties in a 1D photonic crystal containing semiconductor metamaterial photonic quantum-well defect
Physica E 79 (2016) 20–25
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Near-infrared tunable narrow filter properties in a 1D photonic crystal containing semiconductor metamaterial photonic quantum-well defect Mahmood Barati n, Alireza Aghajamali Department of Physics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
H I G H L I G H T S
G R A P H I C A L
Study the tunable NIR narrow filters with semiconductor metamaterial PQW. The number of the defect mode is independent of the periods of the PQW. Defect mode's frequency gets redshifted as the filling factor increases. Defect mode's frequency gets blueshifted for both TE and TM waves.
We perform a theoretical investigation of the near-infrared narrow filter properties in the transmission spectrum of a one-dimensional photonic crystal doped with semiconductor metamaterial photonic quantum-well defect (PQW). It is found that the defect mode’s frequency can be tuned by the variations of the defect structure’s period, polarization, incidence angle, and also the filling factor corresponding to the semiconductor metamaterial layer. In addition, the number of the defect mode is independent of the periods of the PQW defect structure, which is in sharp contrast to the case of using usual dielectric or metamaterial defect.
art ic l e i nf o
a b s t r a c t
Article history: Received 26 October 2015 Received in revised form 5 December 2015 Accepted 12 December 2015 Available online 15 December 2015
The near-infrared (NIR) narrow filter properties in the transmission spectra of a one-dimensional photonic crystal doped with semiconductor metamaterial photonic quantum-well defect (PQW) were theoretically studied. The behavior of the defect mode as a function of the stack number of the PQW defect structure, the filling factor of semiconductor metamaterial layer, the polarization and the angle of incidence were investigated for Al-doped ZnO (AZO) and ZnO as the semiconductor metamaterial layer. It is found that the frequency of the defect mode can be tuned by variation of the period of the defect structure, polarization, incidence angle, and the filling factor of the semiconductor metamaterial layer. It is also shown that the number of the defect mode is independent of the period of the PQW defect structure and is in sharp contrast with the case where a common dielectric or metamaterial defect are used. The results also show that for both polarizations the defect mode is red-shifted as the number of the defect period and filling factor increase. An opposite trend is observed as the angle of incidence increases. The proposed structure could provide useful information for designing new types of tuneable narrowband filters at NIR region. & 2015 Published by Elsevier B.V.
Keywords: 1D photonic crystal Defect mode Semiconductor metamaterial Photonic quantum-well defect
A B S T R A C T
1. Introduction Photonic crystals (PCs), possessing some significant optical properties, are artificial materials having periodic multilayer structure in dielectric constant. One of the most interesting optical properties of PCs is existence of forbidden frequency regions in n
Corresponding author. E-mail address: [email protected] (M. Barati).
http://dx.doi.org/10.1016/j.physe.2015.12.012 1386-9477/& 2015 Published by Elsevier B.V.
their transmission spectra. The regions known as photonic band gaps (PBGs) or Bragg gaps [1–4]. The possibility of producing metamaterials with negative permittivity and permeability in the past decade [5], the optical properties of PCs with metamaterials known as metamaterial photonic crystals (MetaPCs) have been attracted by many authors [6–8]. In recent years MetaPCs have attracted extensive attention of many researchers for their unique electromagnetic properties and their scientific and microwave engineering applications [9–27]. PCs composed of semiconductors
M. Barati, A. Aghajamali / Physica E 79 (2016) 20–25
have received considerable attention for their several superior features such as tunable PBG which is achieved by thermal tuning of the dielectric function of the semiconductor material which is applicable in optoelectronic investigations. Another feature of semiconductor PCs, is appearance of their PBG in the near-infrared (NIR) to mid-IR frequency regions [28]. In 2012, Naik et al. [29] reported a new semiconductor-based metamaterial. This so called semiconductor metamaterial is an artificially periodic structure made by alternating layers of Al-doped ZnO (AZO) and ZnO on a silicon substrate with 16 pair layers and 60 nm periodicity. This AZO–ZnO metamaterial has an anisotropic and negative permittivity in NIR region [30]. The presence of negative permittivity is reminiscent of metallic permittivity, which is negative for frequencies less than the plasma frequency. Thus, AZO–ZnO metamaterial can be classified as an epsilon-negative (ENG) material in the NIR region. A defected photonic crystal would be generated by breaking the periodicity of the conventional PC structure. In the defected structure, localized defect modes within the PBG would be generated as a result of a change in the interference behavior of waves at interfaces. A PC of (CD)M can be inserted in the middle of a host PC of (AB)N rather than a defect layer being added to the PC. Therefore, the structure will be (AB)N/2 (CD)M (AB)N/2, where N and M are the stack numbers of the (AB) and (CD) bilayers, respectively. In such case, the defected structure of (CD)M will be called the photonic-quantum-well (PQW) [30–33]. The structure (AB)N/2 (CD)M (AB)N/2 (with MoN) can present multiple filtering feature due to the photonic confinement, contributing to a perception of a multichannel filter [31]. Moreover, the number of channels that can be considered as the basis of the number of peaks in the transmission spectra is equal to the stack number of PQW. Despite the fact that there have been many reports on PQW-based filters [31–38], few studies have been examined in the tuneable multichannel transmission filters. In addition to the above mentioned theoretical studies, some experimental works have been reported [39,40]. The main purpose of this work is the theoretical investigation of the properties of the defect mode in a 1D PC doped by semiconductor metamaterial PQW defect. To this aim, we study the defect mode as a function of the stack number of the PQW defect structure, the filling factor of semiconductor metamaterial layer, the polarization, and the angle of incidence. Our numerical results show that the frequency of the defect modes can be tuned by the above-mentioned parameters as well, which are in sharp contrast with the case where a common dielectric or a metamaterial defect is used. The results also reveal that a new NIR tunable narrow filter can be achieved in this proposed structure. The outline of our paper is as follows: Section 2 presents a 1D PC containing a PQW defect structure, the characteristic matrix method and its formulation, and also the permittivity of semiconductor metamaterial, Section 3 reveals the numerical results and discussions associated with our purpose, and Section 4 describes the conclusions of the investigation.
21
2. Theoretical framework A 1D PC with asymmetric structure immersed in free space with a defect structure (PQW defect) at the center of the host PC is shown in Fig. 1. Layers A and B are respectively considered to be SiO2 (Silica) and InP (Indium Phosphide), layer C is AZO–ZnO (as semiconductor metamaterial), and layer D is air. The number of lattice periods and the number of unit cells (corresponding to the defect structure) are denoted by N and M respectively. In addition, the thickness, the permittivity, and the permeability of the layers are respectively assumed to be di , εi , and μi ( i = A , B , C , D ). First of all, let us describe the permittivity and the permeability functions of AZO–ZnO layer. The permittivity is anisotropic and is given by [30,41–45]:
⎛ ερ 0 0 ⎞ ⎜ ⎟ εC = ⎜ 0 ερ 0 ⎟ ⎜ ⎟ ⎝ 0 0 ευ ⎠
(1)
Here, ερ and ευ are the parallel and perpendicular components, respectively, and they are related to the permittivity functions of AZO and ZnO. The total permittivity functions of AZO–ZnO composite are expressed as
ερ = h εa + (1 − h) εb
(2)
and
ευ =
1 h εa−1 + (1 − h) εb−1
(3)
where h is the filling factor of AZO, h = da/(da + db ) and dC = da + db is the thickness of layer C, where da and db indicate the thickness of the AZO and ZnO layers, respectively. Moreover, the permittivity functions of AZO and ZnO are εa and εb , respectively. Here, εb is a constant and εa can be expressed as a combination of Lorentz and Drude models, that is [41–45]
εa (f ) = εa1 (f ) + εa2 (f ),
(4)
where the Lorentz part is given by
εa1 (f ) = 1 −
2 2 f ap − f ao 1 1 2 f 2 − f ao 1
, (5)
and the Drude part is
εa2 (f ) = 1 −
2 f ap2
f2
(6)
here, fap1, fao1, and fap2 are constants corresponding with three characteristic frequencies. As for the permeabilities of AZO and ZnO, they are taken to be unity because both materials are nonmagnetic. In performed calculations, we have applied the characteristic matrix
Fig. 1. Schematic diagram of one-dimensional defective photonic crystal immersed in free space with photonic quantum-well defect structure at the center. Where layers A and B corresponding to the main structure are respectively SiO2, InP, and also layers C and D corresponding to the defect structure are respectively are AZO–ZnO and air. θ0 is the incident angle and also N and M are respectively the number of the lattice periods of bilayers (AB) and (CD).
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M. Barati, A. Aghajamali / Physica E 79 (2016) 20–25
method [44,45], which is assumed to be the most effective technique to analyze the transmission properties of finite-periodic PCs. The characteristic matrix for asymmetric (AB )N /2 (CD)M (AB )N /2 defective structures is presented by: M [d] = (MA MB )N /2 (MC MD )M (MA MB )N /2, where MA , MB , MC and MD are the characteristic matrices of layers A, B, C and D. The characteristic matrix Mi is calculated through the following equation for the TE wave at the incidence angle θ0 from vacuum to a 1D PC structure [44,45]:
⎡ ⎤ −i sin γi ⎥ ⎢ cos γi p Mi = ⎢ i ⎥, ⎢⎣ − ip sin γ cos γi ⎥⎦ i i
(7)
In this equation, γi = (ω/c ) ni di cos θi , c is the speed of light in vacuum, θi is the ray angle inside layer i with refractive index ni ,
pi = εi /μi cos θi , and cos θi = 1 − (n02 sin2θ0/ni2 ) , in which n0 is the refractive index of the environment wherein the incidence wave tends to enter the structure. The refractive index is given as ni = ± εi μi , in which the positive sign is for the double-positive materials whereas the negative sign is for the negative index materials (NIMs). The characteristic matrix for N periods of structure is, therefore, [M (d )]N . The transmission coefficient of the multilayer system is calculated by:
2p0 t= (m11 + m12 ps ) p0 + (m21 + m22 ps )
(8)
In this equation, mij (i, j = 1, 2) are the matrix elements of
[M (d )]N , p0 = n0 cos θ0 , and ps = ns cos θs , where ns is the refractive index of the environment whose ray angle is θs . The transmissivity of the multilayer is given by T = (ps /p0 ) t 2. The transmissivity of the multilayer for TM wave can be similarly obtained by pi = μi /εi cos θi , p0 = cos θ0/n0 , and ps = cos θs/ns .
3. Numerical results and discussion In our calculations, the material parameters used for the basic equations described in the previous section are fap1 = 180 THz, fap2 = 150 THz, fao1 = 80 THz, nA = 1.45 (SiO2), nB = 3.52 (InP), and nD = 1 (vacuum). The thickness of layers A, B, C, and D are chosen as dA ¼4.54 mm, dB ¼1.22 mm,dC ¼60 nm, and dD ¼80 nm. Before we present the defect mode properties in the defective PC, it should be mentioned that: (a) as it is inferred from Eqs. (5) and (6) the semiconductor metamaterial is assumed to be lossless. (b) For some limitations in the computational calculations, the optimal total number of the lattice period of the main structure (N) should firstly be determined. In this regard the width of the band gap ( Δ) as a function of the total number of the unit cells (N) is calculated. As it is seen from Fig. 2, the band gap is nearly constant for N ≥ 20. Accordingly, the total number of the lattice periods in all numerical calculation is considered as N = 20. 3.1. Defect mode as a function of number of the PQW unit cell (M) The effect of the number of the PQW unit cell on the normal incidence transmission spectrum is reported. As it is seen from Fig. 3, the number of the defect mode appearing in the band gap is independent of the number of the PQW unit cells, where only one defect mode appears in the band gap for different values of M. This observation is in sharp contrast with the previous studies [31,32,35,38], where the number of the defect modes is reported to be the same as the number of the M cells. Another significant observation is that as M increases, the defect mode is red-shifted although the band gap remains nearly constant. This shifting property is indicative of a technical significance of the structure
Fig. 2. Width of the band gap versus the number of the lattice periods of the main structure.
which is quite useful in designing a narrow tuneable filter. 3.2. Defect mode as a function of filling factor (h) The shifting feature of the defect mode as a function of the filling factor of the semiconductor metamaterial, AZO–ZnO, is investigated. Fig. 4 shows the effect of the filling factor (h) on the position of the defect mode for the normal angle of incidence. It is clearly seen that as h increases, the defect mode is red-shifted, its width slightly increases where the height remains constant. Table 1 presents further detail of the changes in the position of the defect mode and the corresponding wavelength (λD). By comparing the numerical results, we observe that although the thickness of the AZO layer (da) has more salient influence on h, in compare with the ZnO thickness layer (db); on the other hand, both da and db have the same effect on the shifting feature of λD. Moreover, it is noticeable that by increasing the thickness of the AZO layer the defect mode starts red-shifting, but an opposite trend is observed as the ZnO thickness layer increases. Taking this unique result into consideration provides us with an opportunity to make a more practical tuneable narrow filter. 3.3. Defect mode as a function of the angle of incidence ( θ 0 ) We now turn our attention to the case of oblique incidence angle. Fig. 5 shows the transmission spectra for different angles of incidence for TE and TM waves. As shown in the figure, the defect mode and the band gap are blue-shifted as the angle of incidence increases. It is also clearly seen that, by increasing the angle of incidence, a noticeable rise in the width and depth of the band gap and a decline in the height of the defect mode is observed. The results for TE and TM waves also reveal that the depth of the band gap and the height of the defect mode are more sensitive for TE waves as θ 0 increases. This feature is sharply in contrast with the previous report by Xu et al. [35] who showed the defect mode was very weakly dependent on the incidence angle and polarization. The effects of the angle of incidence on the wavelength of the defect mode (λD) for TE and TM waves are shown in Fig. 6. It is seen that, by increasing the angle of incidence, the defect mode gets blue-shifted. Moreover, the shift for the TM modes is considerably more than that for the TE waves, which is in agreement with the previous report by Wu et al. [42] for semiconductor metamaterial defect layer at the center of a PC structure, i.e., (AB )N /2 (C )M (AB )N /2.
M. Barati, A. Aghajamali / Physica E 79 (2016) 20–25
23
Fig. 3. Transmission spectra for the 1D PC doped by PQW defect for different number of the PQW unit cell (M).
Table 1 The numerical results for the wavelength of the defect mode shown in Fig. 4 for different filling factor of semiconductor metamaterial, AZO–ZnO, and different thicknesses of the AZO and ZnO layers.
Fig. 4. Transmission spectra for the 1D PC doped by PQW defect for different filling factors of semiconductor metamaterial, AZO–ZnO.
4. Conclusion In this study, the properties of the defect mode of 1D PCs doped by semiconductor metamaterial PQW defect in NIR region for a defect structure composed of AZO–ZnO and air have been analyzed. Our numerical results show that the number of the defect
da (nm)
db (nm)
h
λD (mm)
55 50 45 40 35 30 25 20 15 10 5
5 10 15 20 25 30 35 40 45 50 55
0.92 0.83 0.75 0.67 0.58 0.50 0.42 0.33 0.25 0.17 0.08
2.735 2.733 2.731 2.729 2.728 2.727 2.726 2.725 2.724 2.723 2.722
mode is independent of the stack number of the PQW unit cell (M) and it is red-shifted as M increases. Moreover, by increasing the filling factor of the semiconductor metamaterial, the defect mode gets red-shifted and its width increases. In addition, for different angles of incidence and for both TE and TM polarization waves, the results show that the defect mode is blue-shifted as incidence angle increases. Analysis of the defect modes in photonic crystals with AZO–ZnO PQW defect will certainly provide helpful information for designing and manufacturing new types of tuneable narrow filters and other optical devices applicable in semiconductor optoelectronics at NIR region.
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M. Barati, A. Aghajamali / Physica E 79 (2016) 20–25
Fig. 5. Transmission spectra for the 1D PC doped by PQW defect for different angle of incidence for TE and TM waves.
References
Fig. 6. Wavelength of the defect mode (λD) of the 1D PC doped by PQW defect for different angle of incidence and for TE and TM waves.
Acknowledgment The authors gratefully acknowledge the financial support of Islamic Azad University, Marvdasht Branch of Iran.
[1] Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joannopoulos, E.L. Thomas, Science 282 (1998) 1679. [2] D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, Appl. Phys. 68 (1999) 25. [3] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [4] S. John, Phys. Rev. Lett. 58 (1987) 2486. [5] V.G. Veselago, Phys. Uspekhi 10 (1968) 509. [6] N. Engheta, R.W. Ziolkowski, Metamaterials: Physics and Engineering Explorations, John Wiley & Sons, New Jersey, 2006. [7] C. Caloz, T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, John Wiley & Sons, New Jersey, 2006. [8] W. Cai, V. Shalaev, Optical Metamaterials: Fundamentals and Applications, Springer, New York, 2010. [9] L. Wu, S. He, L. Chen, Opt. Express 11 (2003) 1283. [10] Z.M. Zhang, C.J. Fu, Appl. Phys. Lett. 80 (2002) 1097. [11] J. Gerardin, L. Lakhtakia, Microw. Opt. Technol. Lett. 34 (2002) 409. [12] L. Wu, S. He, L. Shen, Phys. Rev. B 67 (2003) 235103. [13] J.R. Canto, S.A. Matos, C.R. Paiva, A.M. Barbosa, PIERS Online 4 (2008) 546. [14] J. Gerardin, A. Lakhtakia, Microw. Opt. Technol. Lett. 34 (2002) 409. [15] J. Li, L. Zhou, C.T. Chan, P. Sheng, Phys. Rev. Lett. 90 (2003) 083901. [16] H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, S. Zhu, Phys. Rev. E 69 (2004) 066607. [17] I. Shadrivov, A. Sukhorukov, Y. Kivshar, Phys. Rev. Lett. 95 (2005) 193903. [18] W.-H. Lin, C.-J. Wu, S.-J. Chang, Solid State Commun. 150 (2010) 1729. [19] H.-T. Hsu, K.-C. Ting, T.-J. Yang, C.-J. Wu, Solid State Commun. 150 (2010) 644. [20] A. Aghajamali, M. Barati, Phys. B: Condens. Matter 407 (2012) 1287. [21] A. Aghajamali, M. Barati, Commun. Theor. Phys. 60 (2013) 80. [22] A. Aghajamali, M. Hayati, C.-J. Wu, M. Barati, J. Electromagn. Waves Appl. 27 (2013) 2317. [23] A. Aghajamali, B. Javanmardi, M. Barati, C.-J. Wu, Optik 125 (2014) 839. [24] A. Aghajamali, M. Akbarimoosavi, M. Barati, Adv. Opt. Technol. 1 (2014), Article ID 129568. [25] A. Aghajamali, T. Alamfard, M. Barati, Phys. B: Condens. Matter 454 (2014) 170.
M. Barati, A. Aghajamali / Physica E 79 (2016) 20–25
[26] A. Aghajamali, T. Alamfard, M. Hayati, Optik 126 (2015) 3158. [27] A. Aghajamali, A. Zare, C.-J. Wu, Appl. Opt. 54 (2015) 8602. [28] E. Galindo-Linares, P. Halevi, A.S. Sanchesa, Solid State Commun. 142 (2007) 67. [29] G.V. Naik, J. Liu, A.V. Kildishev, V.M. Shalaev, A. Boltasseva, Proc. Natl. Acad. Sci. USA 109 (2012) 8834. [30] T. Tang, Optik 124 (2013) 6657. [31] F. Qiao, C. Zhang, J. Wan, J. Zi, Appl. Phys. Lett. 77 (2000) 3698. [32] X. Chen, W. Lu, S.C. Shen, Solid State Commun. 127 (2013) 541. [33] T. Bian, Y. Zhang, Optik 120 (2008) 736. [34] C.S. Feng, L.-M. Mei, L.Z. Cai, P. Li, X.L. Yang, Solid State Commun. 135 (2005) 330. [35] C. Xu, X. Xu, D. Han, X. Liu, C.P. Liu, C.-J. Wu, Opt. Commun. 280 (2007) 221.
25
[36] S.H. Xu, Z.H. Xiong, L.L. Gu, Y. Liu, X.M. Ding, J. Zi, X.Y. Hou, Solid State Commun. 126 (2003) 125. [37] S. Yano, Y. Segawa, J.S. Bae, Phys. Rev. B 63 (2011) 153316. [38] H.C. Hung, C.-J. Wu, T.J. Yang, S.J. Chang, IEEE Photon. J. 4 (2012) 283. [39] S.H. Xu, X.H. Xiong, L.L. Gu, Y. Liu, X.M. Ding, J. Zi, X.Y. Hou, Solid State Commun. 126 (2003) 125. [40] S. Yano, Y. Segawa, J.S. Bae, Phys. Rev. B 63 (2001) 153316. [41] M.R. Wu, C.-J. Wu, S.J. Chang, Appl. Opt. 53 (2014) 7285. [42] M.R. Wu, C.-J. Wu, S.J. Chang, Physica E 64 (2014) 146. [43] M.R. Wu, C.-J. Wu, S.J. Chang, Superlattice. Microstruct. 80 (2015) 206. [44] P. Yeh, Optical Waves in Layered Media, John Wiley & Sons Inc., NY, 2005. [45] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, UK, 2005.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Near-infrared tunable narrow filter properties in a 1D photonic crystal containing semiconductor metamaterial photonic quantum-well defect Mahmood Barati n, Alireza Aghajamali Department of Physics, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran
H I G H L I G H T S
G R A P H I C A L
Study the tunable NIR narrow filters with semiconductor metamaterial PQW. The number of the defect mode is independent of the periods of the PQW. Defect mode's frequency gets redshifted as the filling factor increases. Defect mode's frequency gets blueshifted for both TE and TM waves.
We perform a theoretical investigation of the near-infrared narrow filter properties in the transmission spectrum of a one-dimensional photonic crystal doped with semiconductor metamaterial photonic quantum-well defect (PQW). It is found that the defect mode’s frequency can be tuned by the variations of the defect structure’s period, polarization, incidence angle, and also the filling factor corresponding to the semiconductor metamaterial layer. In addition, the number of the defect mode is independent of the periods of the PQW defect structure, which is in sharp contrast to the case of using usual dielectric or metamaterial defect.
art ic l e i nf o
a b s t r a c t
Article history: Received 26 October 2015 Received in revised form 5 December 2015 Accepted 12 December 2015 Available online 15 December 2015
The near-infrared (NIR) narrow filter properties in the transmission spectra of a one-dimensional photonic crystal doped with semiconductor metamaterial photonic quantum-well defect (PQW) were theoretically studied. The behavior of the defect mode as a function of the stack number of the PQW defect structure, the filling factor of semiconductor metamaterial layer, the polarization and the angle of incidence were investigated for Al-doped ZnO (AZO) and ZnO as the semiconductor metamaterial layer. It is found that the frequency of the defect mode can be tuned by variation of the period of the defect structure, polarization, incidence angle, and the filling factor of the semiconductor metamaterial layer. It is also shown that the number of the defect mode is independent of the period of the PQW defect structure and is in sharp contrast with the case where a common dielectric or metamaterial defect are used. The results also show that for both polarizations the defect mode is red-shifted as the number of the defect period and filling factor increase. An opposite trend is observed as the angle of incidence increases. The proposed structure could provide useful information for designing new types of tuneable narrowband filters at NIR region. & 2015 Published by Elsevier B.V.
Keywords: 1D photonic crystal Defect mode Semiconductor metamaterial Photonic quantum-well defect
A B S T R A C T
1. Introduction Photonic crystals (PCs), possessing some significant optical properties, are artificial materials having periodic multilayer structure in dielectric constant. One of the most interesting optical properties of PCs is existence of forbidden frequency regions in n
Corresponding author. E-mail address: [email protected] (M. Barati).
http://dx.doi.org/10.1016/j.physe.2015.12.012 1386-9477/& 2015 Published by Elsevier B.V.
their transmission spectra. The regions known as photonic band gaps (PBGs) or Bragg gaps [1–4]. The possibility of producing metamaterials with negative permittivity and permeability in the past decade [5], the optical properties of PCs with metamaterials known as metamaterial photonic crystals (MetaPCs) have been attracted by many authors [6–8]. In recent years MetaPCs have attracted extensive attention of many researchers for their unique electromagnetic properties and their scientific and microwave engineering applications [9–27]. PCs composed of semiconductors
M. Barati, A. Aghajamali / Physica E 79 (2016) 20–25
have received considerable attention for their several superior features such as tunable PBG which is achieved by thermal tuning of the dielectric function of the semiconductor material which is applicable in optoelectronic investigations. Another feature of semiconductor PCs, is appearance of their PBG in the near-infrared (NIR) to mid-IR frequency regions [28]. In 2012, Naik et al. [29] reported a new semiconductor-based metamaterial. This so called semiconductor metamaterial is an artificially periodic structure made by alternating layers of Al-doped ZnO (AZO) and ZnO on a silicon substrate with 16 pair layers and 60 nm periodicity. This AZO–ZnO metamaterial has an anisotropic and negative permittivity in NIR region [30]. The presence of negative permittivity is reminiscent of metallic permittivity, which is negative for frequencies less than the plasma frequency. Thus, AZO–ZnO metamaterial can be classified as an epsilon-negative (ENG) material in the NIR region. A defected photonic crystal would be generated by breaking the periodicity of the conventional PC structure. In the defected structure, localized defect modes within the PBG would be generated as a result of a change in the interference behavior of waves at interfaces. A PC of (CD)M can be inserted in the middle of a host PC of (AB)N rather than a defect layer being added to the PC. Therefore, the structure will be (AB)N/2 (CD)M (AB)N/2, where N and M are the stack numbers of the (AB) and (CD) bilayers, respectively. In such case, the defected structure of (CD)M will be called the photonic-quantum-well (PQW) [30–33]. The structure (AB)N/2 (CD)M (AB)N/2 (with MoN) can present multiple filtering feature due to the photonic confinement, contributing to a perception of a multichannel filter [31]. Moreover, the number of channels that can be considered as the basis of the number of peaks in the transmission spectra is equal to the stack number of PQW. Despite the fact that there have been many reports on PQW-based filters [31–38], few studies have been examined in the tuneable multichannel transmission filters. In addition to the above mentioned theoretical studies, some experimental works have been reported [39,40]. The main purpose of this work is the theoretical investigation of the properties of the defect mode in a 1D PC doped by semiconductor metamaterial PQW defect. To this aim, we study the defect mode as a function of the stack number of the PQW defect structure, the filling factor of semiconductor metamaterial layer, the polarization, and the angle of incidence. Our numerical results show that the frequency of the defect modes can be tuned by the above-mentioned parameters as well, which are in sharp contrast with the case where a common dielectric or a metamaterial defect is used. The results also reveal that a new NIR tunable narrow filter can be achieved in this proposed structure. The outline of our paper is as follows: Section 2 presents a 1D PC containing a PQW defect structure, the characteristic matrix method and its formulation, and also the permittivity of semiconductor metamaterial, Section 3 reveals the numerical results and discussions associated with our purpose, and Section 4 describes the conclusions of the investigation.
21
2. Theoretical framework A 1D PC with asymmetric structure immersed in free space with a defect structure (PQW defect) at the center of the host PC is shown in Fig. 1. Layers A and B are respectively considered to be SiO2 (Silica) and InP (Indium Phosphide), layer C is AZO–ZnO (as semiconductor metamaterial), and layer D is air. The number of lattice periods and the number of unit cells (corresponding to the defect structure) are denoted by N and M respectively. In addition, the thickness, the permittivity, and the permeability of the layers are respectively assumed to be di , εi , and μi ( i = A , B , C , D ). First of all, let us describe the permittivity and the permeability functions of AZO–ZnO layer. The permittivity is anisotropic and is given by [30,41–45]:
⎛ ερ 0 0 ⎞ ⎜ ⎟ εC = ⎜ 0 ερ 0 ⎟ ⎜ ⎟ ⎝ 0 0 ευ ⎠
(1)
Here, ερ and ευ are the parallel and perpendicular components, respectively, and they are related to the permittivity functions of AZO and ZnO. The total permittivity functions of AZO–ZnO composite are expressed as
ερ = h εa + (1 − h) εb
(2)
and
ευ =
1 h εa−1 + (1 − h) εb−1
(3)
where h is the filling factor of AZO, h = da/(da + db ) and dC = da + db is the thickness of layer C, where da and db indicate the thickness of the AZO and ZnO layers, respectively. Moreover, the permittivity functions of AZO and ZnO are εa and εb , respectively. Here, εb is a constant and εa can be expressed as a combination of Lorentz and Drude models, that is [41–45]
εa (f ) = εa1 (f ) + εa2 (f ),
(4)
where the Lorentz part is given by
εa1 (f ) = 1 −
2 2 f ap − f ao 1 1 2 f 2 − f ao 1
, (5)
and the Drude part is
εa2 (f ) = 1 −
2 f ap2
f2
(6)
here, fap1, fao1, and fap2 are constants corresponding with three characteristic frequencies. As for the permeabilities of AZO and ZnO, they are taken to be unity because both materials are nonmagnetic. In performed calculations, we have applied the characteristic matrix
Fig. 1. Schematic diagram of one-dimensional defective photonic crystal immersed in free space with photonic quantum-well defect structure at the center. Where layers A and B corresponding to the main structure are respectively SiO2, InP, and also layers C and D corresponding to the defect structure are respectively are AZO–ZnO and air. θ0 is the incident angle and also N and M are respectively the number of the lattice periods of bilayers (AB) and (CD).
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method [44,45], which is assumed to be the most effective technique to analyze the transmission properties of finite-periodic PCs. The characteristic matrix for asymmetric (AB )N /2 (CD)M (AB )N /2 defective structures is presented by: M [d] = (MA MB )N /2 (MC MD )M (MA MB )N /2, where MA , MB , MC and MD are the characteristic matrices of layers A, B, C and D. The characteristic matrix Mi is calculated through the following equation for the TE wave at the incidence angle θ0 from vacuum to a 1D PC structure [44,45]:
⎡ ⎤ −i sin γi ⎥ ⎢ cos γi p Mi = ⎢ i ⎥, ⎢⎣ − ip sin γ cos γi ⎥⎦ i i
(7)
In this equation, γi = (ω/c ) ni di cos θi , c is the speed of light in vacuum, θi is the ray angle inside layer i with refractive index ni ,
pi = εi /μi cos θi , and cos θi = 1 − (n02 sin2θ0/ni2 ) , in which n0 is the refractive index of the environment wherein the incidence wave tends to enter the structure. The refractive index is given as ni = ± εi μi , in which the positive sign is for the double-positive materials whereas the negative sign is for the negative index materials (NIMs). The characteristic matrix for N periods of structure is, therefore, [M (d )]N . The transmission coefficient of the multilayer system is calculated by:
2p0 t= (m11 + m12 ps ) p0 + (m21 + m22 ps )
(8)
In this equation, mij (i, j = 1, 2) are the matrix elements of
[M (d )]N , p0 = n0 cos θ0 , and ps = ns cos θs , where ns is the refractive index of the environment whose ray angle is θs . The transmissivity of the multilayer is given by T = (ps /p0 ) t 2. The transmissivity of the multilayer for TM wave can be similarly obtained by pi = μi /εi cos θi , p0 = cos θ0/n0 , and ps = cos θs/ns .
3. Numerical results and discussion In our calculations, the material parameters used for the basic equations described in the previous section are fap1 = 180 THz, fap2 = 150 THz, fao1 = 80 THz, nA = 1.45 (SiO2), nB = 3.52 (InP), and nD = 1 (vacuum). The thickness of layers A, B, C, and D are chosen as dA ¼4.54 mm, dB ¼1.22 mm,dC ¼60 nm, and dD ¼80 nm. Before we present the defect mode properties in the defective PC, it should be mentioned that: (a) as it is inferred from Eqs. (5) and (6) the semiconductor metamaterial is assumed to be lossless. (b) For some limitations in the computational calculations, the optimal total number of the lattice period of the main structure (N) should firstly be determined. In this regard the width of the band gap ( Δ) as a function of the total number of the unit cells (N) is calculated. As it is seen from Fig. 2, the band gap is nearly constant for N ≥ 20. Accordingly, the total number of the lattice periods in all numerical calculation is considered as N = 20. 3.1. Defect mode as a function of number of the PQW unit cell (M) The effect of the number of the PQW unit cell on the normal incidence transmission spectrum is reported. As it is seen from Fig. 3, the number of the defect mode appearing in the band gap is independent of the number of the PQW unit cells, where only one defect mode appears in the band gap for different values of M. This observation is in sharp contrast with the previous studies [31,32,35,38], where the number of the defect modes is reported to be the same as the number of the M cells. Another significant observation is that as M increases, the defect mode is red-shifted although the band gap remains nearly constant. This shifting property is indicative of a technical significance of the structure
Fig. 2. Width of the band gap versus the number of the lattice periods of the main structure.
which is quite useful in designing a narrow tuneable filter. 3.2. Defect mode as a function of filling factor (h) The shifting feature of the defect mode as a function of the filling factor of the semiconductor metamaterial, AZO–ZnO, is investigated. Fig. 4 shows the effect of the filling factor (h) on the position of the defect mode for the normal angle of incidence. It is clearly seen that as h increases, the defect mode is red-shifted, its width slightly increases where the height remains constant. Table 1 presents further detail of the changes in the position of the defect mode and the corresponding wavelength (λD). By comparing the numerical results, we observe that although the thickness of the AZO layer (da) has more salient influence on h, in compare with the ZnO thickness layer (db); on the other hand, both da and db have the same effect on the shifting feature of λD. Moreover, it is noticeable that by increasing the thickness of the AZO layer the defect mode starts red-shifting, but an opposite trend is observed as the ZnO thickness layer increases. Taking this unique result into consideration provides us with an opportunity to make a more practical tuneable narrow filter. 3.3. Defect mode as a function of the angle of incidence ( θ 0 ) We now turn our attention to the case of oblique incidence angle. Fig. 5 shows the transmission spectra for different angles of incidence for TE and TM waves. As shown in the figure, the defect mode and the band gap are blue-shifted as the angle of incidence increases. It is also clearly seen that, by increasing the angle of incidence, a noticeable rise in the width and depth of the band gap and a decline in the height of the defect mode is observed. The results for TE and TM waves also reveal that the depth of the band gap and the height of the defect mode are more sensitive for TE waves as θ 0 increases. This feature is sharply in contrast with the previous report by Xu et al. [35] who showed the defect mode was very weakly dependent on the incidence angle and polarization. The effects of the angle of incidence on the wavelength of the defect mode (λD) for TE and TM waves are shown in Fig. 6. It is seen that, by increasing the angle of incidence, the defect mode gets blue-shifted. Moreover, the shift for the TM modes is considerably more than that for the TE waves, which is in agreement with the previous report by Wu et al. [42] for semiconductor metamaterial defect layer at the center of a PC structure, i.e., (AB )N /2 (C )M (AB )N /2.
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Fig. 3. Transmission spectra for the 1D PC doped by PQW defect for different number of the PQW unit cell (M).
Table 1 The numerical results for the wavelength of the defect mode shown in Fig. 4 for different filling factor of semiconductor metamaterial, AZO–ZnO, and different thicknesses of the AZO and ZnO layers.
Fig. 4. Transmission spectra for the 1D PC doped by PQW defect for different filling factors of semiconductor metamaterial, AZO–ZnO.
4. Conclusion In this study, the properties of the defect mode of 1D PCs doped by semiconductor metamaterial PQW defect in NIR region for a defect structure composed of AZO–ZnO and air have been analyzed. Our numerical results show that the number of the defect
da (nm)
db (nm)
h
λD (mm)
55 50 45 40 35 30 25 20 15 10 5
5 10 15 20 25 30 35 40 45 50 55
0.92 0.83 0.75 0.67 0.58 0.50 0.42 0.33 0.25 0.17 0.08
2.735 2.733 2.731 2.729 2.728 2.727 2.726 2.725 2.724 2.723 2.722
mode is independent of the stack number of the PQW unit cell (M) and it is red-shifted as M increases. Moreover, by increasing the filling factor of the semiconductor metamaterial, the defect mode gets red-shifted and its width increases. In addition, for different angles of incidence and for both TE and TM polarization waves, the results show that the defect mode is blue-shifted as incidence angle increases. Analysis of the defect modes in photonic crystals with AZO–ZnO PQW defect will certainly provide helpful information for designing and manufacturing new types of tuneable narrow filters and other optical devices applicable in semiconductor optoelectronics at NIR region.
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Fig. 5. Transmission spectra for the 1D PC doped by PQW defect for different angle of incidence for TE and TM waves.
References
Fig. 6. Wavelength of the defect mode (λD) of the 1D PC doped by PQW defect for different angle of incidence and for TE and TM waves.
Acknowledgment The authors gratefully acknowledge the financial support of Islamic Azad University, Marvdasht Branch of Iran.
[1] Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joannopoulos, E.L. Thomas, Science 282 (1998) 1679. [2] D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, Appl. Phys. 68 (1999) 25. [3] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [4] S. John, Phys. Rev. Lett. 58 (1987) 2486. [5] V.G. Veselago, Phys. Uspekhi 10 (1968) 509. [6] N. Engheta, R.W. Ziolkowski, Metamaterials: Physics and Engineering Explorations, John Wiley & Sons, New Jersey, 2006. [7] C. Caloz, T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications, John Wiley & Sons, New Jersey, 2006. [8] W. Cai, V. Shalaev, Optical Metamaterials: Fundamentals and Applications, Springer, New York, 2010. [9] L. Wu, S. He, L. Chen, Opt. Express 11 (2003) 1283. [10] Z.M. Zhang, C.J. Fu, Appl. Phys. Lett. 80 (2002) 1097. [11] J. Gerardin, L. Lakhtakia, Microw. Opt. Technol. Lett. 34 (2002) 409. [12] L. Wu, S. He, L. Shen, Phys. Rev. B 67 (2003) 235103. [13] J.R. Canto, S.A. Matos, C.R. Paiva, A.M. Barbosa, PIERS Online 4 (2008) 546. [14] J. Gerardin, A. Lakhtakia, Microw. Opt. Technol. Lett. 34 (2002) 409. [15] J. Li, L. Zhou, C.T. Chan, P. Sheng, Phys. Rev. Lett. 90 (2003) 083901. [16] H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, S. Zhu, Phys. Rev. E 69 (2004) 066607. [17] I. Shadrivov, A. Sukhorukov, Y. Kivshar, Phys. Rev. Lett. 95 (2005) 193903. [18] W.-H. Lin, C.-J. Wu, S.-J. Chang, Solid State Commun. 150 (2010) 1729. [19] H.-T. Hsu, K.-C. Ting, T.-J. Yang, C.-J. Wu, Solid State Commun. 150 (2010) 644. [20] A. Aghajamali, M. Barati, Phys. B: Condens. Matter 407 (2012) 1287. [21] A. Aghajamali, M. Barati, Commun. Theor. Phys. 60 (2013) 80. [22] A. Aghajamali, M. Hayati, C.-J. Wu, M. Barati, J. Electromagn. Waves Appl. 27 (2013) 2317. [23] A. Aghajamali, B. Javanmardi, M. Barati, C.-J. Wu, Optik 125 (2014) 839. [24] A. Aghajamali, M. Akbarimoosavi, M. Barati, Adv. Opt. Technol. 1 (2014), Article ID 129568. [25] A. Aghajamali, T. Alamfard, M. Barati, Phys. B: Condens. Matter 454 (2014) 170.
M. Barati, A. Aghajamali / Physica E 79 (2016) 20–25
[26] A. Aghajamali, T. Alamfard, M. Hayati, Optik 126 (2015) 3158. [27] A. Aghajamali, A. Zare, C.-J. Wu, Appl. Opt. 54 (2015) 8602. [28] E. Galindo-Linares, P. Halevi, A.S. Sanchesa, Solid State Commun. 142 (2007) 67. [29] G.V. Naik, J. Liu, A.V. Kildishev, V.M. Shalaev, A. Boltasseva, Proc. Natl. Acad. Sci. USA 109 (2012) 8834. [30] T. Tang, Optik 124 (2013) 6657. [31] F. Qiao, C. Zhang, J. Wan, J. Zi, Appl. Phys. Lett. 77 (2000) 3698. [32] X. Chen, W. Lu, S.C. Shen, Solid State Commun. 127 (2013) 541. [33] T. Bian, Y. Zhang, Optik 120 (2008) 736. [34] C.S. Feng, L.-M. Mei, L.Z. Cai, P. Li, X.L. Yang, Solid State Commun. 135 (2005) 330. [35] C. Xu, X. Xu, D. Han, X. Liu, C.P. Liu, C.-J. Wu, Opt. Commun. 280 (2007) 221.
25
[36] S.H. Xu, Z.H. Xiong, L.L. Gu, Y. Liu, X.M. Ding, J. Zi, X.Y. Hou, Solid State Commun. 126 (2003) 125. [37] S. Yano, Y. Segawa, J.S. Bae, Phys. Rev. B 63 (2011) 153316. [38] H.C. Hung, C.-J. Wu, T.J. Yang, S.J. Chang, IEEE Photon. J. 4 (2012) 283. [39] S.H. Xu, X.H. Xiong, L.L. Gu, Y. Liu, X.M. Ding, J. Zi, X.Y. Hou, Solid State Commun. 126 (2003) 125. [40] S. Yano, Y. Segawa, J.S. Bae, Phys. Rev. B 63 (2001) 153316. [41] M.R. Wu, C.-J. Wu, S.J. Chang, Appl. Opt. 53 (2014) 7285. [42] M.R. Wu, C.-J. Wu, S.J. Chang, Physica E 64 (2014) 146. [43] M.R. Wu, C.-J. Wu, S.J. Chang, Superlattice. Microstruct. 80 (2015) 206. [44] P. Yeh, Optical Waves in Layered Media, John Wiley & Sons Inc., NY, 2005. [45] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, UK, 2005.