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Tunable defect mode in a semiconductor-dielectric photonic crystal containing extrinsic semiconductor defect
Solid State Communications 152 (2012) 2189–2192
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Tunable defect mode in a semiconductor-dielectric photonic crystal containing extrinsic semiconductor defect Tzu-Chyang King a, Ya-Po Yang a, Yi-Shiou Liou a, Chien-Jang Wu b,n a b
Department of Applied Physics, National Pingtung University of Education, Pingtung 900, Taiwan Institute of Electro-Optical Science and Technology, National Taiwan Normal University, Taipei 116, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 23 August 2012 Received in revised form 20 September 2012 Accepted 6 October 2012 by X.C. Shen Available online 12 October 2012
In this work, we theoretically investigate the tunable properties in the defect mode for a defective semiconductor-dielectric photonic crystal (SDPC). Here, the defect layer is an extrinsic semiconductor of n type InSb, n-InSb, which has a strong temperature- and concentration-dependent dielectric constant. With the use of n-InSb, the defect mode can be tuned as a function of temperature and concentration. The defect mode frequency is blue-shifted as the concentration or the temperature increases. In addition, this defect mode frequency shows a nearly linear dependence on the concentration but a nonlinear dependence on the temperature. Finally, by increasing the defect thickness leads to the presence of multiple defect modes which can be used to design a multichanneled filter. & 2012 Published by Elsevier Ltd.
Keywords: A. Semiconductors A. Photonic crystals D. Wave properties
1. Introduction A familiar application of one-dimensional photonic crystals (PCs) is to design a multilayer Fabry–Perot transmission narrowband filter. Such a filter can be achieved inserting by N a defect layer in a N host PC, that is, its structure is H=L p D L=H p or ðH=LÞNp DðH=LÞNp where Np is the number of periods, H and L are the high- and lowindex layers in the host PC, and D is the defect layer [1,2]. In this filter, there is a transmission peak or a reflection dip within the certain photonic band gap (PBG). The peak or dip shape can be very sharp by increasing Np, which can in turn make an extremely high-Q filter. The transmission in the transmission spectrum is known as the defect mode which is similar to the defect state in the electronic energy band gap in semiconductors. From practical applications in optical electronics, tunable filters will be of technical use. To make the above-mentioned narrowband filter tunable, one can use certain particular material for the defect layer. This particular material itself has a permittivity which can be changed by some tuning agents like electric field, magnetic field, and temperature. For instance, if the defect layer is a liquid crystal or a ferroelectric material, the defect mode can be tuned by the external electric field and it is referred to as E-tuning [3–6]. If the tuning is temperature-dependent, then it is called T-tuning which can be achieved by using superconducting material as a defect layer because the permittivity of a superconductor is known to be a strong function of the temperature [7–10]. Another tuning called
n
Corresponding author. Tel.: þ886 2 77346724; fax: þ 886 2 86631954. E-mail address: [email protected] (C.-J. Wu).
0038-1098/$ - see front matter & 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ssc.2012.10.004
the N-tuning arises from the use of a semiconducting material as a defect layer. Semiconductors are materials whose permittivity is strongly dependent on the doping concentration [11–14]. Also, the carrier concentration of an intrinsic or extrinsic semiconductor is generally depends on the temperature. Thus, a semiconductorbased photonic crystal filter is not only an N-tuning but also a Ttuning device. In this paper, we shall consider the filter structure of ðH=LÞNp DðH=LÞNp where H is Si and L is SiO2. The defect layer D is taken to be the n type InSb (n-InSb), which is a dispersive and absorptive medium. The dielectric function of n-InSb will be contributed by three terms, including the dispersions from electrons, holes, and phonons. The inclusion of phonon effect is due to the fact that nInSb is a polar semiconductor. The main reason to adopt n-InSb as a tunable agent is that its dielectric function is a strong function of the temperature and the doping concentration. We shall investigate how the defect mode is affected by the temperature and the concentration. The tuning feature, i.e., the shifting in the defect mode with the variations in these two factors, will be numerically demonstrated from the calculated transmission and reflection spectrum. The analysis will be made based on the use of the transfer matrix method (TMM) [15]. The study of this work could be of technical use in the semiconductor optoelectronics.
2. Basic equations Let us first describe the dielectric constant of n-InSb. Since InSb is a polar semiconductor, it is necessary to incorporate the phonon dispersion in the dielectric function in addition to the
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contributions by electrons and holes. The complete dielectric function can be explicitly expressed as [16] ! o2ph o2pe o2L o2T eD ðoÞ ¼ e1 1þ 2 2 2 ð1Þ oT o jog o2 jot1 o jot1 e h here eN is the high-frequency limit of the dielectric constant. The second term in the bracket is contributed by the phonon dispersion, in which oT and oL are respectively the transverse and the longitudinal optical phonon frequency, and g is the damping constant for phonons. The third and fourth terms in the bracket are from the electrons and holes, in which te and th are the scattering times for electrons and holes, respectively, and they are related to the carrier mobilities me,h by te,h ¼ me,h me,h =e. In addition, ope and oph are the electron and hole plasma frequencies given by 1=2 ne,h e2 ope,h ¼ , ð2Þ me,h e1
hole mass is given by mh ¼ mhh
1 þr 3=2 , r þ r 3=2
ð9Þ
where mhh ¼0.45m0, and r¼ mhh/mlh ¼30, where mhh and mlh are the masses of heavy hole and light hole, respectively [18]. The mobilities of electron and hole are temperature-dependent given by
me ¼ 7:7 104
T 1:66 , 300
mh ¼ 850
T 300
1:8 ð10Þ
where me,h are the effective masses of electron and hole, ne,h are the electron concentration and the hole concentration, respectively, which can be written by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N2 N ne,h ¼ n2i þ ð3Þ 7 , 2 4 where ni is the intrinsic concentration, N is the doping concentration, and the sign ‘‘ þ’’ is for electron concentration and ‘‘ ’’ is for the hole concentration. With the dielectric function in Eq. (1), the pffiffiffiffiffi index of refraction is nD ¼ eD . The defect mode will be examined through the transmittance and reflectance responses calculated by making use of the transfer matrix method (TMM) [15]. In this work, we limit to the case of normal incidence. According to TMM, the total transfer matrix of the system is given by ! N p M 11 M12 M¼ ¼ DA 1 DH P H DH 1 DL P L DL 1 M 21 M22 N p DD P D DD 1 DH PH DH 1 DL PL DL 1 DA ð4Þ here the propagation matrix in layer i, Pi (i¼H, L, and D), is ! expðjki di Þ 0 Pi ¼ , 0 expðjki di Þ
Fig. 1. (Color online) Calculated transmittance and reflectance spectra for the defective PC of (Si/SiO2)Np/n-InSb/(Si/SiO2)Np, in which the thicknesses of Si and SiO2, dH ¼ dL ¼dD ¼ 0.3 mm, and Np ¼ 10 are used. The defect mode represented by the transmission peak (or reflectance dip) can be produced within the photonic band gap.
ð5Þ
where di is the thickness and ki ¼ ni o=c is the wave number in layer i, where c is the speed of light in vacuum. In addition, the dynamical matrix in medium q is written by ! 1 1 Dq ¼ , ð6Þ nq nq where q¼A is for air with nA ¼1. The transmittance G and the reflectance R can be obtained by the following Eq. 2 1 2 ; R ¼ M 21 G ¼ ð7Þ M11 M 11
Fig. 2. Calculated defect mode in the reflectance spectrum at N ¼1 1023, 2 1023, and 3 1023 m 3. The defect mode is blue-shifted as the concentration N increases.
3. Numerical results and discussion For the material parameters of n-InSb in Eqs. (1) and (2), we take
e1 ¼ 15:6, g ¼ 0.0539 s 1, oT ¼3.5 1013 s 1, oL ¼ 3.7 1013 s 1 [17]. The intrinsic concentration is temperature-dependent given by [18] ni ¼ 5:76 1020 T 3=2 exp 0:26=2kB T ðm3Þ, ð8Þ where the constant 0.26 is in unit of eV. In addition, the electron mass is me ¼0.014m0, where m0 is the true electron mass, and the
Fig. 3. The dependence of defect mode frequency on the doping concentration. A nearly linear dependence is obtained.
T.-C. King et al. / Solid State Communications 152 (2012) 2189–2192
The refractive indices for Si and SiO2 are nH ¼3.3, nL ¼1.46. Both thicknesses are taken equal to dH ¼dL ¼0.3 mm, and Np ¼ 10 are taken. The thickness of defect layer is also equal to dD ¼0.3 mm. Fig. 1 shows the calculated transmittance and reflectance spectra at the conditions of T¼ 300 K and N ¼1023 m 3. It can be seen that there exists a defect mode (near 90 THz) within the
2191
photonic band gap (80–124 THz). The defect mode is not a complete resonance because of the existence of partial reflectance as well as the inclusion of loss factors g, te, and th in the dielectric function of n-InSb. In Fig. 2, the defect mode shown in the reflectance spectrum indicates a blue-shift as the concentration increases. The frequency, at which the defect mode is located, is called the defect frequency (or dip frequency), fD. The dependence of fD on the doping concentration is shown in Fig. 3. It is of interest to note that the dependence is nearly linear. In Fig. 4, we show the defect mode at three different temperatures, T¼300, 400, and 500 K. The defect frequency is also blueshifted when the temperature is increased. The dependence fD on T is shown in Fig. 5, in which a nonlinear dependence is seen instead
Fig. 4. (Color online) Calculated defect mode in the reflectance spectrum at T¼ 300, 400, and 500 K. The defect mode is blue-shifted as the temperature T increases.
Fig. 5. The dependence of defect mode frequency on the temperature. A nonlinear dependence is seen.
Fig. 7. (Color online) Calculated reflectance vs. frequency at a fixed temperature T¼300 K and concentration N ¼ 1 1023 m 3 for three different thicknesses of defect layer. The number of defect modes increases when the defect thickness increases.
Fig. 6. Calculated real part of nD for n-InSb vs. N at frequency f ¼88.7 THz and temperature T¼ 300 K (a), and vs. T at frequency f¼ 88.7 THz and concentration N ¼1 1023 m 3(b).
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of a nearly linear dependence shown in Fig. 3. In addition, Fig. 4 illustrates a stronger resonant transmission at low temperature because of the low reflectance dip. This indicates a relatively high transmission filter can be preferably obtained at a lower temperature when the concentration is fixed. The nearly linear dependence in Fig. 3 and nonlinear dependence in Fig. 5 can be physically explained as follows. The occurrence defect mode can be regarded as an effective Fabry– Perot resonator where the condition of maximum transmission is nD dD ¼ lD ¼
c : fD
i.e., it is blue-shifted as the temperature and concentration increase. It is found the shift in the defect mode is nearly linear dependence on the concentration but a nonlinear dependence on the temperature. We have also illustrated that the number of defect modes can be increased by widening the thickness of the defect layer. Our simulation is more realistic because we have incorporated the complete expression for the dielectric function of n-InSb and the loss factors are included there. The analysis of properties of defect mode provides some useful information to the design of a tunable and/or multichanneled filter.
ð11Þ
It can be seen that the defect frequency (or dip frequency), fD is inversely proportional to the refractive index of defect layer nD. Since the imaginary part of nD is negligibly small compared to its real part, we can safely replace nD by the real part Re(nD) in Eq. (11). Fig. 6 shows the dependence of Re(nD) on concentration N (a) and on temperature T (b), respectively. Fig. 6(a) shows a nearly linear decrease in Re(nD), which in turn makes the defect frequency fD a nearly increasing function of N, as illustrated in Fig. 3. Similarly, a nonlinear decrease in Re(nD) as a function of T is displayed in Fig. 6(b). As a result, fD will be increased nonlinearly versus T according to Eq. (11), which is consistent with that shown in Fig. 5. Finally, let us vary the thickness of the defect layer and investigate the change in the number of defect modes. Fig. 7 depicts three different thicknesses of defect layer. It can be seen that there is one defect mode at 0.3 mm. However, two and three defect modes (marked by the gray arrows) are found at 1 and 3 mm, respectively. The presence of multiple defects enables us to design a multichannel filter with a thicker defect layer. This multichannel filter can further be made tunable by changing the temperature as well as the concentration.
4. Conclusion Base on the use of Si–SiO2 photonic crystal containing a strongly extrinsic semiconductor, n-InSb, properties of defect mode have been analyzed. Since n-InSb is a strongly dispersive and absorptive medium, the defect mode frequency is tunable,
Acknowledgment C.-J. Wu acknowledges the financial support from the National Science Council of the Republic of China (Taiwan) under Contract no. NSC-100-2112-M-003-005-MY3. References [1] S.J. Orfanidis, Electromagnetic Waves and Antennas, Rutger University, Piscataway, NJ, 2008. (Chapter 7). [2] C.-J. Wu, Z.-H. Wang, Prog. Electromagn. Res. 103 (2010) 169. [3] Y.K. Ha, Y.C. Yang, J.E. Kim, H.Y. Park, Appl. Phys. Lett. 79 (2001) 15. [4] Q. Zhu, Y. Zhang, Optik 120 (2009) 195. [5] Y.-H. Chang, C.-C. Liu, T.-J. Yang, C.-J. Wu, Opt. Quantum Electron. 42 (2011) 359. [6] C.-J. Wu, J.-J. Liao, T.-W. Chang, J. Electromagn. Waves Appl. 24 (2010) 531. [7] I.L. Lyubchanskii, N.N. Dadoenkova, A.E. Zabolotin, Y.P. Lee, Th. Rasing, J. Opt. A: Pure Appl. Opt. 11 (2009) 114014. [8] O.L. Berman, Y.E. Lozovik, S.L. Eiderman, R.D. Coalson, Phys. Rev. B 74 (2005) 092505. [9] W.-H. Lin, C.-J. Wu, T.-J. Yang, S.-J. Chang, Opt. Exp. 18 (2010) 27155. [10] S.M. Anlage, J. Opt. 13 (2011) 024001. [11] E. Galindo-Linares, P. Halevi, A.S. Sanchez, Solid State Commun. 142 (2007) 67. [12] C.-J. Wu, Y.-C. Hsieh, H.-T. Hsu, Prog. Electromagn. Res. 111 (2011) 433. [13] X. Dai, Y. Xiang, S. Wen, H. He, J. Appl. Phys. 109 (2011) 053104. [14] P. Halevi, F. Ramos-Mendieta, Phys. Rev. Lett. 85 (2000) 1875. [15] P. Yeh, Optical Waves in Layered Media, John Wiley & Sons, Singapore, 1991. [16] A.S. Sanchez, P. Halevi, J. Appl. Phys. 94 (2003) 797. [17] E.D. Palik, Handbook of Optical Constants, Academic, New York, 1998. [18] O. Madelung, Semiconductors Group IV Elements and III–V Compounds, Springer, Berlin, 1986.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Tunable defect mode in a semiconductor-dielectric photonic crystal containing extrinsic semiconductor defect Tzu-Chyang King a, Ya-Po Yang a, Yi-Shiou Liou a, Chien-Jang Wu b,n a b
Department of Applied Physics, National Pingtung University of Education, Pingtung 900, Taiwan Institute of Electro-Optical Science and Technology, National Taiwan Normal University, Taipei 116, Taiwan
a r t i c l e i n f o
abstract
Article history: Received 23 August 2012 Received in revised form 20 September 2012 Accepted 6 October 2012 by X.C. Shen Available online 12 October 2012
In this work, we theoretically investigate the tunable properties in the defect mode for a defective semiconductor-dielectric photonic crystal (SDPC). Here, the defect layer is an extrinsic semiconductor of n type InSb, n-InSb, which has a strong temperature- and concentration-dependent dielectric constant. With the use of n-InSb, the defect mode can be tuned as a function of temperature and concentration. The defect mode frequency is blue-shifted as the concentration or the temperature increases. In addition, this defect mode frequency shows a nearly linear dependence on the concentration but a nonlinear dependence on the temperature. Finally, by increasing the defect thickness leads to the presence of multiple defect modes which can be used to design a multichanneled filter. & 2012 Published by Elsevier Ltd.
Keywords: A. Semiconductors A. Photonic crystals D. Wave properties
1. Introduction A familiar application of one-dimensional photonic crystals (PCs) is to design a multilayer Fabry–Perot transmission narrowband filter. Such a filter can be achieved inserting by N a defect layer in a N host PC, that is, its structure is H=L p D L=H p or ðH=LÞNp DðH=LÞNp where Np is the number of periods, H and L are the high- and lowindex layers in the host PC, and D is the defect layer [1,2]. In this filter, there is a transmission peak or a reflection dip within the certain photonic band gap (PBG). The peak or dip shape can be very sharp by increasing Np, which can in turn make an extremely high-Q filter. The transmission in the transmission spectrum is known as the defect mode which is similar to the defect state in the electronic energy band gap in semiconductors. From practical applications in optical electronics, tunable filters will be of technical use. To make the above-mentioned narrowband filter tunable, one can use certain particular material for the defect layer. This particular material itself has a permittivity which can be changed by some tuning agents like electric field, magnetic field, and temperature. For instance, if the defect layer is a liquid crystal or a ferroelectric material, the defect mode can be tuned by the external electric field and it is referred to as E-tuning [3–6]. If the tuning is temperature-dependent, then it is called T-tuning which can be achieved by using superconducting material as a defect layer because the permittivity of a superconductor is known to be a strong function of the temperature [7–10]. Another tuning called
n
Corresponding author. Tel.: þ886 2 77346724; fax: þ 886 2 86631954. E-mail address: [email protected] (C.-J. Wu).
0038-1098/$ - see front matter & 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ssc.2012.10.004
the N-tuning arises from the use of a semiconducting material as a defect layer. Semiconductors are materials whose permittivity is strongly dependent on the doping concentration [11–14]. Also, the carrier concentration of an intrinsic or extrinsic semiconductor is generally depends on the temperature. Thus, a semiconductorbased photonic crystal filter is not only an N-tuning but also a Ttuning device. In this paper, we shall consider the filter structure of ðH=LÞNp DðH=LÞNp where H is Si and L is SiO2. The defect layer D is taken to be the n type InSb (n-InSb), which is a dispersive and absorptive medium. The dielectric function of n-InSb will be contributed by three terms, including the dispersions from electrons, holes, and phonons. The inclusion of phonon effect is due to the fact that nInSb is a polar semiconductor. The main reason to adopt n-InSb as a tunable agent is that its dielectric function is a strong function of the temperature and the doping concentration. We shall investigate how the defect mode is affected by the temperature and the concentration. The tuning feature, i.e., the shifting in the defect mode with the variations in these two factors, will be numerically demonstrated from the calculated transmission and reflection spectrum. The analysis will be made based on the use of the transfer matrix method (TMM) [15]. The study of this work could be of technical use in the semiconductor optoelectronics.
2. Basic equations Let us first describe the dielectric constant of n-InSb. Since InSb is a polar semiconductor, it is necessary to incorporate the phonon dispersion in the dielectric function in addition to the
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T.-C. King et al. / Solid State Communications 152 (2012) 2189–2192
contributions by electrons and holes. The complete dielectric function can be explicitly expressed as [16] ! o2ph o2pe o2L o2T eD ðoÞ ¼ e1 1þ 2 2 2 ð1Þ oT o jog o2 jot1 o jot1 e h here eN is the high-frequency limit of the dielectric constant. The second term in the bracket is contributed by the phonon dispersion, in which oT and oL are respectively the transverse and the longitudinal optical phonon frequency, and g is the damping constant for phonons. The third and fourth terms in the bracket are from the electrons and holes, in which te and th are the scattering times for electrons and holes, respectively, and they are related to the carrier mobilities me,h by te,h ¼ me,h me,h =e. In addition, ope and oph are the electron and hole plasma frequencies given by 1=2 ne,h e2 ope,h ¼ , ð2Þ me,h e1
hole mass is given by mh ¼ mhh
1 þr 3=2 , r þ r 3=2
ð9Þ
where mhh ¼0.45m0, and r¼ mhh/mlh ¼30, where mhh and mlh are the masses of heavy hole and light hole, respectively [18]. The mobilities of electron and hole are temperature-dependent given by
me ¼ 7:7 104
T 1:66 , 300
mh ¼ 850
T 300
1:8 ð10Þ
where me,h are the effective masses of electron and hole, ne,h are the electron concentration and the hole concentration, respectively, which can be written by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N2 N ne,h ¼ n2i þ ð3Þ 7 , 2 4 where ni is the intrinsic concentration, N is the doping concentration, and the sign ‘‘ þ’’ is for electron concentration and ‘‘ ’’ is for the hole concentration. With the dielectric function in Eq. (1), the pffiffiffiffiffi index of refraction is nD ¼ eD . The defect mode will be examined through the transmittance and reflectance responses calculated by making use of the transfer matrix method (TMM) [15]. In this work, we limit to the case of normal incidence. According to TMM, the total transfer matrix of the system is given by ! N p M 11 M12 M¼ ¼ DA 1 DH P H DH 1 DL P L DL 1 M 21 M22 N p DD P D DD 1 DH PH DH 1 DL PL DL 1 DA ð4Þ here the propagation matrix in layer i, Pi (i¼H, L, and D), is ! expðjki di Þ 0 Pi ¼ , 0 expðjki di Þ
Fig. 1. (Color online) Calculated transmittance and reflectance spectra for the defective PC of (Si/SiO2)Np/n-InSb/(Si/SiO2)Np, in which the thicknesses of Si and SiO2, dH ¼ dL ¼dD ¼ 0.3 mm, and Np ¼ 10 are used. The defect mode represented by the transmission peak (or reflectance dip) can be produced within the photonic band gap.
ð5Þ
where di is the thickness and ki ¼ ni o=c is the wave number in layer i, where c is the speed of light in vacuum. In addition, the dynamical matrix in medium q is written by ! 1 1 Dq ¼ , ð6Þ nq nq where q¼A is for air with nA ¼1. The transmittance G and the reflectance R can be obtained by the following Eq. 2 1 2 ; R ¼ M 21 G ¼ ð7Þ M11 M 11
Fig. 2. Calculated defect mode in the reflectance spectrum at N ¼1 1023, 2 1023, and 3 1023 m 3. The defect mode is blue-shifted as the concentration N increases.
3. Numerical results and discussion For the material parameters of n-InSb in Eqs. (1) and (2), we take
e1 ¼ 15:6, g ¼ 0.0539 s 1, oT ¼3.5 1013 s 1, oL ¼ 3.7 1013 s 1 [17]. The intrinsic concentration is temperature-dependent given by [18] ni ¼ 5:76 1020 T 3=2 exp 0:26=2kB T ðm3Þ, ð8Þ where the constant 0.26 is in unit of eV. In addition, the electron mass is me ¼0.014m0, where m0 is the true electron mass, and the
Fig. 3. The dependence of defect mode frequency on the doping concentration. A nearly linear dependence is obtained.
T.-C. King et al. / Solid State Communications 152 (2012) 2189–2192
The refractive indices for Si and SiO2 are nH ¼3.3, nL ¼1.46. Both thicknesses are taken equal to dH ¼dL ¼0.3 mm, and Np ¼ 10 are taken. The thickness of defect layer is also equal to dD ¼0.3 mm. Fig. 1 shows the calculated transmittance and reflectance spectra at the conditions of T¼ 300 K and N ¼1023 m 3. It can be seen that there exists a defect mode (near 90 THz) within the
2191
photonic band gap (80–124 THz). The defect mode is not a complete resonance because of the existence of partial reflectance as well as the inclusion of loss factors g, te, and th in the dielectric function of n-InSb. In Fig. 2, the defect mode shown in the reflectance spectrum indicates a blue-shift as the concentration increases. The frequency, at which the defect mode is located, is called the defect frequency (or dip frequency), fD. The dependence of fD on the doping concentration is shown in Fig. 3. It is of interest to note that the dependence is nearly linear. In Fig. 4, we show the defect mode at three different temperatures, T¼300, 400, and 500 K. The defect frequency is also blueshifted when the temperature is increased. The dependence fD on T is shown in Fig. 5, in which a nonlinear dependence is seen instead
Fig. 4. (Color online) Calculated defect mode in the reflectance spectrum at T¼ 300, 400, and 500 K. The defect mode is blue-shifted as the temperature T increases.
Fig. 5. The dependence of defect mode frequency on the temperature. A nonlinear dependence is seen.
Fig. 7. (Color online) Calculated reflectance vs. frequency at a fixed temperature T¼300 K and concentration N ¼ 1 1023 m 3 for three different thicknesses of defect layer. The number of defect modes increases when the defect thickness increases.
Fig. 6. Calculated real part of nD for n-InSb vs. N at frequency f ¼88.7 THz and temperature T¼ 300 K (a), and vs. T at frequency f¼ 88.7 THz and concentration N ¼1 1023 m 3(b).
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of a nearly linear dependence shown in Fig. 3. In addition, Fig. 4 illustrates a stronger resonant transmission at low temperature because of the low reflectance dip. This indicates a relatively high transmission filter can be preferably obtained at a lower temperature when the concentration is fixed. The nearly linear dependence in Fig. 3 and nonlinear dependence in Fig. 5 can be physically explained as follows. The occurrence defect mode can be regarded as an effective Fabry– Perot resonator where the condition of maximum transmission is nD dD ¼ lD ¼
c : fD
i.e., it is blue-shifted as the temperature and concentration increase. It is found the shift in the defect mode is nearly linear dependence on the concentration but a nonlinear dependence on the temperature. We have also illustrated that the number of defect modes can be increased by widening the thickness of the defect layer. Our simulation is more realistic because we have incorporated the complete expression for the dielectric function of n-InSb and the loss factors are included there. The analysis of properties of defect mode provides some useful information to the design of a tunable and/or multichanneled filter.
ð11Þ
It can be seen that the defect frequency (or dip frequency), fD is inversely proportional to the refractive index of defect layer nD. Since the imaginary part of nD is negligibly small compared to its real part, we can safely replace nD by the real part Re(nD) in Eq. (11). Fig. 6 shows the dependence of Re(nD) on concentration N (a) and on temperature T (b), respectively. Fig. 6(a) shows a nearly linear decrease in Re(nD), which in turn makes the defect frequency fD a nearly increasing function of N, as illustrated in Fig. 3. Similarly, a nonlinear decrease in Re(nD) as a function of T is displayed in Fig. 6(b). As a result, fD will be increased nonlinearly versus T according to Eq. (11), which is consistent with that shown in Fig. 5. Finally, let us vary the thickness of the defect layer and investigate the change in the number of defect modes. Fig. 7 depicts three different thicknesses of defect layer. It can be seen that there is one defect mode at 0.3 mm. However, two and three defect modes (marked by the gray arrows) are found at 1 and 3 mm, respectively. The presence of multiple defects enables us to design a multichannel filter with a thicker defect layer. This multichannel filter can further be made tunable by changing the temperature as well as the concentration.
4. Conclusion Base on the use of Si–SiO2 photonic crystal containing a strongly extrinsic semiconductor, n-InSb, properties of defect mode have been analyzed. Since n-InSb is a strongly dispersive and absorptive medium, the defect mode frequency is tunable,
Acknowledgment C.-J. Wu acknowledges the financial support from the National Science Council of the Republic of China (Taiwan) under Contract no. NSC-100-2112-M-003-005-MY3. References [1] S.J. Orfanidis, Electromagnetic Waves and Antennas, Rutger University, Piscataway, NJ, 2008. (Chapter 7). [2] C.-J. Wu, Z.-H. Wang, Prog. Electromagn. Res. 103 (2010) 169. [3] Y.K. Ha, Y.C. Yang, J.E. Kim, H.Y. Park, Appl. Phys. Lett. 79 (2001) 15. [4] Q. Zhu, Y. Zhang, Optik 120 (2009) 195. [5] Y.-H. Chang, C.-C. Liu, T.-J. Yang, C.-J. Wu, Opt. Quantum Electron. 42 (2011) 359. [6] C.-J. Wu, J.-J. Liao, T.-W. Chang, J. Electromagn. Waves Appl. 24 (2010) 531. [7] I.L. Lyubchanskii, N.N. Dadoenkova, A.E. Zabolotin, Y.P. Lee, Th. Rasing, J. Opt. A: Pure Appl. Opt. 11 (2009) 114014. [8] O.L. Berman, Y.E. Lozovik, S.L. Eiderman, R.D. Coalson, Phys. Rev. B 74 (2005) 092505. [9] W.-H. Lin, C.-J. Wu, T.-J. Yang, S.-J. Chang, Opt. Exp. 18 (2010) 27155. [10] S.M. Anlage, J. Opt. 13 (2011) 024001. [11] E. Galindo-Linares, P. Halevi, A.S. Sanchez, Solid State Commun. 142 (2007) 67. [12] C.-J. Wu, Y.-C. Hsieh, H.-T. Hsu, Prog. Electromagn. Res. 111 (2011) 433. [13] X. Dai, Y. Xiang, S. Wen, H. He, J. Appl. Phys. 109 (2011) 053104. [14] P. Halevi, F. Ramos-Mendieta, Phys. Rev. Lett. 85 (2000) 1875. [15] P. Yeh, Optical Waves in Layered Media, John Wiley & Sons, Singapore, 1991. [16] A.S. Sanchez, P. Halevi, J. Appl. Phys. 94 (2003) 797. [17] E.D. Palik, Handbook of Optical Constants, Academic, New York, 1998. [18] O. Madelung, Semiconductors Group IV Elements and III–V Compounds, Springer, Berlin, 1986.