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Optical properties of MgxZn1−xO nanowire photonic crystals
Solid State Communications 142 (2007) 195–199 www.elsevier.com/locate/ssc
Optical properties of Mgx Zn1−x O nanowire photonic crystals Chul-Sik Kee ∗ , Do-Kyeong Ko, Jongmin Lee Nanophotonics Laboratory, Advanced Photonics Research Institute, GIST, Gwangju 500-712, Republic of Korea Received 15 June 2006; received in revised form 12 January 2007; accepted 14 February 2007 by J.W.P. Hsu Available online 25 February 2007
Abstract We investigate the optical properties of two-dimensional periodic arrays of well-aligned Mgx Zn1−x O nanowires, i.e., Mgx Zn1−x O nanowire photonic crystals. The nanowire photonic crystal can exhibit a photonic band gap in the visible range. As the mole fraction of Mg, x, increases, the edge frequencies of the band gap increase and the band gap size decreases. The characteristics of relative band gap and vacant point defect mode are also studied with varying x. From the finite-difference time-domain simulations, we show that the light extraction from nanowires can be controlled by varying the distance between optically excited nanowires and a waveguide, and the mole fraction of Mg. Controlling the light extraction from nanostructures can be useful in the implementation of nanoscale light emitting devices. c 2007 Elsevier Ltd. All rights reserved.
PACS: 42.70.Qs; 42.60.Da Keywords: A. Photonic crystal; B. ZnO nanowire
1. Introduction Artificial structures with periodic dielectric modulation, i.e., photonic crystals, can exhibit frequency ranges in which the propagation of photons is prohibited, i.e., photonic band gaps, and highly dispersive properties causing the anomalous refraction phenomena such as super-prism effects and negative refraction [1–4]. Those properties have been demonstrated to be useful in controlling the propagation and emission of photons [5–9]. Nanostructures with high aspect ratio, such as nanotubes, nanowires, and nanorods, have attracted much attention because they are expected to improve the performance of electronic devices, data storage, biochemical and chemical sensors, optoelectronic devices, and so on. Very recently, the successful fabrications of periodic arrays of InAs nanowires and carbon nanotubes have been reported [10,11]. These periodically aligned nanostructures are very attractive optical materials because they exhibit a periodic modulation of the dielectric constant. Thus, photonic crystals based on the nanostructures could offer unprecedented properties playing a ∗ Corresponding author. Tel.: +82 62 970 3426; fax: +82 62 970 3419.
E-mail address: [email protected] (C.-S. Kee). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.02.021
crucial role in developing future nanophotonics [12,13]. Among the nanostructures, a Mgx Zn1−x O nanowire is an attractive candidate for an efficient ultraviolet light emitting nanodevice. Recently, well-aligned nanowire arrays have been synthesized and their optical properties have been investigated [14,15]. However, the optical properties of periodic arrays of well-aligned nanowires, i.e., nanowire photonic crystals, have been rarely studied. In this paper, we investigate the optical properties of Mgx Zn1−x O nanowire photonic crystals using an effective numerical method to treat frequency-dependent materials. The nanowire photonic crystals can exhibit photonic band gaps in the visible range. As the mole fraction of Mg, x, increases, the edge frequencies of the band gap increase and the band gap decreases because the refractive index of the nanowire decreases when x increases. The relative size and center frequency of the band gap, and the frequency of defect mode caused by a vacant point defect, are expressed as a linear function of x. We also show from the finite-difference timedomain simulations that the light extraction from the nanowire could be controlled by the distance between the optically excited wire and a waveguide. Controlling the light extraction from the nanowires could give an opportunity to implement nanoscale light emitting devices.
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2. Computational model and method The height of a nanowire is in general much longer than its diameter [15,16]. Thus a periodic array of well-aligned nanowires in air can be treated as a two-dimensional photonic crystal. In a two-dimensional photonic crystal, there are two independent modes, i.e., E-polarized mode (E k z) and H polarized mode (E ⊥ z), where E is the electric field of the mode and z is the longitudinal axis of the nanowire. Usually, a Mgx Zn1−x O nanowire has a wurtzite crystal structure and its preferred growth direction is along the c-axis of Mgx Zn1−x O, that is, the c-axis is parallel to the z-direction. Previous works have shown that a two-dimensional triangular photonic crystal composed of dielectric rods in air exhibits a large band gap between the first and second bands for E-polarized modes even though the refractive index of the rod is small [17,18]. Note that the luminescence of the E-polarized mode is much stronger than that of the H -polarized mode, due to the geometry of the nanowire. This means that the band gap for the E-polarized mode affects mainly the luminescent properties of nanowire photonic crystals. Thus our attention is restricted to the Epolarized mode. The refractive index for polarization parallel to the c-axis of Mgx Zn1−x O was recently measured in the infrared and visible ranges when the mole fraction of Mg, x, is lower than 0.29 [19]. The refractive index was dependent of both frequency ω and x, and was fitted in the three-term Cauchy approximation type formula as follows: n k (ω, x) = 1.844 + ax + +
(0.0181 + bx)(h¯ ω)2 (1.24 eV)2
(0.0036 + cx)(h¯ ω)4 , (1.24 eV)4
(1)
where a = −0.574, b = −0.0451, and c = −0.0049. The eigenvalue equation for E-polarized modes of a twodimensional photonic crystal is as follows: ω2 0 (2) det M(K, K ) − 2 δK,K0 = 0, c where M(K, K0 ) = |K||K0 |Q(K − K0 ).
(3)
Here, K = k + G, K0 = k + G0 , and Q(K − K0 ) is the Fourier transformation of 1/n 2k , where k is the wave vector in the first Brillouin zone, and G, G0 the reciprocal vectors. We employed an effective numerical method to treat arbitrary frequency-dependent media in the photonic band calculations. We calculated the matrix solutions of Eq. (2) as follows: we took a few thousand mesh points for 1 eV ≤ h¯ ω ≤ 3 eV for a given k and then calculated the determinants of the matrix equation at these mesh points until we found the region where the determinants of the neighboring mesh points change sign. This method has been successfully employed in the calculation of photonic band structures of photonic crystals with frequency-dependent or µ [20–22].
Fig. 1. The first and second photonic bands of a two-dimensional triangular photonic crystal composed of Mgx Zn1−x O nanowires with the mole fraction of Mg, x = 0, when the period of the nanowire array is 240 nm and the diameter of the wires is 100 nm (left), and the dependence of the band gap edge frequencies on x (right). The inset denotes the first Brillouin zone of a two-dimensional triangular lattice.
3. Results and discussion Fig. 1 shows the first and second photonic bands of a two-dimensional triangular photonic crystal composed of Mgx Zn1−x O nanowires with x = 0 in air when the period of the nanowire array is 240 nm and the diameter of the wires is 100 nm. The inset denotes the first Brillouin zone of a two-dimensional triangular lattice. One can see that there is a photonic band gap between the K point of the first band and the J point of the second band. The gap range is from about 2.17 to 2.63 eV (471–571 nm). Mg is usually doped in ZnO to control the electronic band gap of Mgx Zn1−x O. So, the dependence of the photonic band gap on the mole fraction of Mg would be interesting. The dependence of the band gap edge frequencies on x are represented in the right of Fig. 1. As x increases, the edge frequencies increase and the band gap size decreases. This is easily understood from the fact that the refractive index of a Mgx Zn1−x O nanowire decreases as x increases (see Eq. (1)). Note that there is an x-independent band gap region which is from about 2.37 to 2.63 eV (471–523 nm). We also investigated the dependence of the band gap on the diameter of the wire for all studied values of x. We found that the relative band gap defined as 1ω/ωmid has a maximum value when the diameter is between 122 and 127 nm, where 1ω is the absolute band gap size and ωmid is the center frequency of the band gap. Fig. 2 shows the dependence of the maximum value of relative band gap, 1ω/ωmid,MAX (blue circle), and ωmid (red circle) on x. One can see that 1ω/ωmid,MAX decreases and ωmid increases as x increases. This is because the refractive index of the wire decreases when x increases. 1ω/ωmid,MAX and ωmid can be expressed by a linear function of x, 1ω/ωmid,MAX (x) = −2.88x + 0.214, and h¯ ωmid = (0.76x + 2.35) eV. Introducing various defects into photonic crystals can create allowed modes within the band gaps and give rise to various applications such as control of spontaneous emission, efficient bent guiding of lights, and ultrahigh quality cavities [5–7]. We studied the dependence of the defect mode frequency,
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Fig. 2. Dependence of the maximum value of the relative band gap 1ω/ωmid,MAX (blue circle) and ωmid (red circle) on x, where 1ω is the absolute band gap size and ωmid is the center frequency of the band gap. 1ω/ωmid,MAX and ωmid can be expressed by a linear function of x, 1ω/ωmid,MAX (x) = −2.88x + 0.214, and h¯ ωmid = (0.76x + 2.37) eV, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Dependence of the frequency of the defect mode created by removing one wire, ωd , on x. ωd is linearly proportional to x: h¯ ωd (x) = (0.5x + 2.35) eV.
ωd , created by removing one wire, on x. Fig. 3 shows that ωd is linearly proportional to x: h¯ ωd (x) = (0.5x + 2.37) eV. The empirical formulas would be useful in estimating 1ω/ωmid,MAX , the band gap position, and the defect frequency for arbitrary values of x (x < 0.25). The light emission from a Mgx Zn1−x O nanowire can be altered by the photonic band gap. Our previous work has shown that the photons emitted from the nanowire do not propagate but are strongly localized around the nanowire when the frequencies of emitted photons are within the photonic band gap [23]. However, it is worth noting that the photons localized around the nanowire can escape from the nanowire to a waveguide when a waveguide is near the nanowire because of the optical coupling between the photons localized around the wire and the guided modes of the waveguide [24]. To observe the light escape from the nanowire to a waveguide, we created a waveguide by removing a few nanowires, as shown in Fig. 4, and simulated the radiations from the nanowire.
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The simulations were performed by the finite-difference timedomain method with the perfectly matched layer absorbing boundary condition [25]. Each unit cell was divided into 16×16 discretization grid cells. The total number of time steps was 215 , with each time step 1t = 1x/(2c). The radiative light source was located at the center of the nanowire and simulated as a Gaussian in the time and space domain. The structural parameters employed in the simulations are identical to those of Fig. 1, and the spectral maximum frequency of the emitted photon is matched to the center frequency of the band gap. The guiding range of the waveguide includes the center frequency. To see the coupling characteristics of the photons localized around the nanowire and the guiding modes of the waveguide, we changed the position of the radiative nanowire that emits photons. Fig. 4(a)–(c) show the spatial distributions of the electric fields radiated from the radiative nanowires located at three different positions. In the figures, solid circles denote the nanowires and d is the distance between the radiative nanowire and the fixed detector. Λ is the period of the photonic crystal. One can see from the figures that the photons localized around the radiative nanowire due to the photonic band gap effect (see Fig. 4(a)) can escape from the radiative nanowire to the waveguide when the radiative nanowire becomes close to the waveguide (see Fig. 4(b) and (c)). It is shown in Fig. 4(d) that the measured intensity at the detector linearly increases when d decreases. This shows clearly that the coupling between the photon localized around the radiative nanowire and the guided mode of the waveguide becomes strong when the distance between the radiative nanowire and the waveguide decreases, because the coupling depends on the overlapping of the localized photon and the guided mode. We expect that the light extraction from the nanowire along the waveguide could be observed by exciting the nanowire located around the waveguide optically. The light extraction should depend on the position of excited nanowire. Controlling the light extraction from nanostructures could be useful in implementing new types of nanoscale light emitting devices. It would be interesting to investigate the influence of changing the mole fraction of Mg on the light extraction from the nanowire. To study the influence qualitatively, we assumed that the spectral maximum frequency of the point source is unchanged when x varies. Fig. 5 shows the photon intensity measured at the detector when x = 0, 0.05, 0.1 0.15, 0.2. The distance between the point source and the detector is 13Λ. One can see that the measured intensity at the detector increases as x increases. The increase of light extraction from the nanowire to the waveguide is qualitatively sought from the dependence of the coupling between the localized photon and the guided modes on x. The spectral maximum frequency of the point source downshifts from the center of the band gap to the bottom edge of the band gap as x increases because the band gap range shifts up to higher frequency when x increases, as seen in Fig. 1. Since the optical confinement for a photon localized around the wire becomes weak when the frequency of the localized photon gets away from the center frequency of the band gap, the coupling depending on the overlapping of the localized photon and the guided mode becomes strong. As a result, the
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Fig. 4. Spatial distributions of the electric fields radiated from the radiative nanowire when the distance between the point source and a detector d is 14Λ (a), 13Λ (b), and 12Λ (c), and the measured intensity at the detector (d), where Λ is the period of the photonic crystal. The simulations were performed by the finite-difference time-domain method. Each unit cell was divided into 16 × 16 discretization grid cells. The total number of time steps was 215 , with each time step 1t = 1x/(2c). The radiative source was simulated as a Gaussian in the time and space domain. The spectral maximum frequency of the source is matched to the center frequency of the band gap, 2.36 eV.
photonic crystals can exhibit photonic band gaps in the visible range. As the mole fraction of Mg, x, increases, the relative band gap decreases and the position of the band gap increases because the refractive index of the nanowire decreases when x increases. The relative size and the position of the band gap, and the defect frequency of vacant point defects can be expressed as linear functions of x (x < 0.25). We also showed from the finite-difference time-domain simulations that the light extraction from the nanowire could be controlled by varying the distance between the optically excited wire and the waveguide, and the mole fraction of Mg. Controlling the light extraction from nanostructures could give an opportunity to implement nanoscale light emitting devices. Fig. 5. Dependence of photon intensity measured at the detector on x. The distance between the point source and the detector is 13Λ. The spectral maximum frequency of point source is assumed to be 2.36 eV, which is the center frequency of the band gap when x = 0. One can see that the measured intensity at the detector increases as x increases.
strong coupling between the localized photon and the guided modes enhances the light extraction from the nanowire to the waveguide. 4. Conclusion In conclusion, we theoretically investigated the optical properties of Mgx Zn1−x O nanowire photonic crystals. Nanowire
Acknowledgments This work was supported by the Korea Science and Engineering Foundation through the Quantum Photonic Science Research Center. References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486. [3] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, S. Kawakami, Phys. Rev. B 58 (1998) 10096. [4] S. Foteinopoulou, E.N. Economou, C.M. Soukoulis, Phys. Rev. Lett. 90 (2003) 107402.
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[15] D. Banerjee, J.Y. Lao, D.Z. Wang, J.Y. Huang, Z.F. Ren, D. Steeves, B. Kimball, M. Sennett, Appl. Phys. Lett. 83 (2003) 2061. [16] T. Nobis, E.M. Kaidashev, A. Rahm, M. Lorenz, M. Grundmann, Phys. Rev. Lett. 93 (2004) 103903. [17] M. Plihal, A.A. Maradudin, Phys. Rev. B 44 (1991) 8565. [18] R. Padjen, J.M. Gerard, J.Y. Marzin, J. Mod. Optic. 41 (1994) 295. [19] R. Schmdit, B. Rheinlander, M. Schubert, D. Spemann, T. Butz, J. Lenzner, E.M. Kaidashev, M. Lorenz, A. Rahm, H.C. Semmelhack, M. Grundmann, Appl. Phys. Lett. 82 (2003) 2260. [20] W. Zhang, A. Hu, X. Lei, N. Xu, N. Ming, Phys. Rev. B 54 (1996) 10280. [21] C.-S. Kee, J.-E. Kim, H.Y. Park, I. Park, H. Lim, Phys. Rev. B 61 (2000) 15523. [22] C.-S. Kee, H. Lim, Phys. Rev. B 64 (2001) 121103(R). [23] C.-S. Kee, D.-K. Ko, J. Lee, J. Phys. D: Appl. Phys. 38 (2005) 3850. [24] M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, S. Noda, Science 308 (2005) 1296. [25] A. Taflove, Computational Electrodynamics: The Finite-Difference TimeDomain Method, Artech House, Boston, 1995.
Optical properties of Mgx Zn1−x O nanowire photonic crystals Chul-Sik Kee ∗ , Do-Kyeong Ko, Jongmin Lee Nanophotonics Laboratory, Advanced Photonics Research Institute, GIST, Gwangju 500-712, Republic of Korea Received 15 June 2006; received in revised form 12 January 2007; accepted 14 February 2007 by J.W.P. Hsu Available online 25 February 2007
Abstract We investigate the optical properties of two-dimensional periodic arrays of well-aligned Mgx Zn1−x O nanowires, i.e., Mgx Zn1−x O nanowire photonic crystals. The nanowire photonic crystal can exhibit a photonic band gap in the visible range. As the mole fraction of Mg, x, increases, the edge frequencies of the band gap increase and the band gap size decreases. The characteristics of relative band gap and vacant point defect mode are also studied with varying x. From the finite-difference time-domain simulations, we show that the light extraction from nanowires can be controlled by varying the distance between optically excited nanowires and a waveguide, and the mole fraction of Mg. Controlling the light extraction from nanostructures can be useful in the implementation of nanoscale light emitting devices. c 2007 Elsevier Ltd. All rights reserved.
PACS: 42.70.Qs; 42.60.Da Keywords: A. Photonic crystal; B. ZnO nanowire
1. Introduction Artificial structures with periodic dielectric modulation, i.e., photonic crystals, can exhibit frequency ranges in which the propagation of photons is prohibited, i.e., photonic band gaps, and highly dispersive properties causing the anomalous refraction phenomena such as super-prism effects and negative refraction [1–4]. Those properties have been demonstrated to be useful in controlling the propagation and emission of photons [5–9]. Nanostructures with high aspect ratio, such as nanotubes, nanowires, and nanorods, have attracted much attention because they are expected to improve the performance of electronic devices, data storage, biochemical and chemical sensors, optoelectronic devices, and so on. Very recently, the successful fabrications of periodic arrays of InAs nanowires and carbon nanotubes have been reported [10,11]. These periodically aligned nanostructures are very attractive optical materials because they exhibit a periodic modulation of the dielectric constant. Thus, photonic crystals based on the nanostructures could offer unprecedented properties playing a ∗ Corresponding author. Tel.: +82 62 970 3426; fax: +82 62 970 3419.
E-mail address: [email protected] (C.-S. Kee). c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.02.021
crucial role in developing future nanophotonics [12,13]. Among the nanostructures, a Mgx Zn1−x O nanowire is an attractive candidate for an efficient ultraviolet light emitting nanodevice. Recently, well-aligned nanowire arrays have been synthesized and their optical properties have been investigated [14,15]. However, the optical properties of periodic arrays of well-aligned nanowires, i.e., nanowire photonic crystals, have been rarely studied. In this paper, we investigate the optical properties of Mgx Zn1−x O nanowire photonic crystals using an effective numerical method to treat frequency-dependent materials. The nanowire photonic crystals can exhibit photonic band gaps in the visible range. As the mole fraction of Mg, x, increases, the edge frequencies of the band gap increase and the band gap decreases because the refractive index of the nanowire decreases when x increases. The relative size and center frequency of the band gap, and the frequency of defect mode caused by a vacant point defect, are expressed as a linear function of x. We also show from the finite-difference timedomain simulations that the light extraction from the nanowire could be controlled by the distance between the optically excited wire and a waveguide. Controlling the light extraction from the nanowires could give an opportunity to implement nanoscale light emitting devices.
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2. Computational model and method The height of a nanowire is in general much longer than its diameter [15,16]. Thus a periodic array of well-aligned nanowires in air can be treated as a two-dimensional photonic crystal. In a two-dimensional photonic crystal, there are two independent modes, i.e., E-polarized mode (E k z) and H polarized mode (E ⊥ z), where E is the electric field of the mode and z is the longitudinal axis of the nanowire. Usually, a Mgx Zn1−x O nanowire has a wurtzite crystal structure and its preferred growth direction is along the c-axis of Mgx Zn1−x O, that is, the c-axis is parallel to the z-direction. Previous works have shown that a two-dimensional triangular photonic crystal composed of dielectric rods in air exhibits a large band gap between the first and second bands for E-polarized modes even though the refractive index of the rod is small [17,18]. Note that the luminescence of the E-polarized mode is much stronger than that of the H -polarized mode, due to the geometry of the nanowire. This means that the band gap for the E-polarized mode affects mainly the luminescent properties of nanowire photonic crystals. Thus our attention is restricted to the Epolarized mode. The refractive index for polarization parallel to the c-axis of Mgx Zn1−x O was recently measured in the infrared and visible ranges when the mole fraction of Mg, x, is lower than 0.29 [19]. The refractive index was dependent of both frequency ω and x, and was fitted in the three-term Cauchy approximation type formula as follows: n k (ω, x) = 1.844 + ax + +
(0.0181 + bx)(h¯ ω)2 (1.24 eV)2
(0.0036 + cx)(h¯ ω)4 , (1.24 eV)4
(1)
where a = −0.574, b = −0.0451, and c = −0.0049. The eigenvalue equation for E-polarized modes of a twodimensional photonic crystal is as follows: ω2 0 (2) det M(K, K ) − 2 δK,K0 = 0, c where M(K, K0 ) = |K||K0 |Q(K − K0 ).
(3)
Here, K = k + G, K0 = k + G0 , and Q(K − K0 ) is the Fourier transformation of 1/n 2k , where k is the wave vector in the first Brillouin zone, and G, G0 the reciprocal vectors. We employed an effective numerical method to treat arbitrary frequency-dependent media in the photonic band calculations. We calculated the matrix solutions of Eq. (2) as follows: we took a few thousand mesh points for 1 eV ≤ h¯ ω ≤ 3 eV for a given k and then calculated the determinants of the matrix equation at these mesh points until we found the region where the determinants of the neighboring mesh points change sign. This method has been successfully employed in the calculation of photonic band structures of photonic crystals with frequency-dependent or µ [20–22].
Fig. 1. The first and second photonic bands of a two-dimensional triangular photonic crystal composed of Mgx Zn1−x O nanowires with the mole fraction of Mg, x = 0, when the period of the nanowire array is 240 nm and the diameter of the wires is 100 nm (left), and the dependence of the band gap edge frequencies on x (right). The inset denotes the first Brillouin zone of a two-dimensional triangular lattice.
3. Results and discussion Fig. 1 shows the first and second photonic bands of a two-dimensional triangular photonic crystal composed of Mgx Zn1−x O nanowires with x = 0 in air when the period of the nanowire array is 240 nm and the diameter of the wires is 100 nm. The inset denotes the first Brillouin zone of a two-dimensional triangular lattice. One can see that there is a photonic band gap between the K point of the first band and the J point of the second band. The gap range is from about 2.17 to 2.63 eV (471–571 nm). Mg is usually doped in ZnO to control the electronic band gap of Mgx Zn1−x O. So, the dependence of the photonic band gap on the mole fraction of Mg would be interesting. The dependence of the band gap edge frequencies on x are represented in the right of Fig. 1. As x increases, the edge frequencies increase and the band gap size decreases. This is easily understood from the fact that the refractive index of a Mgx Zn1−x O nanowire decreases as x increases (see Eq. (1)). Note that there is an x-independent band gap region which is from about 2.37 to 2.63 eV (471–523 nm). We also investigated the dependence of the band gap on the diameter of the wire for all studied values of x. We found that the relative band gap defined as 1ω/ωmid has a maximum value when the diameter is between 122 and 127 nm, where 1ω is the absolute band gap size and ωmid is the center frequency of the band gap. Fig. 2 shows the dependence of the maximum value of relative band gap, 1ω/ωmid,MAX (blue circle), and ωmid (red circle) on x. One can see that 1ω/ωmid,MAX decreases and ωmid increases as x increases. This is because the refractive index of the wire decreases when x increases. 1ω/ωmid,MAX and ωmid can be expressed by a linear function of x, 1ω/ωmid,MAX (x) = −2.88x + 0.214, and h¯ ωmid = (0.76x + 2.35) eV. Introducing various defects into photonic crystals can create allowed modes within the band gaps and give rise to various applications such as control of spontaneous emission, efficient bent guiding of lights, and ultrahigh quality cavities [5–7]. We studied the dependence of the defect mode frequency,
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Fig. 2. Dependence of the maximum value of the relative band gap 1ω/ωmid,MAX (blue circle) and ωmid (red circle) on x, where 1ω is the absolute band gap size and ωmid is the center frequency of the band gap. 1ω/ωmid,MAX and ωmid can be expressed by a linear function of x, 1ω/ωmid,MAX (x) = −2.88x + 0.214, and h¯ ωmid = (0.76x + 2.37) eV, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Dependence of the frequency of the defect mode created by removing one wire, ωd , on x. ωd is linearly proportional to x: h¯ ωd (x) = (0.5x + 2.35) eV.
ωd , created by removing one wire, on x. Fig. 3 shows that ωd is linearly proportional to x: h¯ ωd (x) = (0.5x + 2.37) eV. The empirical formulas would be useful in estimating 1ω/ωmid,MAX , the band gap position, and the defect frequency for arbitrary values of x (x < 0.25). The light emission from a Mgx Zn1−x O nanowire can be altered by the photonic band gap. Our previous work has shown that the photons emitted from the nanowire do not propagate but are strongly localized around the nanowire when the frequencies of emitted photons are within the photonic band gap [23]. However, it is worth noting that the photons localized around the nanowire can escape from the nanowire to a waveguide when a waveguide is near the nanowire because of the optical coupling between the photons localized around the wire and the guided modes of the waveguide [24]. To observe the light escape from the nanowire to a waveguide, we created a waveguide by removing a few nanowires, as shown in Fig. 4, and simulated the radiations from the nanowire.
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The simulations were performed by the finite-difference timedomain method with the perfectly matched layer absorbing boundary condition [25]. Each unit cell was divided into 16×16 discretization grid cells. The total number of time steps was 215 , with each time step 1t = 1x/(2c). The radiative light source was located at the center of the nanowire and simulated as a Gaussian in the time and space domain. The structural parameters employed in the simulations are identical to those of Fig. 1, and the spectral maximum frequency of the emitted photon is matched to the center frequency of the band gap. The guiding range of the waveguide includes the center frequency. To see the coupling characteristics of the photons localized around the nanowire and the guiding modes of the waveguide, we changed the position of the radiative nanowire that emits photons. Fig. 4(a)–(c) show the spatial distributions of the electric fields radiated from the radiative nanowires located at three different positions. In the figures, solid circles denote the nanowires and d is the distance between the radiative nanowire and the fixed detector. Λ is the period of the photonic crystal. One can see from the figures that the photons localized around the radiative nanowire due to the photonic band gap effect (see Fig. 4(a)) can escape from the radiative nanowire to the waveguide when the radiative nanowire becomes close to the waveguide (see Fig. 4(b) and (c)). It is shown in Fig. 4(d) that the measured intensity at the detector linearly increases when d decreases. This shows clearly that the coupling between the photon localized around the radiative nanowire and the guided mode of the waveguide becomes strong when the distance between the radiative nanowire and the waveguide decreases, because the coupling depends on the overlapping of the localized photon and the guided mode. We expect that the light extraction from the nanowire along the waveguide could be observed by exciting the nanowire located around the waveguide optically. The light extraction should depend on the position of excited nanowire. Controlling the light extraction from nanostructures could be useful in implementing new types of nanoscale light emitting devices. It would be interesting to investigate the influence of changing the mole fraction of Mg on the light extraction from the nanowire. To study the influence qualitatively, we assumed that the spectral maximum frequency of the point source is unchanged when x varies. Fig. 5 shows the photon intensity measured at the detector when x = 0, 0.05, 0.1 0.15, 0.2. The distance between the point source and the detector is 13Λ. One can see that the measured intensity at the detector increases as x increases. The increase of light extraction from the nanowire to the waveguide is qualitatively sought from the dependence of the coupling between the localized photon and the guided modes on x. The spectral maximum frequency of the point source downshifts from the center of the band gap to the bottom edge of the band gap as x increases because the band gap range shifts up to higher frequency when x increases, as seen in Fig. 1. Since the optical confinement for a photon localized around the wire becomes weak when the frequency of the localized photon gets away from the center frequency of the band gap, the coupling depending on the overlapping of the localized photon and the guided mode becomes strong. As a result, the
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Fig. 4. Spatial distributions of the electric fields radiated from the radiative nanowire when the distance between the point source and a detector d is 14Λ (a), 13Λ (b), and 12Λ (c), and the measured intensity at the detector (d), where Λ is the period of the photonic crystal. The simulations were performed by the finite-difference time-domain method. Each unit cell was divided into 16 × 16 discretization grid cells. The total number of time steps was 215 , with each time step 1t = 1x/(2c). The radiative source was simulated as a Gaussian in the time and space domain. The spectral maximum frequency of the source is matched to the center frequency of the band gap, 2.36 eV.
photonic crystals can exhibit photonic band gaps in the visible range. As the mole fraction of Mg, x, increases, the relative band gap decreases and the position of the band gap increases because the refractive index of the nanowire decreases when x increases. The relative size and the position of the band gap, and the defect frequency of vacant point defects can be expressed as linear functions of x (x < 0.25). We also showed from the finite-difference time-domain simulations that the light extraction from the nanowire could be controlled by varying the distance between the optically excited wire and the waveguide, and the mole fraction of Mg. Controlling the light extraction from nanostructures could give an opportunity to implement nanoscale light emitting devices. Fig. 5. Dependence of photon intensity measured at the detector on x. The distance between the point source and the detector is 13Λ. The spectral maximum frequency of point source is assumed to be 2.36 eV, which is the center frequency of the band gap when x = 0. One can see that the measured intensity at the detector increases as x increases.
strong coupling between the localized photon and the guided modes enhances the light extraction from the nanowire to the waveguide. 4. Conclusion In conclusion, we theoretically investigated the optical properties of Mgx Zn1−x O nanowire photonic crystals. Nanowire
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