April 15, 2015 / Vol. 40, No. 8 / OPTICS LETTERS
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Conditions for stronger field enhancement of semiconductor bowtie nanoantennas Mitsuharu Uemoto1 and Hiroshi Ajiki2,* 1
Department of Materials Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan 2 Photon Pioneers Center, Osaka University, Yamadaoka, Suita Osaka 565-0871, Japan *Corresponding author: [email protected]‑u.ac.jp Received December 16, 2014; revised March 10, 2015; accepted March 16, 2015; posted March 20, 2015 (Doc. ID 230818); published April 8, 2015 A semiconductor bowtie nanoantenna acts as a high-quality cavity because a strongly enhanced field with a narrow spectral width appears at a nanogap region owing to exciton resonance. We theoretically investigate suitable antenna structures to obtain a strong field enhancement, and the following conditions are found: (i) the antenna structure is long in the direction of light polarization, and (ii) the tip structure near the nanogap is blunt. Condition (ii) is opposite to that for a metallic bowtie nanoantenna because the exciton wave function is distributed to avoid a narrow area near the sharp tip. These conditions are expected to be guidelines for designing efficient semiconductor nanoantennas for various applications. © 2015 Optical Society of America OCIS codes: (130.3990) Micro-optical devices; (130.5990) Semiconductors; (160.6000) Semiconductor materials; (240.3990) Micro-optical devices; (240.5420) Polaritons. http://dx.doi.org/10.1364/OL.40.001695
The surface plasmon resonance of a metallic nanostructure aggregates optical-frequency light into a subwavelength-scale area (optical antenna effect) [1], and the field intensity is considerably enhanced at an area called a “hotspot.” To obtain a strong field intensity, various metallic nanostructures have been studied extensively. A typical structure for an optical antenna is a dimer of metallic islands separated by a few nanometers (metallic bowtie nanoantenna) [2–5]. At the nanogap region between the metallic islands, the light field is strongly localized and enhanced because the hotspot area is smaller than light wavelength beyond the diffraction limit [6]. Such a strongly enhanced and localized field has attracted much attention because of its potential for a wide range of applications, e.g., chemical and biological sensing [2,7] and optical manipulations [8,9]. The strong light confinement at the hotspot indicates that the bowtie nanoantenna acts as an optical cavity. Because the hotspot region or mode volume is small beyond the diffraction limit, the vacuum Rabi splitting of a single quantum dot or molecules placed at the hotspot is expected to be much larger than that inside a conventional microcavity. In fact, giant Rabi splitting using the metallic nanoantenna were reported in a hyperspectral transmission microscope [10] and Raman scattering measurements [11,12]. However, the quality factor (Q factor) of the metallic nanostructure is limited to a small value in the range of 10–100 because of its large dephasing due to electron–electron [13], electron– phonon [14], and electron-surface scattering [15]. Thus, the observed Rabi splitting spectra are very broad. In order to increase the Q factor and maintain a small mode volume, the authors proposed a semiconductor nanogap cavity utilizing the exciton resonance of a bowtie nanoantenna [16]. The semiconductor nanogap cavity has a large Q factor of ∼104 and a small mode volume beyond the diffraction limit; as a result, a large and well-defined Rabi splitting has been theoretically demonstrated [16]. 0146-9592/15/081695-04$15.00/0
In this Letter, we theoretically investigate the structure dependence of the field enhancement of a semiconductor bowtie nanoantenna. Because the field enhancement is attributed to an exciton, the enhancement condition is qualitatively different from that of the metallic nanoantenna. The present study leads to guidelines for the design of a semiconductor nanogap cavity or nanoantenna providing a more strongly enhanced field. Here, we consider two types of semiconductor nanoantennas consisting of (a) two nanorods for studying the dependence of the antenna arm length and (b) two nanoprisms for studying the edge-angle dependence. Schematic illustrations of the nanoantenna structures and their parameters are shown in Fig. 1. Incident plane-wave light with x polarization, i.e., parallel polarization to the long axis of the nanoantenna, is applied in the z direction. We consider a CuCl nanoantenna and its Z 3 exciton with an exciton band-edge energy of 3.202 eV, a longitudinal-transverse splitting energy of 5.7 meV, an
Fig. 1. Schematic illustrations and structural parameters of (a) a nanorod antenna and (b) a nanoprism antenna, where L is the length of the antenna arms, R is the radius of curvature at the edges near the nanogap, and θ is the angle of the tip head of the nanoprism. © 2015 Optical Society of America
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exciton translation mass of 2.3 me (where me is the free electron mass), and a background dielectric constant of 5.56 [17]. The exciton damping constant is assumed to be 100 μeV, which was observed at T 40 K [18]. For simplicity, the surrounding dielectric constant including a substrate is set to 1 in vacuum because the dielectric constant does not affect the structure dependence on the field enhancement. If the substrate is considered, the field enhancement is expected to become stronger for higher dielectric substrate. Conventional electromagnetic (EM) simulation methods cannot be directly applied to exciton-active semiconductor nanostructures because light propagates as two-mode exciton polaritons inside semiconductors. In addition, a longitudinal exciton mode should be considered because the longitudinal-transverse mixed modes of the exciton strongly couple with light in a nanostructure [19]. Although an EM-field simulation for the exciton polariton resonance was developed using a finite difference time-domain (FDTD) method [20], the longitudinal component of the exciton was ignored. The time sequential analysis of the FDTD method is not suitable for the EM simulation under exciton resonance because the longitudinal field provides an instantaneous Coulomb interaction. To overcome these difficulties, the authors have developed a new EM-simulation method based on a finite-element method (FEM) accounting for the exciton effects [21] and have applied the method to the Rabi splitting spectra of a semiconductor nanogap cavity [16]. We use the new method in the following calculations. First, we study the arm-length dependence of the field enhancement using the nanorod antenna [see Fig. 1(a)]. Figure 2(a) shows the electric-field distribution around the nanorod antenna with L 20 nm and R 5 nm under exciton resonance energy. Such a field distribution can be measured using a scattering-type near-field scanning optical microscopy (NSOM) with a very sharp silicon tip [22]. A strongly enhanced field appears at the curved edges around the nanogap region. It is noted that a field enhancement occurs owing to the exciton resonance instead of the surface plasmon resonance of a metallic nanostructure. The enhanced field is much stronger than that at the edges on opposite sides, indicating considerable coupling between the exciton resonances of the two nanorods. Figure 2(b) shows the near-field spectra at the center of the nanogap for R 5 nm (red solid line) and R 7 nm (green solid line) and for L 18, 20, and 22 nm. The near-field intensity is estimated by averaging the electric-field intensity over a cubic region with 3 nm on each side. As R and/or L decrease, the spectral peak is blue-shifted owing to the quantum confinement effect. A similar peak shift as a function of L has been found for metallic nanoantennas [23,24]. The near-field enhancement factor (the peak value of jE∕Einc j2 ) is nearly independent of R. This fact indicates that the length of the nanoantenna in the direction perpendicular to the light polarization does not affect the field enhancement. On the other hand, the enhancement factor linearly increases with L, as shown in Fig. 2(c). The similar behavior can be seen for metallic nanoantennas [23,24]. The long antenna arms parallel to the light polarization result in a strong field enhancement. We find a Q factor of 1.5 × 104 for a semiconductor nanorod antenna
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Fig. 2. (a) Near-field distribution around the nanorod antenna with L 20 nm and R 5 nm under an exciton resonant condition. (b) Near-field spectra of a nanorod antenna at the center of the nanogap for R 5 nm (red solid line) and R 7 nm (green solid line). The field intensity is normalized by that of the incident field jEinc j2 . (c) Enhancement factor under the exciton resonance as a function of the arm length L. The red circles and green triangles represent the enhancement factor for nanorod antennas with R 5 nm and R 7 nm, respectively.
with R 5 nm and L 20 nm, which is much larger than that of a metallic nanoantenna [16]. Next, we study the tip-angle dependence of the field enhancement for the nanoprism antenna [see Fig. 1(b)]. We consider various tip angles θ and curvature radii R. The arm length L for each set of the parameters is determined so that the base area of each nanoprism becomes a constant of L2 sin θ∕2 700 nm2 to fix the oscillator strength of the confined exciton. Figure 3(a) shows the near-field spectra at the center of the nanogap of the semiconductor nanoprism antenna with R 0.5 nm and θ 60°. Three resonant peaks appear at the energies of the three quantized levels of the confined exciton. Figure 3(b) shows the enhancement factor as a function of θ for R 0.5 , 1, and 3 nm. Let us focus on the enhancement factor for the lowest exciton level. The enhancement factor of the nanoprisms with R 0.5 nm increases with θ, reaches a maximum, and then decreases. The tip-angle dependence in the small-angle region can be understood from the exciton distribution. The wave function of a low-energy exciton is distributed to avoid the narrow area near the sharp tip because of the position-momentum uncertainty relationship. In fact, the near-field intensity is distributed away from the tips for θ 30°, as shown in Fig. 3(c). Consequently, a strong field enhancement occurs at the opposite-side edges of the tips, and the field enhancement is weak around the nanogap. On the other hand, the field enhancement originating from the arm-length effect decreases with the increase in θ because the arm length becomes shorter according to L2 sin θ∕2 700 nm2 . For large θ, this effect becomes dominant; thus, the field
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Fig. 3. (a) Near-field spectrum of a semiconductor nanoprism antenna at the center of the nanogap. (b) Near-field enhancement factor of a semiconductor nanoprism antenna under the exciton resonance as a function of the tip angle θ for R 0.5 nm (red squares), 1 nm (green circles), and 3 nm (blue triangles). (c) Near-field distribution around the semiconductor nanoprism antenna with R 0.5 nm under the resonant excitation of the lowest exciton. (d) Near-field spectrum of a metallic nanoprism antenna at the center of the nanogap. (e) Near-field distribution around the metallic nanoprism antenna with R 0.5 nm under the surface plasmon resonance.
enhancement totally decreases with the increase in θ > 50°. For the nanoprism with R 1 nm, the enhancement factors of the lowest level indicated by the green circles become larger than those for R 0.5 nm over the entire range of θ, and the tip angle that results in the maximum enhancement factor becomes smaller (θ 40°). This is because the exciton wave function tends to spread near the tip area for larger values of R. This tendency becomes more remarkable for R 3 nm; the enhancement factor monotonically increases with decreasing tip angle and reaches 700 owing to the wave function distribution near the tip and the long arm length. As a reference, we calculate the enhancement factor of a gold nanoprism antenna with R 0.5 nm in Fig. 3(d). We use the Drude-type dielectric function with a plasma frequency of 8.9 meV and a damping energy of 50 meV observed at T 40 K [25]. For a metallic nanoprism antenna, the enhancement spectrum exhibits a single peak originating from the surface plasmon resonance.
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The maximum enhancement factor reaches 3.5 × 104 , which is approximately 100 times larger than that of the semiconductor nanoprism antenna. However, the spectral width is very broad, and the corresponding Q factor is approximately 50. The tip-angle dependence of the enhancement factor is shown in Fig. 3(e). The peak intensity monotonically increases with the decrease in θ; namely, a strongly enhanced field can be obtained by sharpening the tip heads. This is because the electron charge density of the surface plasmon is more concentrated on the sharper tip, which leads to a strong field enhancement. It is noteworthy that the tip-angle dependence of the enhancement factor for the semiconductor nanoprism antenna with R 0.5 nm and 1 nm is very different from that of the metallic one; for a sharper tip, the enhancement factor becomes larger for a metallic nanoantenna but smaller for a semiconductor nanoantenna. In conclusion, we have investigated the suitable structure of a semiconductor bowtie nanoantenna for realizing a more strongly enhanced EM field at its nanogap region. The field enhancement becomes large for a long-arm nanoantenna along the polarization direction of light and for blunt tips near the nanogap. The latter condition is very different from that for a metallic bowtie nanoantenna, in which a strong field enhancement occurs for a sharp-tip structure. The difference originates from the different features of the spatial distribution of the exciton and surface plasmon. The great advantage of the semiconductor bowtie nanoantenna is the narrow spectral peak of the strongly enhanced field; thus, the nanoantenna acts as a high-Q cavity with a small mode volume beyond the diffraction limit. Such an optical cavity can be applied to various applications in quantum information processing, e.g., a highly efficient entangled photon generator [26,27]. Our results provide guidelines for designing desirable semiconductor nanoantennas for various applications. The authors thank Prof. H. Ishihara and Prof. H. Yoshida for fruitful discussions and comments. This work was supported by the Japan Society for the Promotion of Science (JSPS) “Core to Core” Program, the Japan Science and Technology Agency (JST) Advanced Low Carbon (ALCA) Research and Development Program, and a Grant-in-Aid for Scientific Research (C), No. 25400325. References 1. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005). 2. R. D. Grober, R. J. Schoelkopf, and D. E. Prober, Appl. Phys. Lett. 70, 1354 (1997). 3. D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. Moerner, Nano Lett. 4, 957 (2004). 4. Y. Sawai, B. Takimoto, H. Nabika, K. Ajito, and K. Murakoshi, J. Am. Chem. Soc. 129, 1658 (2007). 5. K. Ueno, S. Juodkazis, and T. Shibuya, J. Am. Chem. Soc. 130, 6928 (2008). 6. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010). 7. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. Moerner, Nat. Photonics 3, 654 (2009). 8. A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
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9. Y. Tsuboi, T. Shoji, N. Kitamura, M. Takase, K. Murakoshi, Y. Mizumoto, and H. Ishihara, J. Phys. Chem. Lett. 1, 2327 (2010). 10. A. E. Schlather, N. Large, A. S. Urban, P. Nordlander, and N. J. Halas, Nano Lett. 13, 3281 (2013). 11. F. Nagasawa, M. Takase, and K. Murakoshi, J. Phys. Chem. Lett. 5, 14 (2014). 12. T. Itoh, Y. S. Yamamoto, H. Tamaru, V. Biju, S.-I. Wakida, and Y. Ozaki, Phys. Rev. B 89, 195436 (2014). 13. W. Lawrence and J. Wilkins, Phys. Rev. B 7, 2317 (1973). 14. T. Holstein, Ann. Phys. 29, 410 (1964). 15. C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H. Petrova, M. Reismann, P. Mulvaney, and G. V. Hartland, Phys. Chem. Chem. Phys. 8, 3540 (2006). 16. M. Uemoto and H. Ajiki, Opt. Express 22, 22470 (2014). 17. K. Cho and H. Ishihara, J. Phys. Soc. Jpn. 59, 754 (1990).
18. M. Kalm and C. Uihlein, J. Phys. Stat. Solidi B 87, 575 (1978). 19. H. Ajiki and K. Cho, Phys. Rev. B 62, 7402 (2000). 20. Y. Zeng, Y. Fu, M. Bengtsson, X. Chen, W. Lu, and H. Ågren, Opt. Express 16, 4507 (2008). 21. M. Uemoto and H. Ajiki, Opt. Express 22, 9450 (2014). 22. Y. Tanaka, H. Ishiguro, H. Fujiwara, Y. Yokota, K. Ueno, H. Misawa, and K. Sasaki, Opt. Express 19, 7726 (2011). 23. E. J. Smythe, E. Cubukcu, and F. Capasso, Opt. Express 15, 7439 (2007). 24. G. W. Bryant, F. J. Garca de Abajo, and J. Aizpurua, Nano Lett. 8, 631 (2008). 25. M. Liu, M. Pelton, and P. Guyot-Sionnest, Phys. Rev. B 79, 035418 (2009). 26. H. Ajiki, H. Ishihara, and K. Edamatsu, New J. Phys. 11, 033033 (2009). 27. K. Shibata and H. Ajiki, Phys. Rev. A 89, 042319 (2014).
1695
Conditions for stronger field enhancement of semiconductor bowtie nanoantennas Mitsuharu Uemoto1 and Hiroshi Ajiki2,* 1
Department of Materials Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan 2 Photon Pioneers Center, Osaka University, Yamadaoka, Suita Osaka 565-0871, Japan *Corresponding author: [email protected]‑u.ac.jp Received December 16, 2014; revised March 10, 2015; accepted March 16, 2015; posted March 20, 2015 (Doc. ID 230818); published April 8, 2015 A semiconductor bowtie nanoantenna acts as a high-quality cavity because a strongly enhanced field with a narrow spectral width appears at a nanogap region owing to exciton resonance. We theoretically investigate suitable antenna structures to obtain a strong field enhancement, and the following conditions are found: (i) the antenna structure is long in the direction of light polarization, and (ii) the tip structure near the nanogap is blunt. Condition (ii) is opposite to that for a metallic bowtie nanoantenna because the exciton wave function is distributed to avoid a narrow area near the sharp tip. These conditions are expected to be guidelines for designing efficient semiconductor nanoantennas for various applications. © 2015 Optical Society of America OCIS codes: (130.3990) Micro-optical devices; (130.5990) Semiconductors; (160.6000) Semiconductor materials; (240.3990) Micro-optical devices; (240.5420) Polaritons. http://dx.doi.org/10.1364/OL.40.001695
The surface plasmon resonance of a metallic nanostructure aggregates optical-frequency light into a subwavelength-scale area (optical antenna effect) [1], and the field intensity is considerably enhanced at an area called a “hotspot.” To obtain a strong field intensity, various metallic nanostructures have been studied extensively. A typical structure for an optical antenna is a dimer of metallic islands separated by a few nanometers (metallic bowtie nanoantenna) [2–5]. At the nanogap region between the metallic islands, the light field is strongly localized and enhanced because the hotspot area is smaller than light wavelength beyond the diffraction limit [6]. Such a strongly enhanced and localized field has attracted much attention because of its potential for a wide range of applications, e.g., chemical and biological sensing [2,7] and optical manipulations [8,9]. The strong light confinement at the hotspot indicates that the bowtie nanoantenna acts as an optical cavity. Because the hotspot region or mode volume is small beyond the diffraction limit, the vacuum Rabi splitting of a single quantum dot or molecules placed at the hotspot is expected to be much larger than that inside a conventional microcavity. In fact, giant Rabi splitting using the metallic nanoantenna were reported in a hyperspectral transmission microscope [10] and Raman scattering measurements [11,12]. However, the quality factor (Q factor) of the metallic nanostructure is limited to a small value in the range of 10–100 because of its large dephasing due to electron–electron [13], electron– phonon [14], and electron-surface scattering [15]. Thus, the observed Rabi splitting spectra are very broad. In order to increase the Q factor and maintain a small mode volume, the authors proposed a semiconductor nanogap cavity utilizing the exciton resonance of a bowtie nanoantenna [16]. The semiconductor nanogap cavity has a large Q factor of ∼104 and a small mode volume beyond the diffraction limit; as a result, a large and well-defined Rabi splitting has been theoretically demonstrated [16]. 0146-9592/15/081695-04$15.00/0
In this Letter, we theoretically investigate the structure dependence of the field enhancement of a semiconductor bowtie nanoantenna. Because the field enhancement is attributed to an exciton, the enhancement condition is qualitatively different from that of the metallic nanoantenna. The present study leads to guidelines for the design of a semiconductor nanogap cavity or nanoantenna providing a more strongly enhanced field. Here, we consider two types of semiconductor nanoantennas consisting of (a) two nanorods for studying the dependence of the antenna arm length and (b) two nanoprisms for studying the edge-angle dependence. Schematic illustrations of the nanoantenna structures and their parameters are shown in Fig. 1. Incident plane-wave light with x polarization, i.e., parallel polarization to the long axis of the nanoantenna, is applied in the z direction. We consider a CuCl nanoantenna and its Z 3 exciton with an exciton band-edge energy of 3.202 eV, a longitudinal-transverse splitting energy of 5.7 meV, an
Fig. 1. Schematic illustrations and structural parameters of (a) a nanorod antenna and (b) a nanoprism antenna, where L is the length of the antenna arms, R is the radius of curvature at the edges near the nanogap, and θ is the angle of the tip head of the nanoprism. © 2015 Optical Society of America
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exciton translation mass of 2.3 me (where me is the free electron mass), and a background dielectric constant of 5.56 [17]. The exciton damping constant is assumed to be 100 μeV, which was observed at T 40 K [18]. For simplicity, the surrounding dielectric constant including a substrate is set to 1 in vacuum because the dielectric constant does not affect the structure dependence on the field enhancement. If the substrate is considered, the field enhancement is expected to become stronger for higher dielectric substrate. Conventional electromagnetic (EM) simulation methods cannot be directly applied to exciton-active semiconductor nanostructures because light propagates as two-mode exciton polaritons inside semiconductors. In addition, a longitudinal exciton mode should be considered because the longitudinal-transverse mixed modes of the exciton strongly couple with light in a nanostructure [19]. Although an EM-field simulation for the exciton polariton resonance was developed using a finite difference time-domain (FDTD) method [20], the longitudinal component of the exciton was ignored. The time sequential analysis of the FDTD method is not suitable for the EM simulation under exciton resonance because the longitudinal field provides an instantaneous Coulomb interaction. To overcome these difficulties, the authors have developed a new EM-simulation method based on a finite-element method (FEM) accounting for the exciton effects [21] and have applied the method to the Rabi splitting spectra of a semiconductor nanogap cavity [16]. We use the new method in the following calculations. First, we study the arm-length dependence of the field enhancement using the nanorod antenna [see Fig. 1(a)]. Figure 2(a) shows the electric-field distribution around the nanorod antenna with L 20 nm and R 5 nm under exciton resonance energy. Such a field distribution can be measured using a scattering-type near-field scanning optical microscopy (NSOM) with a very sharp silicon tip [22]. A strongly enhanced field appears at the curved edges around the nanogap region. It is noted that a field enhancement occurs owing to the exciton resonance instead of the surface plasmon resonance of a metallic nanostructure. The enhanced field is much stronger than that at the edges on opposite sides, indicating considerable coupling between the exciton resonances of the two nanorods. Figure 2(b) shows the near-field spectra at the center of the nanogap for R 5 nm (red solid line) and R 7 nm (green solid line) and for L 18, 20, and 22 nm. The near-field intensity is estimated by averaging the electric-field intensity over a cubic region with 3 nm on each side. As R and/or L decrease, the spectral peak is blue-shifted owing to the quantum confinement effect. A similar peak shift as a function of L has been found for metallic nanoantennas [23,24]. The near-field enhancement factor (the peak value of jE∕Einc j2 ) is nearly independent of R. This fact indicates that the length of the nanoantenna in the direction perpendicular to the light polarization does not affect the field enhancement. On the other hand, the enhancement factor linearly increases with L, as shown in Fig. 2(c). The similar behavior can be seen for metallic nanoantennas [23,24]. The long antenna arms parallel to the light polarization result in a strong field enhancement. We find a Q factor of 1.5 × 104 for a semiconductor nanorod antenna
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Fig. 2. (a) Near-field distribution around the nanorod antenna with L 20 nm and R 5 nm under an exciton resonant condition. (b) Near-field spectra of a nanorod antenna at the center of the nanogap for R 5 nm (red solid line) and R 7 nm (green solid line). The field intensity is normalized by that of the incident field jEinc j2 . (c) Enhancement factor under the exciton resonance as a function of the arm length L. The red circles and green triangles represent the enhancement factor for nanorod antennas with R 5 nm and R 7 nm, respectively.
with R 5 nm and L 20 nm, which is much larger than that of a metallic nanoantenna [16]. Next, we study the tip-angle dependence of the field enhancement for the nanoprism antenna [see Fig. 1(b)]. We consider various tip angles θ and curvature radii R. The arm length L for each set of the parameters is determined so that the base area of each nanoprism becomes a constant of L2 sin θ∕2 700 nm2 to fix the oscillator strength of the confined exciton. Figure 3(a) shows the near-field spectra at the center of the nanogap of the semiconductor nanoprism antenna with R 0.5 nm and θ 60°. Three resonant peaks appear at the energies of the three quantized levels of the confined exciton. Figure 3(b) shows the enhancement factor as a function of θ for R 0.5 , 1, and 3 nm. Let us focus on the enhancement factor for the lowest exciton level. The enhancement factor of the nanoprisms with R 0.5 nm increases with θ, reaches a maximum, and then decreases. The tip-angle dependence in the small-angle region can be understood from the exciton distribution. The wave function of a low-energy exciton is distributed to avoid the narrow area near the sharp tip because of the position-momentum uncertainty relationship. In fact, the near-field intensity is distributed away from the tips for θ 30°, as shown in Fig. 3(c). Consequently, a strong field enhancement occurs at the opposite-side edges of the tips, and the field enhancement is weak around the nanogap. On the other hand, the field enhancement originating from the arm-length effect decreases with the increase in θ because the arm length becomes shorter according to L2 sin θ∕2 700 nm2 . For large θ, this effect becomes dominant; thus, the field
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Fig. 3. (a) Near-field spectrum of a semiconductor nanoprism antenna at the center of the nanogap. (b) Near-field enhancement factor of a semiconductor nanoprism antenna under the exciton resonance as a function of the tip angle θ for R 0.5 nm (red squares), 1 nm (green circles), and 3 nm (blue triangles). (c) Near-field distribution around the semiconductor nanoprism antenna with R 0.5 nm under the resonant excitation of the lowest exciton. (d) Near-field spectrum of a metallic nanoprism antenna at the center of the nanogap. (e) Near-field distribution around the metallic nanoprism antenna with R 0.5 nm under the surface plasmon resonance.
enhancement totally decreases with the increase in θ > 50°. For the nanoprism with R 1 nm, the enhancement factors of the lowest level indicated by the green circles become larger than those for R 0.5 nm over the entire range of θ, and the tip angle that results in the maximum enhancement factor becomes smaller (θ 40°). This is because the exciton wave function tends to spread near the tip area for larger values of R. This tendency becomes more remarkable for R 3 nm; the enhancement factor monotonically increases with decreasing tip angle and reaches 700 owing to the wave function distribution near the tip and the long arm length. As a reference, we calculate the enhancement factor of a gold nanoprism antenna with R 0.5 nm in Fig. 3(d). We use the Drude-type dielectric function with a plasma frequency of 8.9 meV and a damping energy of 50 meV observed at T 40 K [25]. For a metallic nanoprism antenna, the enhancement spectrum exhibits a single peak originating from the surface plasmon resonance.
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The maximum enhancement factor reaches 3.5 × 104 , which is approximately 100 times larger than that of the semiconductor nanoprism antenna. However, the spectral width is very broad, and the corresponding Q factor is approximately 50. The tip-angle dependence of the enhancement factor is shown in Fig. 3(e). The peak intensity monotonically increases with the decrease in θ; namely, a strongly enhanced field can be obtained by sharpening the tip heads. This is because the electron charge density of the surface plasmon is more concentrated on the sharper tip, which leads to a strong field enhancement. It is noteworthy that the tip-angle dependence of the enhancement factor for the semiconductor nanoprism antenna with R 0.5 nm and 1 nm is very different from that of the metallic one; for a sharper tip, the enhancement factor becomes larger for a metallic nanoantenna but smaller for a semiconductor nanoantenna. In conclusion, we have investigated the suitable structure of a semiconductor bowtie nanoantenna for realizing a more strongly enhanced EM field at its nanogap region. The field enhancement becomes large for a long-arm nanoantenna along the polarization direction of light and for blunt tips near the nanogap. The latter condition is very different from that for a metallic bowtie nanoantenna, in which a strong field enhancement occurs for a sharp-tip structure. The difference originates from the different features of the spatial distribution of the exciton and surface plasmon. The great advantage of the semiconductor bowtie nanoantenna is the narrow spectral peak of the strongly enhanced field; thus, the nanoantenna acts as a high-Q cavity with a small mode volume beyond the diffraction limit. Such an optical cavity can be applied to various applications in quantum information processing, e.g., a highly efficient entangled photon generator [26,27]. Our results provide guidelines for designing desirable semiconductor nanoantennas for various applications. The authors thank Prof. H. Ishihara and Prof. H. Yoshida for fruitful discussions and comments. This work was supported by the Japan Society for the Promotion of Science (JSPS) “Core to Core” Program, the Japan Science and Technology Agency (JST) Advanced Low Carbon (ALCA) Research and Development Program, and a Grant-in-Aid for Scientific Research (C), No. 25400325. References 1. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, Science 308, 1607 (2005). 2. R. D. Grober, R. J. Schoelkopf, and D. E. Prober, Appl. Phys. Lett. 70, 1354 (1997). 3. D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. Moerner, Nano Lett. 4, 957 (2004). 4. Y. Sawai, B. Takimoto, H. Nabika, K. Ajito, and K. Murakoshi, J. Am. Chem. Soc. 129, 1658 (2007). 5. K. Ueno, S. Juodkazis, and T. Shibuya, J. Am. Chem. Soc. 130, 6928 (2008). 6. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, Nat. Mater. 9, 193 (2010). 7. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. Moerner, Nat. Photonics 3, 654 (2009). 8. A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, Nat. Photonics 2, 365 (2008).
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9. Y. Tsuboi, T. Shoji, N. Kitamura, M. Takase, K. Murakoshi, Y. Mizumoto, and H. Ishihara, J. Phys. Chem. Lett. 1, 2327 (2010). 10. A. E. Schlather, N. Large, A. S. Urban, P. Nordlander, and N. J. Halas, Nano Lett. 13, 3281 (2013). 11. F. Nagasawa, M. Takase, and K. Murakoshi, J. Phys. Chem. Lett. 5, 14 (2014). 12. T. Itoh, Y. S. Yamamoto, H. Tamaru, V. Biju, S.-I. Wakida, and Y. Ozaki, Phys. Rev. B 89, 195436 (2014). 13. W. Lawrence and J. Wilkins, Phys. Rev. B 7, 2317 (1973). 14. T. Holstein, Ann. Phys. 29, 410 (1964). 15. C. Novo, D. Gomez, J. Perez-Juste, Z. Zhang, H. Petrova, M. Reismann, P. Mulvaney, and G. V. Hartland, Phys. Chem. Chem. Phys. 8, 3540 (2006). 16. M. Uemoto and H. Ajiki, Opt. Express 22, 22470 (2014). 17. K. Cho and H. Ishihara, J. Phys. Soc. Jpn. 59, 754 (1990).
18. M. Kalm and C. Uihlein, J. Phys. Stat. Solidi B 87, 575 (1978). 19. H. Ajiki and K. Cho, Phys. Rev. B 62, 7402 (2000). 20. Y. Zeng, Y. Fu, M. Bengtsson, X. Chen, W. Lu, and H. Ågren, Opt. Express 16, 4507 (2008). 21. M. Uemoto and H. Ajiki, Opt. Express 22, 9450 (2014). 22. Y. Tanaka, H. Ishiguro, H. Fujiwara, Y. Yokota, K. Ueno, H. Misawa, and K. Sasaki, Opt. Express 19, 7726 (2011). 23. E. J. Smythe, E. Cubukcu, and F. Capasso, Opt. Express 15, 7439 (2007). 24. G. W. Bryant, F. J. Garca de Abajo, and J. Aizpurua, Nano Lett. 8, 631 (2008). 25. M. Liu, M. Pelton, and P. Guyot-Sionnest, Phys. Rev. B 79, 035418 (2009). 26. H. Ajiki, H. Ishihara, and K. Edamatsu, New J. Phys. 11, 033033 (2009). 27. K. Shibata and H. Ajiki, Phys. Rev. A 89, 042319 (2014).
