NANO LETTERS
Full Three-Dimensional Subwavelength High-Q Surface-Plasmon-Polariton Cavity
2009 Vol. 9, No. 12 4078-4082
Min-Kyo Seo, Soon-Hong Kwon, Ho-Seok Ee, and Hong-Gyu Park* Department of Physics, Korea UniVersity, Seoul 136-701, Korea Received July 15, 2009
ABSTRACT We propose a full three-dimensional subwavelength surface-plasmon-polariton cavity based on a metal-coated dielectric nanowire with an axial heterostructure. Surface plasmon-polaritons are strongly confined at the nanowire-metal interface sandwiched by an effective plasmonic mirror that consists of lower-index nanowire core and metal shell. Numerical simulations show for a cavity 36000 is achieved at 20 K. This ultrasmall plasmonic cavity can be used as a plasmonic emitter or laser device coupled to a plasmonic waveguide with a high coupling efficiency in deep-subwavelength photonic systems.
Investigations of ultrasmall high-Q cavities open up the possibility of demonstrating efficient quantum optical devices,1 optoelectronic devices with low power consumption,2,3 and sensitive optical sensors.4 Many dielectric nanocavities such as photonic crystals and microdisks exhibit high Q/V values;1-5 however, the diffraction-limited size of these cavities, a cubic half-wavelength in material,3 restricts further miniaturization of photonic devices. In contrast to the dielectric cavities, plasmonic cavities do not face this same limitation and can in principle support a subwavelength mode volume.6 Although several structures have been suggested to effectively confine surface plasmon-polaritons (SPPs) at a metal-dielectric interface,6-14 the three-dimensional (3D) confinement of SPPs in a subwavelength volume still remains a challenge. An abrupt change of metal surface such as a metal grating causes strong scattering of SPPs11-13 and thus the dramatic reduction of the SPP mode volume is not straightforward in the metallic cavity structure. In this letter, we demonstrate a full 3D SPP cavity based on a dielectriccore/metal-shell nanowire structure. The cavity volume is much smaller than the physical limit allowed in a conventional optical cavity, ∼(λ/2n)3. We further show that the scattering loss can be minimized by using a SPP cavity rationally designed through the modification of the dielectric nanowire core instead of the metallic shell. Our SPP cavity was constructed by introducing an axial heterostructure along the nanowire axis in a two-dimensional (2D) plasmonic waveguide of a metal-coated dielectric nanowire (Figure 1A). The SPPs are excited at the interface between the dielectric nanowire core and the metal shell. * To whom correspondence should be addressed. E-mail: hgpark@ korea.ac.kr. 10.1021/nl902274m CCC: $40.75 Published on Web 07/29/2009
2009 American Chemical Society
To confine the SPPs in a desired region and thus realize the 3D SPP confinement, the effective SPP mirrors also consist of dielectric-core/metal-shell located on both sides of the cavity region of length, L. The refractive index of the dielectric core in the SPP mirror should be lower than that of the dielectric core in the cavity region. Such an axial nanowire heterostructure with a low/high/low refractive index can be formed as follows: a nanowire with a uniform diameter of dNW1 is selectively etched except the cavity region15 and the etched nanowire region with a reduced diameter of dNW2 is filled with a low-index material (Figure 1A). Then the SPP modes with a specific resonant wavelength are strongly confined in the deep subwavelength highindex cavity region. To quantitatively understand the confinement mechanism of SPPs in our cavity, we calculated the dispersion curves of the fundamental-transverse plasmonic guided mode in the square cross-sectional infinitely long waveguide that consists of high-index-core/low-index-shell/metal-shell structure (inset of Figure 1B). The 3D finite-difference time-domain (FDTD) method capable of including a metallic structure was used for the simulations (Supporting Information).16,17 The refractive indices of the high-index nanowire core and low-index shell were set to 2.6 (for example, CdS or InGaN in the visible wavelength18,19) and 1.5 (SiO2), respectively, and Ag was used as the metal shell. The total diameter of the dielectric region, the high-index core and the low-index shell, dNW1, was fixed to 50 nm. With consideration of the cutoff wavelength of the optical guided mode in this dielectric waveguide,20 we note that the optical guided modes cannot be excited in the visible wavelength region of our interest
Figure 1. 3D confinement of SPPs in a subwavelength high-Q cavity. (A) Schematic illustration of the SPP cavity. dNW1, dNW2, and L are the nanowire diameter of the cavity region, the diameter of the etched nanowire arm and the cavity length, respectively. (B) The dispersion curves of the fundamental-transverse SPP modes in the square cross-sectional infinitely long waveguide that consists of high-index-core/low-index-shell/metal-shell structure. Refractive indices of the high-index core and the low-index shell are 2.6 and 1.5, respectively. Ag is used as the outer metal shell. The total diameter of the dielectric region (dNW1) is fixed to 50 nm: dNW1 ) dNW2 + 2dSiO2 ) 50 nm, where dNW2 and dSiO2 are the thicknesses of the high-index core and the low-index shell, respectively (inset). (C) The electric field profiles (E2, Ex and Ey) of the fundamentaltransverse SPP mode at dNW1 ) dNW2 ) 50 nm (no SiO2 shell).
and only SPP modes can be excited (Supporting Information, Figure S1). As shown in Figure 1B, the frequency of the SPP mode is well controlled by introducing the low-index shell with a thickness of dSiO2. A significant frequency jump of the SPP mode was achieved in the waveguide with the low-index shell (dSiO2 ) 5, 10, 15 nm) compared to the one without the low-index shell (dSiO2 ) 0 nm). It is also notable that the frequency of the SPP mode approaches a finite value at Re(β) ) 0. The existence of this cutoff frequency is a unique property of a 2D SPP waveguide21 and distinguishes it from the conventional metal-insulator-metal (MIM) structure with no cutoff frequency (Supporting Information, Figure S1).22,23 Because of these two important properties of the SPP waveguide, the large mode gap due to the low-index Nano Lett., Vol. 9, No. 12, 2009
shell and the cutoff frequency, the SPP modes excited in the waveguide without the low-index shell are not coupled to the SPP modes in the waveguide with the low-index shell. Consequently, without modifying the metallic structure, the index change established in the dielectric region allows strong 3D confinement of SPPs. The electric field profiles of the fundamental-transverse SPP mode were also computed for the metal-coated nanowire waveguide without the low index shell (Figure 1C). The electric fields are strongly localized at the interface between the nanowire and metal, and the electric field direction at the interface is perpendicular to the metal surface. This fundamental SPP mode is doubly degenerate due to the square symmetry of the waveguide. Two more higher-order transverse SPP modes with different Ex-symmetries appear in the waveguide (Supporting Information, Figure S2) and these SPP guided modes with different symmetries are not coupled to each other. Figures 2A to 2C show the electric field intensities (log E2) of the SPPs strongly confined in the cavities with L values of 40, 120, and 200 nm, respectively. Similar to the typical Fabry-Perot cavity, the longitudinal SPP resonant modes can be distinguished by the number of electric field antinodes, m. The value m of an excited SPP mode increases with increasing L. In the ultrasmall cavity with L ) 40 nm, for example, the longitudinal mode of m ) 1 was dominantly observed in the visible spectral region of interest (Figure 2A). The SPPs are strongly confined in the cavity without remarkable scattering and the electric fields at the cavity edge vary gently enough to obtain efficient feedback (Figures 2A-C). As a result, high Q factors of >36000 approaching the metal-loss-limited value (Supporting Information)14 were calculated at the low temperature of 20 K where the reduced absorption loss of Ag was assumed (see Figure 3C).13,24,25 These SPP cavities can be compared with the structure of Figure 2D, which has no axial heterostructure in the nanowire core and a length of 540 nm that is identical to the total structure length of Figure 2A. In this structure similar to a Fabry-Perot cavity, the SPPs spread over at the nanowiremetal interface along the nanowire axis, and the Q factor (∼9600) becomes a few times smaller than the metal-losslimited value due to the significant scattering of SPPs at the nanowire end facets. In addition, we compared the SPP cavity of Figure 2A with the structures of Supporting Information, Figure S3, which also have ultrasmall SPP confinement regions. The abrupt discontinuity of the metal surface in the structures in Supporting Information, Figure S3 causes a significant decrease in the Q factors. Therefore, one can understand that the most important structural feature leading to a high Q factor in the SPP cavity is the smooth metal surface covering the entire nanowire core. Both a high Q factor and ultrasmall mode volume can be achieved in our SPP cavity with the axial heterostructure introduced only in the dielectric nanowire core. The optical characteristics of the SPP cavity are shown in Figure 3. Using 3D FDTD simulation, the resonant wavelengths, mode volumes, Q factors and confinement factors of the SPP modes with m ) 1, 2, and 3 were calculated as 4079
Figure 2. The SPP modes excited in the cavities with different lengths. (A) The electric field intensities (log E2) in the SPP cavities with (A) L ) 40 nm, (B) L ) 120 nm, and (C) L ) 200 nm. The number of electric field antinodes, m, the resonant wavelengths, λ, and the Q factors are as follows: (A) (m ) 1, λ ) 533 nm, Q ) 36100), (B) (m ) 2, λ ) 529 nm, Q ) 38100), and (C) (m ) 3, λ ) 528 nm, Q ) 38100). (D) The SPP mode of m ) 11 in the cavity that has no axial heterostructure in the nanowire core and a length of 540 nm. The Q factor is ∼9600 and the resonant wavelength is 491 nm. All Q factors are computed at 20 K.
a function of L. Initially, we set the thickness of Ag and the length of the nanowire arm acting as an effective mirror to be 125 and 250 nm, respectively. In the cavity with these parameters, the Q factor of the SPP mode started to become saturated at the metal-loss-limited value (Supporting Information, Figure S4). Then, the resonant wavelength and the mode volume of the SPP mode were calculated as shown in Figure 3A. The resonant wavelength covering a wide spectral range can be tuned readily by varying L and thus is well adjusted to the emission spectrum of a dielectric nanowire. In addition, the mode volume was computed using the effective refractive index of Ag (Supporting Information, Methods). The extremely small mode volume in the order of 10-5 µm3 is ∼100 times smaller than those of nanoscale optical cavities previously reported.2-5,24 For example, the mode volume of ∼0.020 (λ/2nNW)3 obtained in a cavity with L ) 40 nm and m ) 1 (Figure 2A), where nNW is 2.6, overcomes the physical limit of the optical cavity size. This true 3D subwavelength SPP cavity was designed successfully for the first time and is the smallest reported thus far. Next, Q factors of the SPP cavities were calculated at 20 K (Figure 3B). Lowering temperature reduces substantially the absorption loss of the metal and thus increases the metalloss-limited Q.13,24,25 As shown in Figure 3B, high-Q factors were obtained with a good tolerance to L and the resonant wavelength, λ. Hence, our cavity design of the axial heterostructure works efficiently in a wide spectral region. A high-Q factor of ∼38000 approaches the metal-loss-limited value because optical radiation loss of the cavity is much smaller (see Supporting Information). In addition, the Q factor at a fixed m tends to decrease with decreasing L and the resonant wavelength. This can be understood by the fact that the electric fields of the SPPs with a short wavelength penetrate the metal further and suffer from stronger absorption. Because of such a high-Q factor and subwavelengthscale mode volume, an extremely high λ3Q/V value of ∼2.6 × 108 was obtained. This value is comparable to those of 4080
the photonic crystal or microdisk cavities.1-3,5,24 The confinement factor, which is defined as the ratio of the electric field energy confined in the dielectric nanowire core of the cavity to the total electric field energy, was also investigated (Figure 3B). Since the cavity has a large confinement factor of >∼0.45, one can demonstrate an active plasmonic emitter to effectively excite the SPPs from the gain media introduced in the cavity. In order to further investigate the effect of the absorption loss of metal, the Q factor of the SPP mode with m ) 1 at L ) 40 nm was calculated as a function of temperature (Figure 3C). The metal-loss-limited Q factor depends predominantly on the damping collision frequency, γ.6,14 To present a low temperature in the FDTD simulation, γ was scaled by a factor of the room-temperature conductivity of a metal divided by the low-temperature conductivity.13,24,25 Then Q factor of the SPP mode increases exponentially with decreasing temperature (Figure 3C). At 80 K, a high Q factor of ∼600 was still obtained. The fact that the Q factor is inversely proportional to γ in Figure 3C suggests that the Q factor is metal-loss limited and the optical loss of the SPP cavity mode is sufficiently low (Supporting Information). Our SPP cavity design based on a nanowire structure is beneficial to the demonstration of a practical active plasmonic emitter or laser device to offer efficient coupling of SPPs to a plasmonic waveguide and excitation of SPPs by current injection. In particular, the SPP cavity can be efficiently coupled to a dielectric-core/metal-shell plasmonic waveguide shown in Figure 4A. The coupling efficiency is quite large due to excellent mode matching26 between the SPP cavity mode and waveguide mode. In Figure 4B, the coupling efficiency was computed to be 90.9% (at 20 K) when the distance between the cavity and waveguide, dc, was 125 nm and the refractive index of the waveguide dielectric core was 2.6 (for example, TiO227). Furthermore, the efficient excitation of SPPs by current injection is achievable in a SPP cavity consisting of a CdS or InGaN nanowire with a n-i-p core/ Nano Lett., Vol. 9, No. 12, 2009
Figure 4. The SPP cavity coupled to a plasmonic waveguide. (A) Schematic illustration of the SPP cavity coupled to a dielectriccore/metal-shell plasmonic waveguide. dc is the distance between the cavity and waveguide. (B) The electric field intensities (E2) at dc ) 125 nm. The refractive index of the dielectric core of the waveguide is 2.6. The coupling efficiency is computed to be 90.9% at dc ) 125 nm and at 20 K. At dc ) 75 nm and dc ) 50 nm, the coupling efficiencies are increased to be 99.0 and 99.7%, respectively.
volume that is much smaller than the lower bound of an optical cavity. We believe that this promising SPP cavity represents a significant step toward the realization of new active plasmonic devices such as an efficient plasmon emitter coupled to a plasmonic waveguide or an electrical injection plasmonic laser for deep subwavelength photonic systems. Acknowledgment. This work was supported by Creative Research Initiatives (2009-0081565) of MEST/KOSEF.
Figure 3. The optical properties of the SPP modes with m ) 1, 2, and 3. (A) The resonant wavelengths and mode volumes are calculated as a function of L. The extremely small mode volume in an order of 10-5 µm3 is obtained. (B) The Q factors and confinement factors are calculated as a function of L. The Q factors are computed at 20 K. (C) The Q factor of the SPP mode of m ) 1 in the cavity with L ) 40 nm is calculated as a function of temperature.
shell/shell doping structure that has been demonstrated previously.28 In this scenario, we envision current injected from one end of the n-doped nanowire core and the p-doped metal shell to excite a SPP mode. Because of the high Q factor and ultrasmall mode volume of the SPP cavity, it should be possible to realize a plasmonic laser14,29 for our design. The conventional rate equation analysis using the parameters in ref 19 suggests that threshold of this plasmonic laser will be extremely low (a few tens of nA at 20 K and a few hundreds of nA at 80 K). Consequently, the SPP cavity might be used as an efficient plasmon source in an ultracompact subwavelength plasmonic circuit. In summary, we proposed a full 3D subwavelength SPP cavity based on a metal-coated dielectric nanowire with an axial heterostructure and show using numerical simulations that this structure yields strong confinement of SPPs in an Nano Lett., Vol. 9, No. 12, 2009
Supporting Information Available: Detailed description of FDTD simulation method and five additional figures are provided. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Vahala, K. J. Nature 2003, 424, 839–846. (2) Park, H.-G.; Kim, S.-H.; Kwon, S.-H.; Ju, Y.-G.; Yang, J.-K.; Baek, J.-H.; Kim, S.-B.; Lee, Y.-H. Science 2004, 305, 1444–1447. (3) Nozaki, K.; Baba, T. Appl. Phys. Lett. 2006, 88, 211101. (4) Armani, A. M.; Kulkarni, R. P.; Fraser, S. E.; Flagan, R. C.; Vahala, K. J. Science 2007, 317, 783–787. (5) Song, B.-S.; Noda, S.; Asano, T.; Akahane, Y. Nat. Mater. 2005, 4, 207–210. (6) Maier, S. A. Plasmonics: Fundamentals and Applications; Springer: New York, 2007. (7) Ditlbacher, H.; Hohenau, A.; Wagner, D.; Kreibig, U.; Rogers, M.; Hofer, F.; Aussenegg, F. R.; Krenn, J. R. Phys. ReV. Lett. 2005, 95, 257403. (8) Bozhevolnyi, S. I.; Volkov, V. S.; Devaux, E.; Laluet, J.-Y.; Ebbesen, T. W. Nature 2006, 440, 508–511. (9) Miyazaki, H. T.; Kurokawa, Y. Phys. ReV. Lett. 2006, 96, 097401. (10) Li, J.; Engheta, N. Phys. ReV. B 2006, 74, 115125. (11) Fedutik, Y.; Temnov, V. V.; Schops, O.; Woggon, U.; Artemyev, M. V. Phys. ReV. Lett. 2007, 99, 136802. (12) Weeber, J.-C.; Bouhelier, A.; Colas des Francs, G.; Markey, L.; Dereux, A. Nano Lett. 2007, 7, 1352–1359. (13) Gong, Y. Y.; Vuckovic, J. Appl. Phys. Lett. 2007, 90, 033113. (14) Min, B.; Ostby, E.; Sorger, V.; Ulin-Avila, E.; Yang, L.; Zhang, X.; Vahala, K. Nature 2009, 457, 455–458. (15) Tian, B.; Zheng, X.; Kempa, T. J.; Fang, Y.; Yu, N.; Yu, G.; Huang, J.; Lieber, C. M. Nature 2007, 449, 885–890. 4081
(16) Taflove, A.; Hagness, S. C. Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed.; Artech House: Boston, 2000. (17) Johnson, P. B.; Christy, R. W. Phys. ReV. B 1972, 6, 4370–4379. (18) Palik, E. D. Handbook of Optical Constants of Solids; Academic Press: New York, 1985. (19) Fang, Q.; Li, Y.; Gradecak, S.; Park, H.-G.; Dong, Y.; Wang, Z. L.; Lieber, C. M. Nat. Mater. 2008, 7, 701–706. (20) Jackson, J. D. Classical Electrodynamics, 3rd ed.; John Wiley & Sons: New York, 1999. (21) Dionne, J. A.; Lezec, H. J.; Atwater, H. A. Nano Lett. 2006, 6, 1928– 1932. (22) Kusunoki, F.; Yotsuya, T.; Takahara, J.; Kobayashi, T. Appl. Phys. Lett. 2005, 86, 211101. (23) Dionne, J. A.; Sweatlock, L. A.; Atwater, H. A.; Polman, A. Phys. ReV. B 2006, 73, 035407. (24) Hill, M. T.; Oei, Y.-S.; Smalbrugge, B.; Zhu, Y.; Vries, T. D.;
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(25) (26) (27) (28) (29)
Veldhoven, P. J. V.; Otten, F. W. M V.; Eijkemans, T. J.; Turkiewicz, J. P.; Waardt, H. D.; Geluk, E. J.; Kwon, S.-H.; Lee, Y.-H.; Notzel, R.; Smit, M. K. Nature Photon. 2007, 1, 589–594. CRC Handbook of Chemistry and Physics, 87th ed., Internet Version 2007; Lide, D. R., Ed.; Taylor & Francis: Boca Raton, FL, 2007. Park, H.-G.; Barrelet, C. J.; Wu, Y.; Tian, B.; Qian, F.; Lieber, C. M. Nature Photon. 2008, 2, 622–626. Kim, S.-K.; Cho, H. K.; Bae, D. K.; Lee, J. S.; Park, H.-G.; Lee, Y.-H. Appl. Phys. Lett. 2008, 92, 241118. Dong, Y.; Tian, B.; Kempa, T.; Lieber, C. M. Nano Lett. 2009, 9, 2183–2187. Hill, M. T.; Marell, M.; Leong, E. S. P.; Smalbrugge, B.; Zhu, Y.; Sun, M.; Veldhoven, P. J.; Geluk, E. J.; Karouta, F.; Oei, Y.-S.; No¨tzel, R.; Ning, C.-Z.; Smit, M. K. Opt. Express 2009, 17, 11107–11112.
NL902274M
Nano Lett., Vol. 9, No. 12, 2009
