Size dependence of band structures in a twodimensional plasmonic crystal with a square lattice Naoki Yamamoto1,2,* and Hikaru Saito2 1
2
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan Quantum Nanoelectronics Reserch Center, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 1528551, Japan *[email protected]
Abstract: A scanning transmission electron microscope (STEM) cathodoluminescence (CL) technique is used to investigate the size dependence of the band structures in two-dimensional plasmonic crystals with a square lattice (SQ-PlCs) composed of cylindrical pillars and holes. The experimentally determined and calculated dependences of the band edge energies of the three SPP modes at the Γ point on the diameter of the cylindrical structure agree well. The photon maps reveal the field strength distributions of the standing SPP waves of the three eigenmodes. Additionally, a mechanism is proposed to explain the dependence of the contrast on the detected light polarization. ©2014 Optical Society of America OCIS codes: (240.6680) Cathodoluminescence.
Surface
plasmons;
(250.5403)
Plasmonics;
(250.1500)
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007). M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer, 2007). M. U. González, J.-C. Weeber, A.-L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006). H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988). S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3–4), 131–314 (2005). Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007). M. Honda and N. Yamamoto, “Size dependence of surface plasmon modes in one-dimensional plasmonic crystal cavities,” Opt. Express 21(10), 11973–11983 (2013). T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission throughsub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). C. Valsecchi and A. G. Brolo, “Periodic metallic nanostructures as plasmonic chemical sensors,” Langmuir 29(19), 5638–5649 (2013). J. Rosenberg, R. V. Shenoi, S. Krishna, and O. Painter, “Design of plasmonic photonic crystal resonant cavities for polarization sensitive infrared photodetectors,” Opt. Express 18(4), 3672–3686 (2010). P. Senanayake, C.-H. Hung, J. Shapiro, A. Lin, B. Liang, B. S. Williams, and D. L. Huffaker, “Surface plasmonenhanced nanopillar photodetectors,” Nano Lett. 11(12), 5279–5283 (2011). H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). S. Y. Chou and W. Ding, “Ultrathin, high-efficiency, broad-band, omniacceptance, organic solar cells enhanced by plasmonic cavity with subwavelength hole array,” Opt. Express 21(S1 Suppl 1), A60–A76 (2013). T. Okamoto, J. Simonen, and S. Kawata, “Plasmonic band gaps of structured metallic thin films evaluated for a surface plasmon laser using the coupled-wave approach,” Phys. Rev. B 77(11), 115425 (2008). A. M. Lakhani, M. K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29761
18. W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, “Lasing action in strongly coupled plasmonic nanocavity arrays,” Nat. Nanotechnol. 8(7), 506–511 (2013). 19. F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110(20), 206802 (2013). 20. H. Watanabe, M. Honda, and N. Yamamoto, “Size dependence of band-gaps in a one-dimensional plasmonic crystal,” Opt. Express 22(5), 5155–5165 (2014). 21. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996). 22. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92(10), 107401 (2004). 23. C. Billaudeau, S. Collin, C. Sauvan, N. Bardou, F. Pardo, and J.-L. Pelouard, “Angle-resolved transmission measurements through anisotropic two-dimensional plasmonic crystals,” Opt. Lett. 33(2), 165–167 (2008). 24. M. Iwanaga, “Polarization-selective transmission in stacked two-dimensional complementary plasmonic crystal slabs,” Appl. Phys. Lett. 96(8), 083106 (2010). 25. T. T. Truong, J. Maria, J. Yao, M. E. Stewart, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Nanopost plasmonic crystals,” Nanotechnology 20(43), 434011 (2009). 26. W. Zhou, H. Gao, and T. W. Odom, “Toward broadband plasmonics: Tuning dispersion in rhombic plasmonic crystals,” ACS Nano 4(2), 1241–1247 (2010). 27. K. Takeuchi and N. Yamamoto, “Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence,” Opt. Express 19(13), 12365–12374 (2011). 28. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3–4), 131–314 (2005). 29. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267– 1290 (2007). 30. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15(15), 9681–9691 (2007). 31. T. Okamoto and S. Kawata, “Dispersion relation and radiation properties of plasmonic crystals with triangular lattices,” Opt. Express 20(5), 5168–5177 (2012). 32. G. Parisi, P. Zilio, and F. Romanato, “Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method,” Opt. Express 20(15), 16690–16703 (2012). 33. T. Suzuki and N. Yamamoto, “Cathodoluminescent spectroscopic imaging of surface plasmon polaritons in a 1dimensional plasmonic crystal,” Opt. Express 17(26), 23664–23671 (2009). 34. N. Yamamoto, S. Ohtani, and F. J. García de Abajo, “Gap and Mie plasmons in individual silver nanospheres near a silver surface,” Nano Lett. 11(1), 91–95 (2011). 35. N. Yamamoto, The Transmission Electron Microscope (InTech, 2011) Chap.15. 36. F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. 100(10), 106804 (2008). 37. M. Kociak and F. J. García de Abajo, “Nanoscale mapping of plasmons, photons, and excitons,” MRS Bull. 37(01), 39–46 (2012). 38. M. Kuttge, E. J. R. Vesseur, A. F. Koenderink, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Local density of states, spectrum, and far-field interference of surface plasmon polaritons probed by cathodoluminescence,” Phys. Rev. B 79(11), 113405 (2009). 39. G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, F. J. G. de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett. 105(25), 255501 (2010).
1. Introduction In the developing field of plasmonics, metal surfaces with periodic structures on the subwavelength scale have attracted much attention due to their valuable properties for nanophotonic and plasmomic applications, and have been recently named “plasmonic crystals (PlCs)” [1, 2]. Due to their band gap structure, PlCs act as mirrors for surface plasmon polaritons (SPPs) with an energy in the band gap region [3], where an SPP is a transverse magnetic (TM) mode electromagnetic wave propagating at a metal/dielectric interface that is evanescently confined in the perpendicular direction [4]. PlCs can be used as waveguides [5, 6] or cavities [7, 8] for SPPs. Since Ebbesen et al. discovered an optical anomalous transmission through a metal film with an array of sub-wavelength sized holes [9], PlCs have been used in interesting applications for chemical sensors [10, 11], photo detectors [12, 13], plasmonic solar cells [14, 15], and plasmonic lasers [16–19]. Knowledge about the properties of the SPP band gaps and band edge states is key to controlling SPPs in plasmonic devices. Previously, the size dependence of the band edge energies at the Γ and X points in one-dimensional (1D) PlCs has been studied in detail [20]; the band edge energies of the symmetric and anti-symmetric modes change dramatically with the terrace width (D) to period (P) ratio. On the other hand, multiple bands cross at the Γ point
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29762
in two-dimensional (2D) PlCs, and the dependence of the band edge states on the surface structure parameters is complicated. Although many studies have examined the band gap properties of the 2D PlCs experimentally [21–27] and theoretically [28–32], our knowledge about the properties of the plasmonic band gap is still insufficient. SPPs on a metal surface can be excited by an incident electron via a scanning transmission electron microscope (STEM; JEM2100F and JEM2000FX) with a spherical aberration (Cs) corrector. The light emitted from a specimen can be detected by a cathodoluminescence (CL) system equipped with a STEM. We have developed a STEM-CL system to study plasmonic structures such as metal particles and plasmonic crystals [20, 27, 33–35]. Our STEM-CL operates with a beam current of 1 nA at acceleration voltages of 80 kV (JEM2100F) and 200 kV (JEM2000FX) with electron beam diameters of 1 nm and 10 nm, respectively. In the present study, we investigate 2D PlCs with a square lattice (SQ-PlCs) composed of cylindrical pillars and holes. The three band edge modes at the Γ point are derived from group theory. Then the influence of the diameters of the cylindrical pillars and holes on the changes in their energies is evaluated. Standing SPP waves of the band edge modes are visualized in the photon maps acquired by the angle resolved measurement of the STEM-CL. 2. Experimental We fabricated SQ-PlCs by electron beam lithography. Cylindrical pillars and holes arranged on a square lattice were produced from the resist layer (ZEP520A) on an InP substrate. The lattice period (P) was fixed at 600 nm, while the pillar (hole) height (depth) (h) was either 50 nm or 100 nm. The diameter (D) of the pillars (holes) was varied from 100 nm to 500 nm in 50-nm increments. The square lattice structure was composed of 50 × 50 pillars (holes) with different diameters, and a set of SQ-PlCs was fabricated on the same substrate. A 200-nm thick silver layer was evaporated onto the structure by thermal evaporation in a vacuum. The photon emission observed in the present experiment is induced by the SPPs excited on the silver layer and air, because the emission by the SPPs at the silver/resist interface cannot go through the silver layer. Figure 1(a) schematically depicts the experimental setup for the angle-resolved CL measurement in the STEM, which is described elsewhere in detail [8, 20]. A parabolic mirror had a 0.6-mm diameter hole above a sample through which the electron beam was incident on the sample. The sample was set with its x-axis parallel to the X-axis, and the surface normal direction (z-axis) was tilted from the incident beam direction by about 15° towards the Y-axis [Fig. 1(a)], where the xyz coordinates were fixed at the sample and the XYZ ones were fixed at the mirror and the X-Y stage. Figure 1(b) shows the equal contours of the polar angle θ and the azimuthal angle φ with respect to the surface normal direction (z-axis). The emitted light was selected by a pinhole in the mask supported by the X-Y stage. The emission spectra were successively recorded by moving the pinhole parallel to the X-axis, as indicated by the red circle along the broken red line in Fig. 1(b), which corresponds to the change of the polar angle θ towards the x-axis. These spectra create an angle-resolved spectral (ARS) pattern, which shows the emission intensity distribution in the energy (E)–θ plane [20] that approximately represents the dispersion relation of SPP along the Γ–X line. A beam-scan spectral (BSS) image reveals the spatial distribution of the standing SPP wave in the band edge state [20, 27, 33, 34]. The BSS image was obtained by successively recording the emission spectra while scanning the electron beam across the PlC with the pinhole fixed at the θ = 0° position (the Γ point). Lines L1 and L2 in Fig. 1(c) show the beamscan paths to acquire the BSS images, which were used to identify the band edge mode of the observed standing wave. Furthermore, when the pinhole was fixed at the position corresponding to the Γ point, successive acquisition of the emission spectra during twodimensional scanning of the electron beam provided a photon map.
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29763
Fig. 1. (a) Setup for an angle-resolved measurement, and (b) angular map in a parabolic mirror with respect to a tilted specimen. Red circle indicates a pinhole moving along the broken red line. (c) Two linear paths on a SQ-PlC along which the electron beam is scanned to acquire the BSS images.
3. SPP modes at the Γ point of SQ-PlCs An SPP in a plasmonic crystal can be described as a Bloch wave, which forms a standing wave at the Brillouin zone boundary. The SPP wave that represents surface charge density distribution can be expressed as [27, 28]:
Ψn (r , t ) = ψ n (r )e −iωt ,
(1)
ψ n (r ) = ei k ⋅ r φn (r ) = ei k.r Cgn ei g ⋅ r ⋅
(2)
g
The subscript n specifies the mode of the Bloch wave. Because the electric field normal to the surface is proportional to the surface charge, Ψ (r , t ) can be recognized as the surface normal component of the electric field of the SPP. Here the in-plane vector on the sample surface is written as r = x a + y b , where a and b are the basic translational vectors of the square lattice. Figure 2(a) shows the corresponding reciprocal lattice. The dispersion plane of an SPP on the SQ-PlC can be approximated by a set of dispersion planes obtained from a single dispersion plane of an SPP on a flat surface by shifting the SPP dispersion plane by the reciprocal lattice vectors (the empty lattice approximation). The dispersion plane of an SPP on a flat silver surface has a cone-like shape in the E − kp space, which opens around the E axis [33]. The solid lines in Fig. 2(b) indicate SPP dispersion curves along the Γ−X direction. Here we are concerned with band edge modes at the Γ point (kp = 0) indicated by a red circle in Fig. 2(b), where the 4 bands emerging from the 4 reciprocal lattice points (red dots in Fig. 2(a)) cross each other and form three band edge modes. The eigenmodes of an SPP at the Γ point in the SQ-PlC are deduced using group theory. The k group at the Γ point corresponds to the C4 v point group (Table 1). The wave vector for 2π , 0, 0 . This particular the band structure at the Γ point (Fig. 2(b), red circle) is k p = a* = P wave vector can be transformed into other equivalent wave vectors via operations involving the C4 v point group. These transformations yield a set of basic wave vectors,
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29764
g1 = a*, g2 = b*, g3 = −a*, g4 = −b * . The eigenfunctions of the band edge states are
constructed by a linear combination of the basis functions, e
i g j ⋅r
( j = 1 ~ 4) . The
representation based on the four basis functions is {4, 0, 0, 2, 0} for the five classes of C4 v (Table 1, bottom row). This representation is reducible, and can be decomposed into three irreducible representations, which are expressed as their direct sum, Γ red = A1 ⊕ B1 ⊕ E Hence, the four bands should form three eigenstates with three different energies at the Γ point. Table 1. Character table for C4v point symmetry. C4v
E
2C4
C2
2sv
2sd
Eigenfunc.
A1
1
1
1
1
1
ψ A1
A2
1
1
1
−1
−1
B1
1
−1
1
1
−1
B2
1
−1
1
−1
1
E
2
0
−2
0
0
Γred
4
0
0
2
0
ψ B1
ψ E (1) ,ψ E (2) A1⊕B1⊕E
The eigenfunctions of these irreducible representations can be derived by applying the projection operator to one of the basis functions using the characters in Table 1 and are expressed as:
ψ A1 ( x, y ) = cos 2π x + cos π y, ψ B1 ( x, y ) = cos 2π x − cos π y, ψ E (1) ( x, y ) = sin 2π x,
(3)
ψ E (2) ( x, y ) = sin 2π y, where the normalization factors are omitted. Because the E mode in Eq. (3) is energetically doubly degenerate, other eigenfunctions for E(1) and E(2) modes are possible [27]. As shown in Fig. 2(c), these functions represent the surface normal component of the electric field, ψ n (r ) ∝ Ezn (r ) . CL imaging and electron energy loss spectroscopy (EELS) can reveal the spatial distribution of a standing SPP wave, which is given by the field strength of the z component in the electric field [36, 37]. To compare EELS with the observed CL photon maps in a latter section, the time average of the field strength is calculated for each eigenmode,
Re [ψ n (r ) exp(−iω t ) ]
2
, using Eqs. (1) and (3) as t
A1 mod e : (cos 2π x + cos 2π y ) 2 , B1 mod e : (cos 2π x − cos 2π y ) 2 , E (1) mod e : sin 2 2π x,
(4)
E (2) mod e : sin 2 2π y.
Figure 2(d) shows the spatial distributions of the eigenmodes, where the field strength of the E mode is given by the sum of the E(1) and E(2) modes assuming that the two modes are excited equally.
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29765
Fig. 2. (a) Reciprocal lattice and (b) dispersion relation of a SQ-PlC. (c) Electric field and (d) field strength distribution of the three band edge modes at the Γ point. Square in the middle of each pattern indicates a unit cell.
4. Results
4.1 Emission spectra from SQ-PlCs with pillars and holes Figure 3 shows the emission spectra from the SQ-PlCs with various diameters for (a) cylindrical pillars and (b) cylindrical holes taken in the surface normal direction (the Γ point). The electron beam was scanned over a 2 × 2-μm2 area during a 10-sec acquisition time. Two strong peaks appear (Fig. 3(a), red and blue triangles). Their energy difference is maximized near D = 200 nm and D = 500 nm. Similar to the 1D-PlC case [20], these two peaks cross at D = 350 nm. A third peak (black triangles) appears for D ≤ 250 nm. The origin of the peak at 2.07 eV (open triangles) will be discussed later. In the spectra for SQ-PlCs with holes [Fig. 3(b)], three peaks appear over a large D range. The strong peak (red triangles) is at the lowest energy for D ≤ 300 nm, but shifts to higher energies with increasing D. The two peaks located at the higher energy side shift to lower energies as D increases, and cross the strong peak. These three peaks have comparable intensities at D = 500 nm.
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29766
Fig. 3. Emission spectra for the SQ-PlCs with various diameters of (a) cylindrical pillars and (b) cylindrical holes taken in the surface normal direction. (P = 600 nm, h = 100 nm).
4.2 SQ-PlCs with pillars Figure 4 shows the ARS patterns for the SQ-PlCs with cylindrical pillars taken by p-polarized [(a)–(e)], and s-polarized light [(f)–(j)]. The diameters of the pillar are (a, f) 200 nm, (b, g) 300 nm, (c, h) 350 nm, (d, i) 400 nm, and (e, j) 500 nm. Figure 4(k) illustrates the sample. The ARS pattern can approximate the dispersion of the SPP band structure along the Γ–X line. The dispersion curves, which are horizontally elongated, appear in the p-polarized ARS pattern due to the two SPP bands with wave vectors of k p = k x a * ± b * . These SPPs form standing waves in the y direction with different energies, and split the dispersion curves in two. One standing wave is the non-radiative mode connecting the B mode at the Γ point. The other is the radiative mode connecting the E(2) mode, producing an oscillating electric dipole parallel to the y-axis around the cylindrical pillar and emitting p-polarized light. Since k x
2
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan Quantum Nanoelectronics Reserch Center, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 1528551, Japan *[email protected]
Abstract: A scanning transmission electron microscope (STEM) cathodoluminescence (CL) technique is used to investigate the size dependence of the band structures in two-dimensional plasmonic crystals with a square lattice (SQ-PlCs) composed of cylindrical pillars and holes. The experimentally determined and calculated dependences of the band edge energies of the three SPP modes at the Γ point on the diameter of the cylindrical structure agree well. The photon maps reveal the field strength distributions of the standing SPP waves of the three eigenmodes. Additionally, a mechanism is proposed to explain the dependence of the contrast on the detected light polarization. ©2014 Optical Society of America OCIS codes: (240.6680) Cathodoluminescence.
Surface
plasmons;
(250.5403)
Plasmonics;
(250.1500)
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007). M. L. Brongersma and P. G. Kik, Surface Plasmon Nanophotonics (Springer, 2007). M. U. González, J.-C. Weeber, A.-L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006). H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988). S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3–4), 131–314 (2005). Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007). M. Honda and N. Yamamoto, “Size dependence of surface plasmon modes in one-dimensional plasmonic crystal cavities,” Opt. Express 21(10), 11973–11983 (2013). T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission throughsub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7(6), 442–453 (2008). C. Valsecchi and A. G. Brolo, “Periodic metallic nanostructures as plasmonic chemical sensors,” Langmuir 29(19), 5638–5649 (2013). J. Rosenberg, R. V. Shenoi, S. Krishna, and O. Painter, “Design of plasmonic photonic crystal resonant cavities for polarization sensitive infrared photodetectors,” Opt. Express 18(4), 3672–3686 (2010). P. Senanayake, C.-H. Hung, J. Shapiro, A. Lin, B. Liang, B. S. Williams, and D. L. Huffaker, “Surface plasmonenhanced nanopillar photodetectors,” Nano Lett. 11(12), 5279–5283 (2011). H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). S. Y. Chou and W. Ding, “Ultrathin, high-efficiency, broad-band, omniacceptance, organic solar cells enhanced by plasmonic cavity with subwavelength hole array,” Opt. Express 21(S1 Suppl 1), A60–A76 (2013). T. Okamoto, J. Simonen, and S. Kawata, “Plasmonic band gaps of structured metallic thin films evaluated for a surface plasmon laser using the coupled-wave approach,” Phys. Rev. B 77(11), 115425 (2008). A. M. Lakhani, M. K. Kim, E. K. Lau, and M. C. Wu, “Plasmonic crystal defect nanolaser,” Opt. Express 19(19), 18237–18245 (2011).
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29761
18. W. Zhou, M. Dridi, J. Y. Suh, C. H. Kim, D. T. Co, M. R. Wasielewski, G. C. Schatz, and T. W. Odom, “Lasing action in strongly coupled plasmonic nanocavity arrays,” Nat. Nanotechnol. 8(7), 506–511 (2013). 19. F. van Beijnum, P. J. van Veldhoven, E. J. Geluk, M. J. A. de Dood, G. W. ’t Hooft, and M. P. van Exter, “Surface plasmon lasing observed in metal hole arrays,” Phys. Rev. Lett. 110(20), 206802 (2013). 20. H. Watanabe, M. Honda, and N. Yamamoto, “Size dependence of band-gaps in a one-dimensional plasmonic crystal,” Opt. Express 22(5), 5155–5165 (2014). 21. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996). 22. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface plasmon polaritons and their role in the enhanced transmission of light through periodic arrays of subwavelength holes in a metal film,” Phys. Rev. Lett. 92(10), 107401 (2004). 23. C. Billaudeau, S. Collin, C. Sauvan, N. Bardou, F. Pardo, and J.-L. Pelouard, “Angle-resolved transmission measurements through anisotropic two-dimensional plasmonic crystals,” Opt. Lett. 33(2), 165–167 (2008). 24. M. Iwanaga, “Polarization-selective transmission in stacked two-dimensional complementary plasmonic crystal slabs,” Appl. Phys. Lett. 96(8), 083106 (2010). 25. T. T. Truong, J. Maria, J. Yao, M. E. Stewart, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Nanopost plasmonic crystals,” Nanotechnology 20(43), 434011 (2009). 26. W. Zhou, H. Gao, and T. W. Odom, “Toward broadband plasmonics: Tuning dispersion in rhombic plasmonic crystals,” ACS Nano 4(2), 1241–1247 (2010). 27. K. Takeuchi and N. Yamamoto, “Visualization of surface plasmon polariton waves in two-dimensional plasmonic crystal by cathodoluminescence,” Opt. Express 19(13), 12365–12374 (2011). 28. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408(3–4), 131–314 (2005). 29. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267– 1290 (2007). 30. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15(15), 9681–9691 (2007). 31. T. Okamoto and S. Kawata, “Dispersion relation and radiation properties of plasmonic crystals with triangular lattices,” Opt. Express 20(5), 5168–5177 (2012). 32. G. Parisi, P. Zilio, and F. Romanato, “Complex Bloch-modes calculation of plasmonic crystal slabs by means of finite elements method,” Opt. Express 20(15), 16690–16703 (2012). 33. T. Suzuki and N. Yamamoto, “Cathodoluminescent spectroscopic imaging of surface plasmon polaritons in a 1dimensional plasmonic crystal,” Opt. Express 17(26), 23664–23671 (2009). 34. N. Yamamoto, S. Ohtani, and F. J. García de Abajo, “Gap and Mie plasmons in individual silver nanospheres near a silver surface,” Nano Lett. 11(1), 91–95 (2011). 35. N. Yamamoto, The Transmission Electron Microscope (InTech, 2011) Chap.15. 36. F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. 100(10), 106804 (2008). 37. M. Kociak and F. J. García de Abajo, “Nanoscale mapping of plasmons, photons, and excitons,” MRS Bull. 37(01), 39–46 (2012). 38. M. Kuttge, E. J. R. Vesseur, A. F. Koenderink, H. J. Lezec, H. A. Atwater, F. J. García de Abajo, and A. Polman, “Local density of states, spectrum, and far-field interference of surface plasmon polaritons probed by cathodoluminescence,” Phys. Rev. B 79(11), 113405 (2009). 39. G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, F. J. G. de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett. 105(25), 255501 (2010).
1. Introduction In the developing field of plasmonics, metal surfaces with periodic structures on the subwavelength scale have attracted much attention due to their valuable properties for nanophotonic and plasmomic applications, and have been recently named “plasmonic crystals (PlCs)” [1, 2]. Due to their band gap structure, PlCs act as mirrors for surface plasmon polaritons (SPPs) with an energy in the band gap region [3], where an SPP is a transverse magnetic (TM) mode electromagnetic wave propagating at a metal/dielectric interface that is evanescently confined in the perpendicular direction [4]. PlCs can be used as waveguides [5, 6] or cavities [7, 8] for SPPs. Since Ebbesen et al. discovered an optical anomalous transmission through a metal film with an array of sub-wavelength sized holes [9], PlCs have been used in interesting applications for chemical sensors [10, 11], photo detectors [12, 13], plasmonic solar cells [14, 15], and plasmonic lasers [16–19]. Knowledge about the properties of the SPP band gaps and band edge states is key to controlling SPPs in plasmonic devices. Previously, the size dependence of the band edge energies at the Γ and X points in one-dimensional (1D) PlCs has been studied in detail [20]; the band edge energies of the symmetric and anti-symmetric modes change dramatically with the terrace width (D) to period (P) ratio. On the other hand, multiple bands cross at the Γ point
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29762
in two-dimensional (2D) PlCs, and the dependence of the band edge states on the surface structure parameters is complicated. Although many studies have examined the band gap properties of the 2D PlCs experimentally [21–27] and theoretically [28–32], our knowledge about the properties of the plasmonic band gap is still insufficient. SPPs on a metal surface can be excited by an incident electron via a scanning transmission electron microscope (STEM; JEM2100F and JEM2000FX) with a spherical aberration (Cs) corrector. The light emitted from a specimen can be detected by a cathodoluminescence (CL) system equipped with a STEM. We have developed a STEM-CL system to study plasmonic structures such as metal particles and plasmonic crystals [20, 27, 33–35]. Our STEM-CL operates with a beam current of 1 nA at acceleration voltages of 80 kV (JEM2100F) and 200 kV (JEM2000FX) with electron beam diameters of 1 nm and 10 nm, respectively. In the present study, we investigate 2D PlCs with a square lattice (SQ-PlCs) composed of cylindrical pillars and holes. The three band edge modes at the Γ point are derived from group theory. Then the influence of the diameters of the cylindrical pillars and holes on the changes in their energies is evaluated. Standing SPP waves of the band edge modes are visualized in the photon maps acquired by the angle resolved measurement of the STEM-CL. 2. Experimental We fabricated SQ-PlCs by electron beam lithography. Cylindrical pillars and holes arranged on a square lattice were produced from the resist layer (ZEP520A) on an InP substrate. The lattice period (P) was fixed at 600 nm, while the pillar (hole) height (depth) (h) was either 50 nm or 100 nm. The diameter (D) of the pillars (holes) was varied from 100 nm to 500 nm in 50-nm increments. The square lattice structure was composed of 50 × 50 pillars (holes) with different diameters, and a set of SQ-PlCs was fabricated on the same substrate. A 200-nm thick silver layer was evaporated onto the structure by thermal evaporation in a vacuum. The photon emission observed in the present experiment is induced by the SPPs excited on the silver layer and air, because the emission by the SPPs at the silver/resist interface cannot go through the silver layer. Figure 1(a) schematically depicts the experimental setup for the angle-resolved CL measurement in the STEM, which is described elsewhere in detail [8, 20]. A parabolic mirror had a 0.6-mm diameter hole above a sample through which the electron beam was incident on the sample. The sample was set with its x-axis parallel to the X-axis, and the surface normal direction (z-axis) was tilted from the incident beam direction by about 15° towards the Y-axis [Fig. 1(a)], where the xyz coordinates were fixed at the sample and the XYZ ones were fixed at the mirror and the X-Y stage. Figure 1(b) shows the equal contours of the polar angle θ and the azimuthal angle φ with respect to the surface normal direction (z-axis). The emitted light was selected by a pinhole in the mask supported by the X-Y stage. The emission spectra were successively recorded by moving the pinhole parallel to the X-axis, as indicated by the red circle along the broken red line in Fig. 1(b), which corresponds to the change of the polar angle θ towards the x-axis. These spectra create an angle-resolved spectral (ARS) pattern, which shows the emission intensity distribution in the energy (E)–θ plane [20] that approximately represents the dispersion relation of SPP along the Γ–X line. A beam-scan spectral (BSS) image reveals the spatial distribution of the standing SPP wave in the band edge state [20, 27, 33, 34]. The BSS image was obtained by successively recording the emission spectra while scanning the electron beam across the PlC with the pinhole fixed at the θ = 0° position (the Γ point). Lines L1 and L2 in Fig. 1(c) show the beamscan paths to acquire the BSS images, which were used to identify the band edge mode of the observed standing wave. Furthermore, when the pinhole was fixed at the position corresponding to the Γ point, successive acquisition of the emission spectra during twodimensional scanning of the electron beam provided a photon map.
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29763
Fig. 1. (a) Setup for an angle-resolved measurement, and (b) angular map in a parabolic mirror with respect to a tilted specimen. Red circle indicates a pinhole moving along the broken red line. (c) Two linear paths on a SQ-PlC along which the electron beam is scanned to acquire the BSS images.
3. SPP modes at the Γ point of SQ-PlCs An SPP in a plasmonic crystal can be described as a Bloch wave, which forms a standing wave at the Brillouin zone boundary. The SPP wave that represents surface charge density distribution can be expressed as [27, 28]:
Ψn (r , t ) = ψ n (r )e −iωt ,
(1)
ψ n (r ) = ei k ⋅ r φn (r ) = ei k.r Cgn ei g ⋅ r ⋅
(2)
g
The subscript n specifies the mode of the Bloch wave. Because the electric field normal to the surface is proportional to the surface charge, Ψ (r , t ) can be recognized as the surface normal component of the electric field of the SPP. Here the in-plane vector on the sample surface is written as r = x a + y b , where a and b are the basic translational vectors of the square lattice. Figure 2(a) shows the corresponding reciprocal lattice. The dispersion plane of an SPP on the SQ-PlC can be approximated by a set of dispersion planes obtained from a single dispersion plane of an SPP on a flat surface by shifting the SPP dispersion plane by the reciprocal lattice vectors (the empty lattice approximation). The dispersion plane of an SPP on a flat silver surface has a cone-like shape in the E − kp space, which opens around the E axis [33]. The solid lines in Fig. 2(b) indicate SPP dispersion curves along the Γ−X direction. Here we are concerned with band edge modes at the Γ point (kp = 0) indicated by a red circle in Fig. 2(b), where the 4 bands emerging from the 4 reciprocal lattice points (red dots in Fig. 2(a)) cross each other and form three band edge modes. The eigenmodes of an SPP at the Γ point in the SQ-PlC are deduced using group theory. The k group at the Γ point corresponds to the C4 v point group (Table 1). The wave vector for 2π , 0, 0 . This particular the band structure at the Γ point (Fig. 2(b), red circle) is k p = a* = P wave vector can be transformed into other equivalent wave vectors via operations involving the C4 v point group. These transformations yield a set of basic wave vectors,
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29764
g1 = a*, g2 = b*, g3 = −a*, g4 = −b * . The eigenfunctions of the band edge states are
constructed by a linear combination of the basis functions, e
i g j ⋅r
( j = 1 ~ 4) . The
representation based on the four basis functions is {4, 0, 0, 2, 0} for the five classes of C4 v (Table 1, bottom row). This representation is reducible, and can be decomposed into three irreducible representations, which are expressed as their direct sum, Γ red = A1 ⊕ B1 ⊕ E Hence, the four bands should form three eigenstates with three different energies at the Γ point. Table 1. Character table for C4v point symmetry. C4v
E
2C4
C2
2sv
2sd
Eigenfunc.
A1
1
1
1
1
1
ψ A1
A2
1
1
1
−1
−1
B1
1
−1
1
1
−1
B2
1
−1
1
−1
1
E
2
0
−2
0
0
Γred
4
0
0
2
0
ψ B1
ψ E (1) ,ψ E (2) A1⊕B1⊕E
The eigenfunctions of these irreducible representations can be derived by applying the projection operator to one of the basis functions using the characters in Table 1 and are expressed as:
ψ A1 ( x, y ) = cos 2π x + cos π y, ψ B1 ( x, y ) = cos 2π x − cos π y, ψ E (1) ( x, y ) = sin 2π x,
(3)
ψ E (2) ( x, y ) = sin 2π y, where the normalization factors are omitted. Because the E mode in Eq. (3) is energetically doubly degenerate, other eigenfunctions for E(1) and E(2) modes are possible [27]. As shown in Fig. 2(c), these functions represent the surface normal component of the electric field, ψ n (r ) ∝ Ezn (r ) . CL imaging and electron energy loss spectroscopy (EELS) can reveal the spatial distribution of a standing SPP wave, which is given by the field strength of the z component in the electric field [36, 37]. To compare EELS with the observed CL photon maps in a latter section, the time average of the field strength is calculated for each eigenmode,
Re [ψ n (r ) exp(−iω t ) ]
2
, using Eqs. (1) and (3) as t
A1 mod e : (cos 2π x + cos 2π y ) 2 , B1 mod e : (cos 2π x − cos 2π y ) 2 , E (1) mod e : sin 2 2π x,
(4)
E (2) mod e : sin 2 2π y.
Figure 2(d) shows the spatial distributions of the eigenmodes, where the field strength of the E mode is given by the sum of the E(1) and E(2) modes assuming that the two modes are excited equally.
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29765
Fig. 2. (a) Reciprocal lattice and (b) dispersion relation of a SQ-PlC. (c) Electric field and (d) field strength distribution of the three band edge modes at the Γ point. Square in the middle of each pattern indicates a unit cell.
4. Results
4.1 Emission spectra from SQ-PlCs with pillars and holes Figure 3 shows the emission spectra from the SQ-PlCs with various diameters for (a) cylindrical pillars and (b) cylindrical holes taken in the surface normal direction (the Γ point). The electron beam was scanned over a 2 × 2-μm2 area during a 10-sec acquisition time. Two strong peaks appear (Fig. 3(a), red and blue triangles). Their energy difference is maximized near D = 200 nm and D = 500 nm. Similar to the 1D-PlC case [20], these two peaks cross at D = 350 nm. A third peak (black triangles) appears for D ≤ 250 nm. The origin of the peak at 2.07 eV (open triangles) will be discussed later. In the spectra for SQ-PlCs with holes [Fig. 3(b)], three peaks appear over a large D range. The strong peak (red triangles) is at the lowest energy for D ≤ 300 nm, but shifts to higher energies with increasing D. The two peaks located at the higher energy side shift to lower energies as D increases, and cross the strong peak. These three peaks have comparable intensities at D = 500 nm.
#223793 - $15.00 USD Received 24 Sep 2014; revised 6 Nov 2014; accepted 10 Nov 2014; published 20 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.029761 | OPTICS EXPRESS 29766
Fig. 3. Emission spectra for the SQ-PlCs with various diameters of (a) cylindrical pillars and (b) cylindrical holes taken in the surface normal direction. (P = 600 nm, h = 100 nm).
4.2 SQ-PlCs with pillars Figure 4 shows the ARS patterns for the SQ-PlCs with cylindrical pillars taken by p-polarized [(a)–(e)], and s-polarized light [(f)–(j)]. The diameters of the pillar are (a, f) 200 nm, (b, g) 300 nm, (c, h) 350 nm, (d, i) 400 nm, and (e, j) 500 nm. Figure 4(k) illustrates the sample. The ARS pattern can approximate the dispersion of the SPP band structure along the Γ–X line. The dispersion curves, which are horizontally elongated, appear in the p-polarized ARS pattern due to the two SPP bands with wave vectors of k p = k x a * ± b * . These SPPs form standing waves in the y direction with different energies, and split the dispersion curves in two. One standing wave is the non-radiative mode connecting the B mode at the Γ point. The other is the radiative mode connecting the E(2) mode, producing an oscillating electric dipole parallel to the y-axis around the cylindrical pillar and emitting p-polarized light. Since k x
