Size dependence of band structures in a two-dimensional plasmonic crystal with a square lattice

Naoki Yamamoto; Hikaru Saito

doi:10.1364/OE.22.029761

1. Introduction

In the developing field of plasmonics, metal surfaces with periodic structures on the sub-wavelength scale have attracted much attention due to their valuable properties for nano-photonic 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 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.

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.

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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 beam-scan 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 two-dimensional scanning of the electron beam provided a photon map.

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]:

Ψ_{n} (r, t) = ψ_{n} (r) e^{- i ω t},

ψ_{n} (r) = e^{i k \cdot r} ϕ_{n} (r) = e^{i k . r} \sum_{g} C_{g}^{n} e^{i g \cdot r} \cdot

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 −

k_{p}

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 (k_p = 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.

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.

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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 $C_{4 v}$ point group (Table 1). The wave vector for the band structure at the Γ point (Fig. 2(b), red circle) is $k_{p} = a * = (\frac{2 π}{P}, 0, 0)$ . This particular wave vector can be transformed into other equivalent wave vectors via operations involving the $C_{4 v}$ point group. These transformations yield a set of basic wave vectors, $g_{1} = a *, g_{2} = b *, g_{3} = - a *, g_{4} = - b *$ . The eigenfunctions of the band edge states are constructed by a linear combination of the basis functions, $e^{i g_{j} \cdot r} (j = 1 ~ 4)$ . The representation based on the four basis functions is ${4, 0, 0, 2, 0}$ for the five classes of $C_{4 v}$ (Table 1, bottom row). This representation is reducible, and can be decomposed into three irreducible representations, which are expressed as their direct sum, $Γ_{r e d} = A_{1} \oplus B_{1} \oplus E$ Hence, the four bands should form three eigenstates with three different energies at the Γ point.

Table 1. Character table for C_4v point symmetry.

View Table

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:

\begin{array}{l} ψ_{A 1} (x, y) = \cos 2 π x + \cos π y, \\ ψ_{B 1} (x, y) = \cos 2 π x - \cos π y, \\ ψ_{E (1)} (x, y) = \sin 2 π x, \\ ψ_{E (2)} (x, y) = \sin 2 π y, \end{array}

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) \propto E_{z}^{n} (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 (- i ω t)] |}^{2} 〉}_{t}

, using Eqs. (1) and (3) as

\begin{array}{l} A_{1} \mod e : (^{\cos 2 π x + \cos 2 π y) 2}, \\ B_{1} \mod e : (^{\cos 2 π x - \cos 2 π y) 2}, \\ E (1) \mod e : \sin^{2} 2 π x, \\ E (2) \mod e : \sin^{2} 2 π y . \end{array}

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.

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.

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).

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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.

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.

Fig. 4 ARS patterns from the SQ-PlCs with various diameters of cylindrical pillars acquired with [(a)–(e)] p-polarized and [(f)–(j)] s-polarized light. (k) Illustration of the sample. (l) and (m) Schematic dispersion curves for D = 200 nm and D = 500 nm, respectively. (P = 600 nm, h = 100 nm).

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The dispersion curves, which are horizontally elongated, appear in the p-polarized ARS pattern due to the two SPP bands with wave vectors of $k_{p} = k_{x} a * \pm b *$ . 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 $k_{x} < < 1$ near the Γ point, these SPP waves have a long wavelength along the x-axis and produce only a small electric dipole around the pillars. Consequently, the s-polarized component of the emission is weak except at Γ [27].

On the other hand, in the s-polarized ARS patterns, the dispersion curves mainly have a steep slope. The SPP bands with wave vectors of $k_{p} = k_{x} a * \pm a *$ contribute to these dispersion curves as they produce an oscillating electric dipole parallel to the x-axis around the cylindrical pillar to emit s-polarized light. These steep dispersion curves connect the A and E(1) modes at the Γ point as expected from the group theory. It should be noted that different types of dispersion curves can be visualized by appropriately selecting the polarization direction of the emitted light. This polarization dependence of the contrast in the ARS pattern has been mentioned previously [27].

The spectra at the Γ point (θ = 0°) in the p- and s-polarized ARS patterns should be the same. The SPP bands cross near D = 350 nm, corresponding to the result in Fig. 3(a). Considering that the E mode is doubly degenerated, we can draw the SPP band structure around the Γ point from these ARS patterns. Figures 4(l) and 4(m) schematically depict the dispersion curves for D = 200 nm and D = 500 nm, respectively. Assignment of the band edge modes and their energy positions presented in the figures are referred to the results of the BSS images and photon maps to be consistent with each other. The change of these band edge energy positions with diameter D will be discussed in the latter section.

Figure 5 shows the BSS images of the SQ-PlCs with pillar diameters between 200 nm and 500 nm acquired using non-polarized light emitted in the surface normal direction (the Γ point). The BSS images were taken along a 2.4-μm long (4 periods) scanning path of the L1 line in Fig. 1(c) for Figs. 5(a)–5(f), and the L2 line for Figs. 5(g)–5(l) with an acquisition time of 5 sec for each pixel. Each image is composed of about 100 pixels. These images reveal the spatial distribution of the standing SPP waves along each line for the band edge modes at Γ.

Fig. 5 BSS images of the SQ-PlCs with pillar diameters between 200 nm and 500 nm taken by non-polarized light emitted in the surface normal direction. Images in the upper row are taken along the L1 line, while those in the lower row are along the L2 line. (P = 600 nm, h = 100 nm).

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The E mode is the SPP with a strong intensity at the higher energies (~2.0 eV) in the range of D ≤ 300 nm because bright regions are located at both edges of the pillar. On the other hand, the A mode is the SPP with a strong intensity at the lower energies (~1.8 eV) because the bright regions are located at the center of the pillar. The two weak contrasts, which appear at both the higher and lower energy around the E mode, show the same spatial distribution along the x-axis, and based on their positions are assigned to the B mode. A plausible explanation for the splitting is discussed in a later section. The A and E modes have the same energy near D = 350 nm, which is where their emission intensities are maximized. For D ≥ 400 nm, the energy positions of the A and E modes are reversed. The E mode shifts to the lower energies and the FWHM of the peak becomes larger, while the A mode shifts to the higher energies, crossing the B mode near D = 450 nm.

Figures 6(a)–6(e) show the photon maps of the SQ-PlCs of the pillar with D = 200 nm, while Figs. 6(f)–6(i) show those with D = 500 nm taken with non-polarized light emitted in the surface normal direction. Figures 6(a) and 6(f) are panchromatic photon maps measured without the polarizer and pinhole; thus, the emission intensity is integrated over the wavelength and the emission angle. For each photon map, the scanning area measures 2.1 × 2.1 μm² and is composed of 200 × 200 pixels. Figures 6(b)–6(e) are monochromatic photon maps viewed at the band edge energies of 1.77 eV, 1.97 eV, 2.02 eV, and 2.07 eV, respectively. These patterns resemble those in Fig. 2(d) for the A, B, and E modes. Consequently, the corresponding SPP modes can be identified as the A, B, E, and B’ mode in (b) to (e), respectively. As described later, the SPP mode in (e) is assigned as B’. Circular contrasts along the pillar edge appear in the photon maps of the B and E modes. This contrast is attributed to the edge plasmon mode localized at the pillar because a similar pattern is observed at an isolated pillar.

Fig. 6 (a)–(e) Photon maps of the SQ-PlCs of the pillars with D = 200 nm and (f)–(i) those with D = 500 nm taken by non-polarized light emitted in the surface normal direction. (a) and (f) are panchromatic photon maps, and the others are monochromatic ones at the mode energies shown. (P = 600 nm, h = 100 nm).

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Similarly the SPP modes in Figs. 6(g)–6(i) are identified as the E, B, and A modes from the low to the high energy. In the case where D = 500 nm, most of the oscillating surface charges of the E mode are located on the top surface of the pillar, slightly weakening the emission intensity. Although the intensity of the B mode increases, that of the A mode becomes very weak. In addition the contrast due to the edge plasmon appears along the pillar edge in Figs. 6(g) and 6(i).

4.3 SQ-PlCs with holes

Figures 7(a)–7(d) show the ARS patterns from the SQ-PlCs of the cylindrical holes taken by p-polarized light, while (e)–(h) are those by s-polarized light. The hole diameters are (a, e) 200 nm, (b, f) 300 nm, (c, g) 400 nm, and (d, h) 500 nm with a hole depth of 100 nm. The intensity of the E mode is dominant, but the mode energy at Γ shifts to higher energies as the diameter increases. For D ≤ 300 nm, the E mode is located at the lowest energy position and the A mode is at the highest [Fig. 7(i)]. The dispersion curves at D = 400 nm indicates that the energy of the B mode moves below that of the E mode [Fig. 7(j)]. Then at D = 500 nm, the energies of the A and E modes are reversed, and that of the E mode is the highest [Fig. 7(k)].

Fig. 7 (a)–(d) ARS patterns from the SQ-PlCs of cylindrical holes acquired using p-polarized light and (e)–(h) those taken by s-polarized light. (i)–(k) Schematic drawings of the band structures along the Γ–X line for D = 200 nm, 400 nm, and 500 nm, respectively. (P = 600 nm, h = 100 nm).

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Figure 8 shows the BSS images of the SQ-PlCs with hole diameters from 200 nm to 500 nm taken by scanning the electron beam along (a)–(e) the L1 line and (f)–(j) the L2 line in Fig. 1(c) using non-polarized light emitted in the surface normal direction. The scanning length of 2.4 μm corresponds to four periods of the square lattice. The intensity distribution can be used to determine the SPP modes at Γ [Fig. 7(i)–7(k)]. The standing wave pattern of the E mode is easily recognized due to its strong intensity. The emission intensity of the A mode is extremely weak, especially for D ≤ 400 nm, and the emission distribution extends to the higher energies, which correspond to the diffusive peaks (Fig. 3(b), blue triangles. For D = 350 nm, the B mode is split into two with the E mode peak located in the middle. The B mode moves to the lowest energy position for D ≥ 350 nm. The emission intensity of all the modes increases as the hole diameter increases.

Fig. 8 BSS images of the SQ-PlCs with hole diameters ranging from 200 nm to 500 nm taken by scanning the electron beam along (a)–(e) the L1 line and (f)–(j) the L2 line using non-polarized light emitted in the surface normal direction. (P = 600 nm, h = 100 nm).

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Figures 9(a)–9(d) show the photon maps of the SQ-PlCs of the holes with D = 300 nm, while Figs. 9(e)–9(h) show those with D = 500 nm acquired with non-polarized light emitted in the surface normal direction. Figures 9(a) and 9(e) are panchromatic photon maps measured under the same condition as Figs. 6(a) and 6(f). Figures 9(b)–9(d) are monochromatic photon maps viewed at mode energies of 1.92 eV, 1.98 eV, and 2.04 eV, which are identified as the E, B, and A modes, respectively. In the photon maps of D = 300 nm, the oscillating surface charges for every mode are distributed on a flat surface region around the holes. Hence, theintensity distributions are similar to the expected patterns for the field strength shown in Fig. 2(d). In the case of D = 500 nm, the photon maps in Fig. 9(f)–9(h) are attributed to the B, A, and E modes from low to high energies. Most of the oscillating surface charges of the E mode are located on the bottom surfaces of the holes, so the pattern is modified from that in Fig. 9(b). In addition, the emission intensity weakens slightly.

Fig. 9 (a)–(d) Photon maps of the SQ-PlCs of cylindrical holes with D = 300 nm and (e)–(h) those with D = 500 nm acquired using non-polarized light emitted in the surface normal direction. (a) and (e) are panchromatic photon maps, and the others are monochromatic ones at the mode energies shown. (P = 600 nm, h = 100 nm).

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5. Discussion

5.1 Size dependence of the band edge energies

Figure 10 shows the band edge energies at the Γ point of the SQ-PlCs of cylindrical pillars and holes as functions of the pillar diameter with heights of (a) h = 50 nm and (b) h = 100 nm and hole depths of (c) h = 50 nm and (d) h = 100 nm. For the SQ-PlCs with pillars and D ≤ 300 nm, the E mode has the highest energy, while the A mode has the lowest. The SQ-PlCs with holes show the opposite trend. The standing SPP wave of the B mode is always distributed in a flat surface region, especially for D ≤ 300 nm; thus, the B mode contains the energy of the propagating SPP mode on a flat silver surface at the wavenumber of k_p = 2π/600 (i.e., 2.01 eV). Regarding the size dependence of the mode energies, the A and E modes are analogous to the symmetric and anti-symmetric modes at the Г point in 1D-PlCs [20]. The surface charge of the symmetric mode is distributed at the center of the terrace and groove in the 1D-PlC structures; thus, the energy is lower than that of the SPP on a flat surface if the terrace width to period ratio, D/P, is less than 1/2. On the other hand, the energy of the anti-symmetric mode is higher than that of the SPP on a flat surface. The pillars correspond to terraces in the SQ-PlCs, but the holes correspond to grooves in the SQ-PlCs, which explain the reversed energy positions of the A and E modes in these two cases. Although this reversal occurs near D/P = 1/2 in the 1D-PlCs, a similar reversal between the energies of the A and E modes occurs near D = 350 nm (D/P ≒ 3/5) in the SQ-PlCs.

Fig. 10 Size dependence of the band edge energies at the Γ point of the SQ-PlCs with cylindrical pillars and holes. Pillar height is (a) h = 50 nm and (b) h = 100 nm, and the hole depth is (c) h = 50 nm and (d) h = 100 nm. (P = 600 nm).

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In the case of the 1D-PlCs, the value of D/P where the modes cross is deduced based on the observation that the second order Fourier component of the surface shape function becomes zero. As for the SQ-PlC of cylindrical pillars, the second order Fourier coefficient is written as

h_{2} = π h {(\frac{D}{2 P})}^{2} \frac{2 J_{1} (X)}{X},

where J₁ is the Bessel function of the first kind and

X = 2 π D / P

. This corresponds to the frequency component at the wavenumber of 2k_p = 4π/P of the 1D function formed by projecting the shape function of the cylindrical pillar onto the a-axis. Then h₂ becomes zero when X = 3.832 (i.e.,

D = 0.61 P = 366 n m

). For the SQ-PlCs with pillars, this value is very close to the position where the plotted curves of the A and E modes cross [Figs. 10(a) and 10(b)]. On the other hand, for the SQ-PlCs with holes, the crossing position between the A and E modes deviates from this value (D = 475 nm), but that between the B and E modes is close to this value.

Figures 11(a) and 11(b) show the D dependence of the band edge energies at the Γ point of the SQ-PlCs of cylindrical pillars calculated by the FDTD method for pillar heights of 30 nm and 50 nm, respectively. These calculated results well reproduce the observed dependence in Figs. 10(a) and 10(b), but the height used in the calculation is smaller than that in the sample in order to fit the observed values. This discrepancy can be attributed to the fact that the sample shape deviates from the ideal shape of the cylindrical pillar due to rounding of the edges. Figure 11(c) shows the D dependence of the band edge energies of the cylindrical holes calculated for hole depth of 50 nm. This result well reproduces the observed dependence in Fig. 10(d), except that the energy position of the E mode is slightly higher than the observed one.

Fig. 11 Size dependence of the band edge energies and the Q-factors of the SQ-PlCs of (a), (b), (d), (e) pillars and (c), (f) holes calculated by FDTD. Pillar height is h = 30 nm for (a) and (d), and h = 50 nm for (b) and (e), and the hole depth is h = 50 nm for (c) and (f).

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The FDTD calculation reveals an additional peak at an energy higher than the three band edge energies when the SPP is excited by an oscillating electric dipole close to the surface or by the incident light from the surface normal direction. The calculated electric field distribution of this SPP mode is nearly the same as that of the B mode. This peak is plotted by open circles in Fig. 11(b), and is denoted as the B’ mode in Figs. 6(e), 10(a), and 10(b). The peak energy of the B’ mode is independent of the pillar diameter and coincides with a photon energy for a wavenumber of 2π/P (2.067 eV). This fact suggests that the emission of the B’ mode originates from the leaky mode associated with the B mode. As for the SQ-PlCs with the holes, the FDTD calculation shows that a similar peak appears at 2.1 eV in the emission spectrum. This peak corresponds to the diffusive peak seen in Figs. 3(b) and 8. Because the SPP resembles that of the A mode, this emission should originate from the leaky mode associated with the A mode.

Figures 11(d) and 11(e), which correspond to Figs. 11(a) and (b), respectively, show the dependence of the quality factor (Q-factor) on the pillar diameter calculated by the FDTD method. Modes with higher energies tend to have a high Q value. This agrees with the fact that the penetration depth of the electric field is shallower for the higher energy mode, which decreases the ohmic loss. Similar to the case for 1D-PlCs [8], the FDTD calculation predicts that the Q-factor of the SQ-PlCs is mainly affected by ohmic loss rather than radiative loss. This property is confirmed from the change in the peak widths of the three modes with diameter (Fig. 5). Figure 11(f), which corresponds to Fig. 11(c), shows the dependence of the quality factor (Q-factor) on the hole diameter. There is tendency that the Q-factor becomes high with decreasing hole diameter.

5.2 Formation mechanism of contrast in a photon map

Figure 12(a) shows the photon maps of the SQ-PlCs of cylindrical pillars with D = 500 nm and h = 50 nm taken by non-polarized light emitted in the surface normal direction. Because the photon maps correspond to those in Figs. 6(g)–6(h), except for the pillar height, they are identified as the E, B, and A mode patterns from left to right. Figure 12(b) shows the distributions of the field strengths for the three SPP modes derived from group theory [Fig. 2(d)]. The electron beam is assumed to equally excite the E(1) and E(2) modes. The calculated patterns in Fig. 12(b) well reproduce the observed ones in Fig. 12(a) except for the contrast along the pillar edge. Figure 12(c) shows the photon maps acquired in the same way as those in Fig. 12(a) except that polarized light is used with the polarization direction parallel to the horizontal direction. The photon maps of the A and B modes [Figs. 12(a) and 12(c)] showalmost the same pattern, but that of the E mode drastically changes. Figure 12(d) shows the field strength distributions calculated by the FDTD method when the excitation is due to an oscillating electric dipole near the surface for the A and B modes and by the normal incident light polarized in the horizontal direction for the E mode. The calculated patterns [Fig. 12(d)] and experimental results [Fig. 12(c)] agree well.

Fig. 12 Photon maps of the SQ-PlCs of the cylindrical pillars with D = 500 nm and h = 50 nm taken by (a) non-polarized light and (c) polarized light emitted in the surface normal direction. From left to right, the photon maps correspond to the E, B, and A modes. Field strength distributions of the three modes calculated (b) by group theory and (d) by the FDTD method.

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To understand the information provided by a photon map, the contrast formation mechanism must be discussed. An incident electron parallel to the z-axis loses energy by interacting with the induced electric field of the z-component and exciting SPPs. The excited SPPs propagate to the surrounding region, and some SPPs with a wave vector at the zone boundary are Bragg reflections due to the periodic surface structure that form standing waves. SPPs possessing energies within the band gap vanish due to destructive interference and are consequently not excited. The excitation probability of the standing SPP of the band edge state depends on the incident position of the electron. If the incident position is located at the node of the standing SPP wave, the reflected wave cancels the SPP wave, preventing SPP excitation. On the other hand, if the incident position coincides with the anti-node of the standing SPP wave, where the z-component of the electric field is maximized, the excitation probability of the SPP is maximized. The theory of EELS and CL confirms that the excitation efficiency of the SPP mode, $η_{e x} (E_{n}, R)$ , depends on the incident position of electron, $R = (X, Y)$ , and is proportional to the photonic localized density of states (photonic-LDOS) [36, 38]. In other words, it corresponds to the electromagnetic LDOS along the electron trajectory of the z-axis (z-EMLDOS) [37,39]. Because the photonic DOS is proportional to the field strength of the z-component of the electric field, ${| E_{z}^{n} (R) |}^{2}$ , its time average expressed as the excitation efficiency is compared with the observed photon map [Fig. 2(d)].

Furthermore, the excited SPP wave is accompanied by an oscillating in-plane electric field and the surface current, which cause light emission through the surface structure. The normal emission due to the standing SPP wave mode at the Γ point is mainly caused by the in-plane electric dipole formed in the localized surface structure. Assuming that the SPP to photon conversion efficiency is written as $η_{S P P - p h o t o n} (E_{n}, θ, φ)$ , the intensity distribution of the photon map of the n-th SPP mode is given as

I_{P M} (R) \propto η_{e x} (E_{n}, R) \cdot η_{S P P - p h o t o n} (E_{n}, 0, 0) .

Consequently, the spatial pattern of the photon map is determined solely by the excitation efficiency of the SPP mode. On the other hand, the SPP and photon conversion efficiency contribute to the emission intensity.

Here the effect of the polarization of light on the photon map pattern is explained on the basis of the wave function derived from group theory. For a 1D representation, such as the A and B modes, the polarization direction of the normal emission intensity is specified by the vector e, which is generated by the n-th SPP mode excited by the electron incident at the position of $R$ . The normal emission intensity is expressed as

I_{n / / e} (R) \propto {| C_{n} (R) (e \cdot p_{n}) |}^{2} \propto {| C_{n} (R) \int_{}^{} (e \cdot r) ψ_{n} (r) d r |}^{2},

where

C_{n} (R)

is the excitation amplitude of the SPP mode. Here

p_{n}

is the in-plane electric dipole moment formed by the surface charge oscillation of the n-th SPP mode mediated by the surface structure. Because the surface charge distribution,

σ_{n} (r)

, should be proportional to the amplitude of the SPP wave,

ψ_{n} (r)

, the following equation is valid,

p_{n} \equiv \int_{}^{} r σ_{n} (r) d r \propto \int_{}^{} r ψ_{n} (r) d r \cdot

This electric dipole moment formed around the cylindrical pillars and holes at the lattice point can emit light. Due to the cylindrical symmetry of the surface structure, the contribution of the electric dipole moment to the emission intensity, which is equivalent to the SPP to photon conversion efficiency, is proportional to the inner product of e and

p_{n}

, irrespective of the direction of e. Since

η_{e x} (E_{n}, R) \propto {| E_{z} (R) |}^{2} \propto {| C_{n} (R) |}^{2}

, it follows from the comparison of Eq. (6) and Eq. (7) that

| C_{n} (R) | \propto | ψ_{n} (R) |

. Then the photon map should show the same pattern irrespective of the polarization because

p_{n}

in Eq. (7) is independent of

R

. Consequently we can conclude that

I_{n / / e} (R) \propto {| ψ_{n} (R) |}^{2}

. This is consistent with the fact that the observed photon maps of the A and B modes in Fig. 12 coincide with the field strength distribution derived from group theory. We should mention that it results from the calculation of Eq. (8) using the eigenfunctions of the A and B modes in Eq. (3) that

p_{n}

is 0 for these non-radiative modes. This means that the emission intensity at the Γ point cannot be detected. The observed intensity is considered to come from the light emitted in slightly tilted directions from the surface normal direction because of the finite size of the pinhole used in the experiment, or to be due to the surface parallel dipole produced by the imperfection of the surface structure.

On the other hand, for the E mode, the radiative mode with a large $η_{S P P - p h o t o n}$ is doubly degenerate, so the SPP wave is expressed by linear combination of the two basis functions. In this case the interference between emissions due to the two modes must be taken into account, and then the intensity cannot be written in the form of Eq. (6) and (7). The SPP wave of the E mode excited by the electron incident at the position of $R$ is expressed as

ψ_{E} (r, R) = C_{1} (R) ψ_{E (1)} (r) + C_{2} (R) ψ_{E (2)} (r),

where

C_{n} (R)

(n = 1,2) is the excitation amplitude and

| C_{n} (R) | \propto | ψ_{n} (R) |

for each mode. The excited SPP wave forms the in-plane electric dipole moment on the surface structure and emits light. Here it is assumed that

C_{n} (R)

is a real function and

C_{n} (R) \propto ψ_{n} (R)

for simplicity, which results in the emission of linearly polarized light. Similar to Eq. (8), the dipole moment of the E mode is written as

p_{E} (R) \equiv \int_{}^{} r σ_{E} (r) d r \propto \int_{}^{} r ψ_{E} (r, R) d r = C_{1} (R) p_{1} + C_{2} (R) p_{2} \cdot

Here p₁ and p₂ are defined as

p {}_{1}= \int_{}^{} r ψ_{E (1)} (r) d r = (\begin{array}{l} p_{x} \\ 0 \end{array}), p {}_{2}= \int_{}^{} r ψ_{E (2)} (r) d r = (\begin{array}{l} 0 \\ p_{y} \end{array}),

and p_x and p_y are expressed as

p_{x} \equiv \int_{}^{} x ψ_{E (1)} (r) d r, p_{y} \equiv \int_{}^{} y ψ_{E (2)} (r) d r \cdot

These electric dipole moments represented in Eq. (11) emit light that is purely polarized in the x-direction for

p_{1}

and in the y-direction for

p_{2}

, and it is derived from the calculation of Eq. (12) where

p_{x} = p_{y} = p

. The emission amplitudes generated by the two dipoles are the same due to the cylindrical symmetry of the surface structure. Consequently, the Eq. (7) should be modified as

I_{E / / e} (R) \propto {| e \cdot p_{E} (R) |}^{2} = {| C_{1} (R) \cdot (e \cdot p_{1}) + C_{2} (R) \cdot (e \cdot p_{2}) |}^{2} \cdot

The intensity of the photon map acquired using the polarized light in the x direction,

e = (1, 0)

, can be expressed as:

I_{E / / x} (R) \propto {| C_{1} (R) \cdot (e \cdot p_{1}) |}^{2} \propto p^{2} {| ψ_{E (1)} (R) |}^{2} \propto p^{2} \sin^{2} 2 π X .

Thus, the polarized photon map is solely a function expressing the excitation efficiency of the E(1) mode [i.e.,

η_{e x}^{E (1)} (E_{E}, R)

]. This agrees with the observation that the field strength of the E(1) mode [see Fig. 2(c)] is similar to the observed polarized photon map in Fig. 12(c). Next, for the non-polarized photon map of the E mode, the intensity distribution is obtained by averaging Eq. (13) with respect to

e = (\cos θ, \sin θ)

over the in-plane orientation, giving

\begin{array}{l} I_{E, N P} (R) \propto {\int_{0}^{2 π} | ψ_{E (1)} (R) \cdot p \cos θ + ψ_{E (2)} (R) \cdot p \sin θ |}^{2} d θ \\ \propto p^{2} {{| ψ_{E (1)} (R) |}^{2} + {| ψ_{E (2)} (R) |}^{2}} \propto p^{2} (\sin^{2} 2 π X + \sin^{2} 2 π Y) \end{array}

This result corresponds to the sum of the field strength distributions of the E(1) and E(2) modes. The calculated pattern using Eq. (15), which is shown in Fig. 12(b), reproduces well the observed pattern of the E mode in Fig. 12(a). Although a previous paper [27] found that Eq. (12) may not hold for all eigenfunctions, the same results of Eq. (14) and Eq. (15) can be deduced for such eigenfunctions using Eq. (13).

6. Conclusion

We investigated the size dependence of the band structures in the SQ-PlCs composed of cylindrical pillars and holes using a STEM-CL technique equipped with an angle-resolved measurement system. The energies of the three band edge modes at the point Γ derived from group theory were experimentally measured as functions of the diameter of the cylindrical structure. The ARS patterns, BSS images, and photon maps were used to identify the dispersion of the SPP band structure, the modes, and the field strength distributions of the standing SPP waves, respectively. These measurements yielded detailed data of the SPP modes at the Γ point, providing insight on how the structural parameters (e.g., diameter, pillar height, hole depth) affect the energies of the three band edge modes in SQ-PlCs. These results should assist in designing plasmonic devices using SQ-PlCs, such as plasmonic photo-detectors, sensors, and laser. In addition, the STEM-CL technique can effectively evaluate such devices.

Acknowledgments

This work was supported by MEXT Nanotechnology Platform 12025014.

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C_4v	E	2C₄	C₂	2s_v	2s_d	Eigenfunc.
A₁	1	1	1	1	1	$ψ_{A 1}$
A₂	1	1	1	−1	−1
B₁	1	−1	1	1	−1	$ψ_{B 1}$
B₂	1	−1	1	−1	1
E	2	0	−2	0	0	$ψ_{E (1)}, ψ_{E (2)}$
Γ_red	4	0	0	2	0	A₁⊕B₁⊕E

Size dependence of band structures in a two-dimensional plasmonic crystal with a square lattice

Abstract

1. Introduction

2. Experimental

3. SPP modes at the Γ point of SQ-PlCs

4. Results

4.1 Emission spectra from SQ-PlCs with pillars and holes

4.2 SQ-PlCs with pillars

4.3 SQ-PlCs with holes

5. Discussion

5.1 Size dependence of the band edge energies

5.2 Formation mechanism of contrast in a photon map

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (12)

Tables (1)

Equations (15)

Optics Express