## Abstract

Scanning transmission electron microscopy in combination with electron energy-loss spectroscopy is a powerful tool for the spatial and spectral characterization of the plasmonic modes of lithographically defined photonic meta-atoms. As an example, we present a size dependence study of the resonance energies of the plasmonic modes of a series of isolated split-ring resonators. Furthermore, we show that the comparison of the plasmonic maps of a split-ring resonator and the corresponding complementary split-ring resonator allows a direct visualization of Babinet’s principle. Our experiments are in good agreement with numerical calculations based on a discontinuous Galerkin time-domain approach.

©2011 Optical Society of America

## 1. Introduction

Photonic metamaterials offer many new and exciting optical properties not available in natural materials, e.g., magnetism at optical frequencies [1–3], a negative index of refraction [4,5], or strong chirality [6–8]. In most cases, these properties result from the excitation of localized plasmonic modes in the elementary building blocks of the photonic metamaterial, i.e., the photonic meta-atoms. The characteristics of the plasmonic modes, in particular the electromagnetic near-field distributions, are determined by the geometry and the constituting materials of the photonic meta-atoms. For example, the split-ring resonator (SRR) [9], i.e., a sub-wavelength metallic ring with one or several slits, is the paradigm magnetic photonic meta-atom. Its fundamental plasmonic mode is characterized by a circulating current density distribution along the ring which results in a magnetic dipole moment normal to the plane of the SRR.

So far, most experimental studies on photonic metamaterials have concentrated on the optical far-field. However, the in-depth understanding of the optical properties of a given photonic metamaterial requires the knowledge of the corresponding electromagnetic near-field distribution for the frequency range of interest. On this account, electromagnetic field calculations based on efficient numerical schemes have become an essential tool for the metamaterial design process. Additionally, experiments capable of determining the spectral and spatial distribution of plasmonic modes with nanometer spatial resolution are required to verify these calculations and to study the influence of structural imperfections.

Recently, it has been shown that electron energy-loss spectroscopy (EELS) in combination with transmission electron microscopy is a powerful tool for the spatio-spectral characterization of plasmonic modes of metallic nanostructures [10–18]. In the STEM-EELS mode, the tightly focused electron beam of a scanning transmission electron microscope (STEM) is raster scanned across the sample. A fast electron passing near or through the metallic nanostructure can excite one of the plasmonic modes resulting in an energy-loss of the electron equal to the plasmonic resonance energy. The electron energy-loss probability is thereby related to a generalized electromagnetic density of states [19,20]. By recording an energy-loss spectrum for each position of the electron beam, we can map the spectral and spatial distribution of the plasmonic modes with nanometer spatial resolution.

So far, the majority of plasmon related EELS experiments has dealt with chemically synthesized metallic nanoparticles [10–15]. However, in the context of plasmonics and photonic metamaterials, chemical synthesis is often not an option as it offers insufficient control over the size, shape, separation, and relative orientation of the produced metallic nanoparticles. Recently, several groups have addressed this issue by combining EELS with lithographically defined metallic nanostructures [16–18]. In a first metamaterial related study, G. Boudarham et al. reported on spectral imaging of the four lowest-order eigenmodes of a silver SRR in the infrared-visible range [17]. The SRR eigenmodes were identified as plasmonic standing waves which resemble many characteristics of the corresponding plasmonic modes of nanoantennas. G. Boudarham et al. also found first indications for coupling effects between both legs of the SRR that resulted in quantitative differences of the energies’ dispersion of SRRs and nanoantennas.

In this letter, we report on STEM-EELS experiments in the energy range from 0.36 eV to 2.5 eV on SRRs and complementary SRRs (CSRRs) fabricated by electron-beam lithography. We present high quality EELS maps with nanometer spatial resolution of the plasmonic modes of these photonic meta-atoms. In the first part of our paper, we investigate the size dependence of the plasmonic modes of a series of gold SRRs with lateral dimensions varying between 120 nm × 110 nm and 480 nm × 465 nm. For the largest SRRs of our series, we are able to characterize the plasmonic modes up to the seventh order. We observe significant influence of the material dispersion of gold on the size dependence of the resonance energies of the different plasmonic modes. In the second part of our paper, we compare the EELS maps of the three lowest-order plasmonic modes of a SRR with those of the corresponding CSRR. This comparison enables a direct visualization of Babinet’s principle for all three modes.

## 2. Methods

Electron-beam lithography (EBL) is *the* standard method for the fabrication of high quality photonic metamaterials. Here, we use as substrates suspended 30 nm thick and 100 µm × 100 µm large silicon nitride (Si_{3}N_{4}) membranes (Silson Ltd., Northampton (UK)) which are transparent for fast electrons. In a first step, polymethyl methacrylate dissolved in anisole (PMMA 950K A4) is spin coated at 4000 rpm on the substrate. Next, the PMMA film is exposed by a 30 kV electron beam lithography system (Raith eLiNE). After development, a 2 nm thin layer of chromium followed by a 35 nm thin film of gold are deposited via electron-beam evaporation. Finally, the remaining PMMA is removed. Figures 1(a)
and 1(d) depict high angular annular dark field (HAADF) images of a typical SRR and a CSRR, respectively, which have been fabricated by this procedure.

The STEM-EELS experiments are performed with a Zeiss Libra200 MC Cs-STEM (CRISP) operated at 200 kV [21]. The microscope is equipped with a monochromator and a Cs-corrector for the illumination system. For spectroscopy, the CRISP uses a 90° energy filter, fully corrected for second order aberrations. The electron energy-loss spectra are recorded with a 2 k × 2 k SSCCD camera (Gatan, Ultrascan 1000). In STEM-mode, the system is operated with a beam convergence semiangle of 25 mrad and a collection semiangle of 7 mrad. The dispersion of the spectrometer is 0.016 eV/channel and the acquisition time for each spectrum is in the order of a few seconds. For our settings, the spatial resolution of the EELS maps is determined by the pixel step width (3 - 6 nm). The energetic resolution of our experiments as defined by the full width at half maximum of the zero loss peak (ZLP) is 0.18 eV on the gold structures and 0.15 eV on the Si_{3}N_{4}-membranes. For post-processing, each spectrum is first normalized to its total number of counts. Afterwards, the ZLP is centered at 0 eV in each case.

In order to compare our measurements with theory, we also perform numerical ab-initio calculations of the electron energy-loss spectra. Since the near fields of plasmonic nanostructures are highly sensitive to the geometrical details of the system, one needs a simulation technique which allows to accurately model the rounded geometry of the SRR and CSRR (see Figs. 1(c) and 1(f)). For this work, we employ a nodal discontinuous Galerkin time-domain (DGTD) method [22,23], which is well suited for the efficient and accurate simulation of plasmonic nanostructures [24,25]. Very recently, it was demonstrated how to extract electron energy-loss spectra from DGTD calculations [26]. For all numerical calculations in this paper, we follow the procedures described in [26] with the exception that we use a pure scattered-field excitation instead of the total-field/scattered-field source originally proposed. This allows us to calculate electron energy-loss spectra for electron beams penetrating the plasmonic nanostructures. Since the DGTD method is a time-domain approach, we need to employ a suitable material model to describe the dispersive response of gold [25]. For all calculations in this paper, we use a Drude-Lorentz model given by

*ε*

_{∞}= 6.21,

*ω*= 8.794 eV,

_{D}*γ*= 0.066 eV, Δ

_{D}*ε*= 1,

*ω*= 2.646 eV, and

_{L}*γ*= 0.382 eV.

_{L}## 3. EELS on SRRs

In the following, we present our EELS experiments on SRRs. Here, we investigate the size dependence of the resonance energies of the plasmonic modes of different orders. This section also serves as preparation to section 4, in which we experimentally test the generalized Babinet’s principle for SRRs and CSRRs in the near-infrared and visible regime.

Before we present the experimental EELS maps, we discuss the anticipated EELS intensity distributions for the different resonances of a SRR. In a classical picture, the energy loss experienced by an electron results from the work done by the electron against the induced electric field of the excited mode. A strong EELS signal for a given resonance energy and electron beam position occurs if the corresponding mode has a large electric field component *E _{z}* along the trajectory (

*z*-axis) of the electron [17]. For planar metallic nanostructures like SRRs or CSRRs, a large

*E*-component is connected with the antinodes of the charge density oscillation. Hence, the EELS map of a given mode qualitatively resembles the corresponding charge density distribution. The modes of a SRR are expected to be plasmonic resonances of increasing order of the entire structure [17,28]. To a first approximation, the current density distribution of the

_{z}*m*-th plasmonic mode is a simple standing wave with nodes at the ends and

*m*-1 nodes distributed along the SRR. The locations of the current density nodes coincide with the antinodes of the charge density oscillation. For example, the fundamental mode (

*m*= 1) exhibits antinodes of the charge density oscillation at the two ends of the SRR and the second mode (

*m*= 2) has an additional antinode in the middle of the SRR. Hence, we expect that the EELS map of the fundamental mode has a maximum at each end of the SRR. For the EELS map of the second mode, we anticipate three maxima - two at the ends and one in the middle of the SRR. The number and locations of the EELS maxima of the higher-order SRR modes can be deduced from analogous considerations. In particular, we expect an alternating series of symmetric and anti-symmetric modes which exhibit an even and odd number of EELS maxima, respectively [17].

For our STEM-EELS experiments, we have prepared a series of SRRs with lateral dimensions (width *w* × height *h*, see Fig. 1(a)) ranging from 120 nm × 110 nm to 480 nm × 465 nm. In the following, we will first concentrate on the largest SRR with *w* = 480 nm and *h* = 465 nm. Figure 2
(Media 1) depicts a HAADF image and EELS maps of this SRR. The center energies of the EELS maps correspond to the resonance energies of the plasmonic modes. The energy range of the EELS maps is set to 0.032 eV. Each EELS map has an independent color scale in which small EELS signals are represented by dark blue and large EELS signals correspond to yellow. As expected, we find for the fundamental mode (0.36 eV) strong EELS signals at the two ends of the SRR. The EELS map of the second mode (0.67 eV) exhibits an additional strong EELS maximum in the bottom wire. Significant EELS signals at the lower corners partially stem from the low energy tails of the third resonance (0.86 eV). The EELS maps of the third and fourth mode (1.26 eV) exhibit four and five EELS maxima, respectively. According to the model presented above, one would expect that the fifth mode is a symmetric mode with six EELS maxima distributed along the entire wire and a minimum of the EELS signal in the middle of the bottom wire. Instead, we find that the EELS map of the next mode at 1.5 eV better fits the characteristics of the sixth mode, i.e., it features seven EELS maxima with one of them located in the middle of the bottom wire. Numerical calculations of the optical spectra reveal (not shown) that the fifth plasmonic mode of this SRR can be efficiently excited with a plane wave polarized along the bottom wire of the SRR. It is currently not clear why this mode is absent in our EELS experiments. Finally, the seventh plasmonic mode (1.82 eV) is symmetric in nature and its EELS maps features eight maxima. We have also indications of at least one more mode with 2.0 eV resonance energy (not shown). However, the EELS maxima are not clearly separated in the corresponding EELS map which circumvents an unambiguous identification of the nature of this (these) mode(s).

The EELS maps of the other SRRs of the series exhibit a similar behavior (not shown). As expected, the corresponding resonance energies shift to higher energy if we decrease the lateral size of the SRRs. In all these EELS experiments, we are not able to resolve modes with resonances energies above approximately 2.2 eV. These modes are strongly damped because their resonance energies are larger than the onset of the interband transitions in gold [27]. For this reason, we can resolve only the four lowest-order plasmonic modes for the smaller SRRs (*w* < 300 nm) of our series.

The spectroscopic data extracted from these EELS experiments allows us to investigate the size dependence of the resonance energies of the four lowest-order plasmonic modes in the near-infrared and visible regime. Previous size-scaling studies were restricted to the fundamental mode of the SRR [29]. All SRRs of our series possess approximately the same ratio of width and height (see Fig. 3(a) ). However, due to limitations in the fabrication process, we are not able to scale all lateral geometric parameters of the SRR by exactly the same factor when changing the size of the SRR. Furthermore, the thickness of all SRRs of the series is the same (2 nm Cr and 35 nm Au). These experimental constraints should be kept in mind during the following discussion of the spectroscopic data.

Figure 3 summarizes the resonance energies of the different modes as a function of the inverse width 1/*w* of the SRRs. For resonance frequencies much smaller than the plasma frequency of gold, the metal basically acts as a perfect conductor. In this limit, i.e. for large SRRs, the resonance energy of the fundamental mode is inversely proportional to the size of the SRR. This scaling law also holds to a very good approximation if we only change the lateral dimensions of the SRR and keep its thickness fixed [29]. Here, we tacitly assume that the largest SRR of our series is still within this size-scaling limit. Without material dispersion, the fundamental resonance energies of all SRRs would fall on a line defined by the origin and the data point of the largest SRR (see guide to the eye in Fig. 3(b) for the fundamental mode). Deviations of the data points of the smaller SRRs from this straight line can be attributed to the influence of the frequency dependent permittivity of gold on the fundamental resonance energy. In our experiments, we observe that the evolution of the fundamental mode is no longer in accordance with the simple scaling law for *w* < 200 nm. The fundamental resonance frequency grows slower as we further decrease the size. This effect is even more pronounced for the other three modes. We find that these modes only follow the scaling law for the largest SRR sizes (*w* > 300 nm). For smaller SRRs, the resonance frequencies of the second to fourth plasmonic mode start to saturate as we further decrease the size of the SRRs.

## 4. EELS on CSRRs

In the following, we will use EELS to investigate the near-field distributions of complementary photonic meta-atoms. According to the generalized Babinet’s principle [30–32], the modes of a metal particle and those of the corresponding complementary metallic screen are closely connected. Both structures exhibit modes with the same resonance energies and complementary mode profiles which are related to each other by interchanging the *E*-field and *B*-field distributions. For our EELS studies, we are particularly interested in the connection between the *B _{z}*-component of the mode of a SRR and the

*E*-component of the corresponding CSRR-mode.

_{z}Figure 4
depicts calculated field distributions of |*E _{z}|* and |

*B*for the first three modes of a SRR and the corresponding CSRR for optical excitation with a plane wave. The fields are recorded in a plane 20 nm above the structures. Each field plot has an independent color scale in which small field strengths are represented by dark blue and large fields correspond to yellow. Note that the exciting optical fields for SRR and CSRR are in each case polarized orthogonally (see white arrows).

_{z}|We find for all three modes the expected correspondence between the |*E _{z}|* and |

*B*field distributions of the SRR and the CSRR. The fundamental mode of the SRR has a strong

_{z}|*B*-component in the center of the SRR which stems from the oscillating current along the entire ring. Hence, the fundamental CSRR mode exhibits a strong

_{z}*E*-component in the center part of the metallic screen. The second mode of the SRR exhibits three maxima in the |

_{z}*E*field distribution. The maximum in the middle of the bottom wire of the SRR results from the node of the current density at this position. The currents in the left and right part of the SRR are oscillating in phase. The magnetic fields produced by these currents interfere destructively in the center of the SRR and constructively outside of the SRR. As expected, we find that the second mode of the CSRR exhibits a small |

_{z}|*E*component in the center part of the metallic screen and a strong |

_{z}|*E*component on the metal on the left and right side of the SRR-shaped aperture. Finally, the |

_{z}|*B*component of the third mode of the SRR has a pronounced maximum below the bottom wire and two smaller maxima outside the side wires of the SRR. As expected, we find the analogous |

_{z}|*E*field distribution for the third mode of the CSRR.

_{z}|For the test of the generalized Babinet’s principle at optical frequencies, we have chosen a SRR with *w* = 220 nm and *h* = 210 nm. The lateral dimensions of the CSRR do not exactly match those of the SRR (see HAADF images in Fig. 5
) which results in a slight red shift of the resonance energies of the CSRR compared to the SRR resonances. The experimental EELS maps qualitatively reproduce in each case the calculated |*E _{z}|* field distribution of the corresponding mode of the SRR and the CSRR, respectively (compare Fig. 4 and Fig. 5). Our experiments give clear evidence that the generalized Babinet’s principle also holds for near-infrared and visible frequencies. Thus, the combination of EELS experiments on complementary structures provides, at least qualitatively, the distributions of the normal components of the electric and the magnetic field.

Finally, Fig. 6
depicts calculated EELS maps of the first three modes of the SRR and CSRR, respectively. Here, the shape and size of the SRR and CSRR are defined by the following geometry parameters (see Fig. 1): *w* = 210 nm, *h* = 220 nm, *w _{g}* = 80 nm,

*h*= 145 nm, and

_{g}*r*= 32.5 nm. For the DGTD calculation, we expand the electric and magnetic field components into polynomials of third degree. We checked the convergence of our numerical results by repeating some of the calculations with refined meshes and polynomial orders up to 5. A comparison of Fig. 5 and Fig. 6 shows, that the numerical calculations are in excellent agreement with the experiments.

## 5. Conclusions

EELS is a powerful method to characterize the near-field distributions of photonic meta-atoms in the near-infrared and visible regime. As a first example, we have investigated the size dependence of the plasmonic modes of a series of gold SRRs with lateral dimensions varying between 120 nm × 110 nm and 480 nm × 465 nm. For the largest SRRs of this series, we have mapped the plasmonic modes up to the seventh order. These EELS maps can be interpreted on the basis of a simple and intuitive model. The comparison of the resonance energies of SRRs with different sizes shows that the higher order modes are more susceptible to the influence of the material dispersion of gold than the fundamental mode. In the second part of the paper we have investigated EELS maps of a SRR and the corresponding CSRR. Our experiments indicate that the generalized Babinet’s principle still holds for near-infrared and visible frequencies. Thus, combined EELS experiments on complementary structures allow for the visualization of the corresponding electric and magnetic near-field distributions. The experimental results are in excellent agreement with numerical calculations.

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