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We present a comprehensive theoretical and experimental investigation of the plasmon hybridization of coupled split-ring resonators by means of the electron energy-loss spectroscopy. Split-ring resonator is a key element in design of negative refractive index metamaterials, and has been therefore intensively studied in the literature. Here, our aim is the study of hybridization effects for higher-order non-dipolar modes, which have been not investigated beforehand. We provide a complete scheme of the multimodal distribution of the coupled and single-element split-ring resonators, with a precise attention to the hybridization of those modes according to the induced moments. Our study suggests a clear dominance of electric and magnetic dipole moments over higher-order modes in the far-field radiation spectrum.
© 2015 Optical Society of America
1. Introduction
Split-ring resonators (SRRs), the meta-atoms of many magnetic metamaterials with negative permeability [1], have been widely applied to bio-sensing devices [2], antennas [3], perfect metamaterial absorbers [4] and filters [5]. The very wide applicability of SRRs in integrated photonic devices is based on the closed-loop electric current that this element sustains, which in a dual representation effectively builds up a magnetic dipole moment. The concomitant induction of electric and magnetic moments in SRRs is utilized to realize negative refractive index metamaterials [1].
Besides its application in photonic circuitry, the electromagnetic interactions between the meta-atoms are also fundamentally interesting for investigating the coupling effects between induced moments [6–8]. An intuitive understanding of the interaction between the plasmons of adjacent nanoparticles can be provided by a plasmon hybridization method [9], utilizing a hydrodynamic approximation for the conduction electrons. The result of such an approximation can be understood as the formation of bonding and anti-bonding final states. This hybridization method has been investigated in the dipolar limit for several simple geometries [10–12].
In field theory, the electromagnetic potentials due to localized charges can be expanded in multipoles, providing a powerful technique for investigating the interaction between adjacent localized charge distributions. The lowest-order terms in this expansion set, i.e. electric and magnetic dipoles, are usually considered to be the dominating terms. Due to this reason, only the interaction energy between the dipoles is widely investigated in the plasmon hybridization scheme [9, 10, 13]. Besides the theoretical investigations, experimental studies of the coupling effect in coupled SRRs using optical excitation have been widely reported [14–16]. Most of the previous investigations with optical excitation have utilized optical far-field spectroscopy techniques. However, the non-radiative and dark modes with vanishing electric dipole moments are not directly accessible in such experiments. These dark modes with no radiative losses play a substantial role for the coupling of quantum emitters to metallic nanostructures [17, 18]. Moreover, higher-order multipoles also play an important role in engineering the spectral response of solar cells and sensors [19, 20].
In contrast to optical far-field spectroscopy, electron energy-loss spectroscopy (EELS) in scanning transmission electron microscopy (STEM) allows us to visualize a rich set of different plasmonic modes, including the bright and dark modes in metallic nanostructures [7, 21–25] with high spatial and energy resolution. Specifically, EELS can directly measure the photonic local density of states projected along the electron trajectory [26]. Additionally, we can selectively excite particular plasmonic modes by focusing the electron beam onto the hotspot regions (by hotspot we mean the spatially localized surface plasmon resonances where the electric field is locally enhanced) [27, 28], since the plasmonic modes sustain unique spatial distributions with hotspots localized along the circumference of the metallic nanostructures.
Investigations of the size dependence of isolated SRRs [29] and the fundamental modes of the arm-by-arm coupled SRRs [13] on the plasmonic resonance energies with EELS have been recently reported. In this letter, we theoretically and experimentally investigate the plasmon hybridization schemes of the fundamental modes and the higher-order modes of coupled SRRs by using EELS, aiming at a comprehensive understanding of the role of magnetic and electric moments. Various trajectories of the electron beam have been used to selectively excite different plasmonic modes in SRRs. In addition, different relative orientations of adjacent split rings have been introduced to study the hybridization of the coupled SRRs according to the orientation of the induced moments. An in-depth understanding of the fundamental and high-order multipole plasmon coupling effects of coupled SRRs will provide significant perception into the design and optimization of magnetic metamaterials with desirable properties as well as their resonant behavior.
2. Numerical methods
We investigate the coupling phenomena of two adjacent gold SRRs by considering an isolated SRR which is referred to as structure S, and three different coupled SRR structures: a 0°-rotated coupled SRRs structure A, a 180°-rotated coupled SRR structure B, and a 90°-rotated coupled SRR structure C, as depicted in the insets of Fig. 1(a)–1(d), respectively. The lateral dimensions of the adjacent SRRs in the simulations were: thickness H = 30 nm, length L = 120 nm, width W = 30 nm and the gap distance between the left and right split rings is G = 20 nm. In the simulations, gold SRR structures without substrate were considered. The gold dielectric permittivity was implemented into the calculations using experimental data from Johnson and Christy [30]. Numerical calculations of EELS spectra were obtained by using COMSOL Multiphysics software package, which is based on the frequency-domain finite-element method [31]. EELS maps were calculated using the discrete dipole approximation (DDA) method [32]. In COMSOL Multiphysics, the electron beam excitation for an electron moving in the direction at the velocity of is introduced through a current density as:
where is the angular frequency, and is the impact parameter of the electron trajectory. The moving electron creates an incident electric field, , which in the presence of a metallic nanostructure results in an induced field, . The induced electric field acts back on the electron, causing it to lose energy. The amount of the energy loss due to each frequency component of the induced field is given by the action integral [26, 33]:in which is the total volume of the simulation domain. By directly inserting the current distribution given by Eq. (1) into Eq. (2), the action integral can be further simplified to:which yields the same result as shown in [28]. In order to compute the moments directly from the induced field, we need to calculate the induced charge distribution which can be obtained by applying Gauss law to the calculated induced field as:
Fig. 1 Calculated spectra of (a) an isolated SRR structure S, (b) a 0°-rotated coupled SRRs structure A, (c) a 180°-rotated coupled SRRs structure B, and (d) a 90°-rotated coupled SRRs structure C. The parameters of the SRRs in the simulations are: thickness H = 30 nm, length L = 120 nm, width W = 30 nm and gap distance between the left and right SRRs G = 20 nm. The inserts depict the schemes of the SRR structures used in the simulations. The excitation positions of the electron beam are indicated by the “1”, “2”, and “3”. Three maxima at energies of 0.98 eV, 1.68 eV, and 1.98 eV are visible in the calculated EELS spectra of the isolated SRR, corresponding to the fundamental (I), second (II)-, and third (III)-order plasmonic modes. The fundamental, second-, and third-order plasmonic modes split into two new modes in the structures A and B. However, only the third-order plasmonic mode splits into two new modes in structure C.
3. Experimental methods
For our experiments, the samples were fabricated using the following electron beam lithography (EBL) process: polymethyl methacrylate (PMMA) 950k molecular weight 1.5% in anisole was spin-coated at 6000 rpm onto 3 mm-diameter, 30 nm-thick Si3N4 membrane TEM windows. The free-standing membrane occupied a 500 μm-square area for each of the 9 windows (available from UltraSM Silizium TEM Windows). The samples were patterned in a 30 kV EBL system (Raith eLINE) and then developed at 20°C temperature in 3:1 isopropyl alcohol (IPA): methyl isobutyl ketone (MIBK) for 20 sec. Afterwards a 3 nm Cr adhesion layer and 30 nm Au metal were deposited on the samples in a thermal evaporator. Finally, the lift-off was done in N-Ethyl-2-Pyrrdidon (NEP) solvent at 75 °C for 2 h. The characterization experiments were carried out with the Sub-Electronvolt Sub-Angstrom Microscope (SESAM) (Zeiss, Germany) operated at 200 kV. The SESAM is equipped with a Schottky field-emission gun, a symmetric electrostatic Omega-type electron monochromator, and the in-column MANDOLINE filter [34]. The Au SRRs were investigated by energy-filtered transmission electron microscopy (EFTEM). A 0.23-eV monochromator slit and a 0.2-eV energy-selecting slit in the energy filter were used. Series of energy-filtered images were recorded in the energy-loss range from 0.6 to 2.6 eV using a step size of 0.2 eV. The images were recorded on a 2k × 2k CCD camera (Ultrascan, Gatan, USA).
It should be mentioned that in contrast to STEM-EELS, EFTEM leads to reduced contamination of the structure. Moreover it has both a higher spatial sampling and a shorter acquisition time in comparison with STEM-EELS. As a consequence, EFTEM leads to a reduced electron dose on the sample, and therefore minimizes radiation damage.
4. Results and discussions
In order to numerically study the plasmonic modes of the SRRs, various electron trajectories have been used to excite the plasmonic modes of the SRRs. The experimental EFTEM series will be discussed later.
The excitation positions of the electron beam are indicated by the number “1”, “2” and “3” as shown in the insets of the Fig. 1(a)-1(d). Three maxima at energies of 0.98 eV, 1.68 eV, 1.98 eV are visible in the EELS spectra of the isolated SRR, corresponding to the fundamental (I), second (III)-, and third (III)-order plasmonic modes, respectively (Fig. 1(a)).
Considering only a single SRR structure, three electron impact parameters can be introduced to excite selectively different plasmon resonances. By comparing the intensity of the EELS spectra in Fig. 1(a) for various electron impact positions and resonant modes, it is clear that the lowest-energy mode at 0.98 eV has a dominant hotspot along the arms which is more intense at the free end of the arms. When the electron beam is focused at position “3”, we can only see the second-order mode at 1.68 eV, as shown by the green curve in Fig. 1(a).
By comparing with the isolated SRR spectra, it is evident that when the electron excites structures A and B at positions “1”, the fundamental plasmonic mode splits into two new modes with both resonances apparent in the EELS spectra (blue lines in Figs. 1(b) and 1(c)). Although the second- and third-order (at 1.98 eV) resonances also split into two new resonances, the electron probe focused onto position “1” can only sense the higher-order anti-bonding resonances.
All hybridized modes of the structure can be detected when the electron beam excites the structure at position “2”. This shows that at this position none of the modes of structure A exhibits zero intensity of the z-component of the electric field.
In order to investigate the physics beyond the energy splitting in our structures, we have calculated two-dimensional EELS maps, by scanning the electron beam over the structure. The distances between the hotspots, which are visible from the EELS maps, have a direct impact on the interaction energy and the amount of energy splitting in the hybridization of the plasmon modes [10]. Besides that, the relative orientation of the induced dipoles also affects the interaction energy [14]. Both of these parameters, i.e. the distance between the hotspots and the relative orientations of the dipoles can be engineered by simply rotating one SRR element, as shown in the insects of Figs. 1(b) and 1(c). The distance between the hotspots for the fundamental mode is decreased substantially in structure B in comparison with structure A, as shown in Fig. 2(a) and Fig. 3(a). Correspondingly, the energy splitting of the fundamental mode is more than two times higher in comparison with structure A. When calculating the EELS maps at selected resonant energies for structure A, as shown in Fig. 2(a), we observe an interesting locations for hotspots in the coupled structure, in the sense that at the energy loss of 0.92 eV the EELS signal is more concentrated near the left-side element, while at the energy loss of 1 eV the right element exhibits higher EELS intensity. This behavior, which is a direct consequence of the spatial symmetry of the structure, can be also captured from theexperimentally acquired EFTEM images (Fig. 2(b)). Structure B sustains a mirror spatial symmetry, directly observable in the spatial distribution of the electric field as observed in both the calculated EELS maps and experimentally acquired EFTEM images.
Fig. 2 (a) The calculated EELS maps, (b) the experimental EFTEM images, and (c) the calculated charge-density maps and magnetic moment distributions of the isolated SRR structure S and the 0°-rotated coupled SRRs structure A at resonance energies of the fundamental, second- and third-order plasmonic modes. Charges are marked by “+” and “-”, magnetic moments are marked by “⊗” and “☉”. The scale bars of the EFTEM images are 100 nm and the color codes for the intensities in (a)−(c) are presented accordingly.
A direct consequence of the plasmon hybridization effect is understood by calculating the charge distributions (Fig. 2(c)). Apparently, induced charges of structure A demonstrate an anti-symmetric distribution with respect to a plane imposed in the middle of adjacent SRRs at the energy loss of 0.92 eV, while the higher-order anti-bonding resonance at 1 eV sustains a symmetric charge distribution. The amount of energy splitting and therefore the interaction energy of structure A is much less than in the case of structure B. This can also be understood from the fact that the distance between the induced electric dipoles for structure B is less than for structure A, hence facilitating a more enhanced Förster-like energy transfer between individual elements [35, 36].
The experimental resonance energies are inferred from the acquired 3-dimensional data cube, where the two lateral dimensions are formed by the spatial coordinate systems transverse to the electron trajectory and the third axis is the energy-loss axis. We refer to the resonant energies as those energies where the extracted peak maps demonstrate a maximum intensity [21]. It should be noted that despite the perfect agreement between the experimental EFTEM image and the simulations for mode II, there are slight differences between the EFTEM images for the modes I and III compared to the calculated EELS maps. This is originated from the fact that the energy resolution in EFTEM is limited to 0.2 eV, due to the mechanical slit which is used here. The energy splitting of the fundamental and of the third modes is only at the order of 0.1 eV, which cannot be perfectly captured by the EFTEM technique. Moreover, the deterioration of the structures from the idealized geometry and the formation of rough surfaces are accompanied by Rayleigh scattering of the induced plasmonic waves, which can affect the experimentally observed intensity profiles enormously, especially at higher energies.
The magnetic moments induced in structure A at the fundamental resonant energy sustain the same symmetry as the induced electric moments (both transversally arranged and anti-symmetric for the lower-order resonance). An arrangement of coupled SRRs in the form of structure B imposes the same electric moment distribution for the fundamental mode, while the magnetic moments sustain mirror symmetry with comparison to structure A (Fig. 3(c)). Since the size of individual SRR elements in comparison with the effective wavelength suggests an appropriate applicability of the quasi-static theory, it is not surprising that the magnetic moment does not play a fundamental role in the hybridization scheme of the fundamental mode [37]. Shifting our attention to the second-order resonance of an individual SRR at the energy loss of 1.68 eV, it is apparent that this mode sustains three hotspots along the circumference. Since there is no difference in the distance between the hotspots of the second-order modes for both structures A and B, there is no change in the amount of energy splitting for both structures. Interestingly, the new set of hybridized modes at energy losses of 1.5 eV and 1.8 eV are less sensitive to spatial symmetry.
Fig. 3 (a) The calculated EELS maps, (b) the experimental EFTEM images, and (c) the calculated charge-density maps and magnetic moment distributions of the isolated SRR structure S and the 180°-rotated coupled SRRs structure B at resonance energies of the fundamental, second- and third-order plasmonic modes. Charges are marked by “+” and “-”, magnetic moments are marked by “⊗” and “☉”. The scale bars of the EFTEM images are 100 nm and the color codes for the intensities in (a)−(c) are presented accordingly.
The low-energy bonding second-order mode sustains hotspots at the two ends of the combination of coupled SRR structures in x-direction, while the higher-energy antibonding second-order mode exhibits a hotspot at the gap between the two individual elements. This behavior is also perfectly captured in the experimentally acquired EFTEM images, as shown in Figs. 2(b) and 3(b).
Finally, for the third-order resonance of a single SRR structure a quadrupole mode with four antinodes of charge density oscillation is excited at the energy loss of 1.98 eV, as shown in Fig. 2(c) by the charge-density map. However, the hotspots are mainly located at the corners of the SRR, in striking difference to the hotspots in the fundamental mode, as visiblein both, in the calculated EELS map (Fig. 2(a)) and in the EFTEM image (Fig. 2(b)). However, despite similar distances between the induced dipoles for the third-order resonance, the amount of energy splitting of structure B is less than for the same mode excited in structure A. This is understood by considering the locations of the hotspots in both structures, as shown in the calculated EELS maps (Figs. 2(a) and 3(a)). In other words, the dominant dipolar moments are distinguished by considering the location of hotspots.
A clear understanding of the role of magnetic moments in the hybridization scheme can be obtained from the SRR arrangement of structure C (see Fig. 4(a)). In such an arrangement, the induced electric dipoles are oriented perpendicular to each other, while the induced magnetic moments are arranged transversely parallel or antiparallel. In fact, structure C does not demonstrate any energy splitting in the EELS spectra of its fundamental and second-order resonances. As mentioned previously, the size of the introduced SRR here is much smaller than the effective plasmon wavelength at the energy losses of 0.98 eV and 1.68 eV, which renders the electromagnetic response in the quasi-static limit. In contrast, the third-order plasmonic mode demonstrates a clear energy splitting which can be detected by electron probes focused onto positions “1” and “2”. For the third-order resonance at the energies of1.94 eV and 2.04 eV, the induced electric dipoles interact with each other, leading to the energy splitting. More interestingly, the locations of the hotspots also demonstrate rotational symmetry, dictated by the imposed spatial symmetry of the structure, as shown in Fig. 4(b). It is also worth to note that the fundamental mode has been excited when the electron beam is introduced at position “3”. This is understood by considering the rotational symmetry of the structure. The electron beam introduced at position “3” excites an electric-dipolar response in the right split ring, which induces an excitation of the dipolar response in the left split ring.
Fig. 4 (a) The calculated EELS maps, (b) the experimental EFTEM images, and (c) the calculated charge-density maps and magnetic moment distributions of the 90°-rotated SRRs structure C at resonance energies of the fundamental, second- and third-order plasmonic modes. Charges are marked by “+” and “-”, magnetic moments are marked by “⊗” and “☉”. The scale bars of the EFTEM images are 100 nm and the color codes for the intensities in (a)−(c) are presented accordingly.
5. Plasmon hybridization scheme and retardation effect in coupled SRRs
As suggested by the above mentioned investigation, a direct hybridization diagram can be arranged as shown in Fig. 5. The hybridization diagram directly displays the energetic splitting of all plasmon resonances up to the quadrupole arrangement of the charge distributions, for both structure A and B, while for the case of structure C only the quadrupole mode splits into a bonding and an anti-bonding mode. The electric dipole moments are marked by red arrows in the hybridization diagram according to the charge distributions. Dashed arrows for the third-order modes denote weak intensities of the electric dipole moments, by considering that the hotspots of the third-order modes are more intensively located at the corners of the split rings. Since the hotspots show the local concentration of the averaged energy and hence the induced local charge accumulations, we conclude that the relative strength of induced individual moments should be understood by taking into account the hotspot locations. The magnetic moments have also been marked in the hybridization diagram. It is obvious from Fig. 5 that the same charges across the gap between the left and right split rings, are responsible for all modes, the high-energy fundamental, and the second-and third-order modes of structures A and B, as well as for the high-energy third-order mode of structure C. This is because symmetric charge distributions have stronger Coulomb repulsion, and therefore correspond to a higher resonant energy. To get deeper insight into the role of magnetic and electric moments in the plasmon hybridization of the coupled SRRs, the radiation far-field spectra for the electric dipole moment, the magnetic dipole moment, the electric quadruple moment, and the magnetic quadruple moment, of all the structures S, A, B and C were calculated with Comsol Multiphysics, using the approach proposed elsewhere [38].
Fig. 5 Plasmon hybridization schemes of the SRR structure S and the coupled SRRs A, B and C with different relative orientations. Charges are marked by “+” and “-”, strong electric dipoles are marked by solid arrows “↑” and “↓”, magnetic dipole moments are marked by “⊗” and “☉”. The dashed arrows represent weak electric dipoles.
The radiated power of the dominant moment contributions, which are the x-component of the electric dipole moment , the y-component of the electric dipole moment, and the z-component of the magnetic dipole moment of structures S, A, B and C are shown as a function of electron energy loss (Figs. 6(a)–6(d)). It is apparent that the radiated power of the electric dipole is much higher than that of the magnetic dipole. Thus, the electric dipoles contribute more significantly. By directly comparing the contribution of each moment to the far-field spectrum, it becomes evident that only the transverse and longitudinal parallelarrangements of dipolar moments contribute to the far-field radiation. In fact, the near-field spatial distribution of moments forming the dark modes interfere destructively in the far-field, which hinders them to be detected by optical far-field spectroscopy techniques.
Fig. 6 Radiated power spectra of the x-component of the electric dipole moment , the y-component of the electric dipole moment , the z-component of the magnetic dipole moment and the total radiated power for (a) the isolated SRR structure S, (b) the coupled SRRs structure A, (c) the coupled SRRs structure B, and (d) the coupled SRRs structure C. The locations of the electron beam excitation and the origins of the coordinates are at the same positions which is marked in the insets schemes.
In addition, the radiated power of the electric quadrupole moment and the magnetic quadrupole moment are much smaller than that of the electric dipole and the magnetic dipole moments, which is not shown here. Thus, the contribution of the quadrupole moments to the far-field radiation is completely negligible.
6. Conclusion
We have numerically and experimentally investigated the plasmon modes of coupled SRRs by EFTEM, DDA, and Comsol Multiphysics calculations. We demonstrate the plasmon hybridization schemes for the plasmons modes up to quadrupole modes. Furthermore, we have verified the ability of EELS to directly probe the energy splitting in coupled meta-atoms. The calculated radiation spectra from each individual electric and magnetic moment reveal that the role of the dipole moments play a key role in the plasmonic hybridization in coupled SRRs. Here we confirm that the locations of the induced hotspots dictated by the spatial symmetry of the coupled SRRs should be considered for deriving the hybridization scheme for the plasmonic resonances.
Acknowledgments
QL gratefully acknowledges financial support from the Ph.D. student exchange program between the Max-Planck Society and the Chinese Academy of Sciences. NT acknowledges the Alexander von Humboldt Foundation for the research fellowship. The research leading to these results has received funding from the European Union Seventh Framework Program [FP/2007-2013] under grant agreement no. 312483 (ESTEEM2).
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