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A plasmonic molecule showing strong magnetic resonance modes and flexible tunability is proposed. The molecule is composed of two elements, a crescent shaped metallic disk and a smaller one embedded in the cavity of the larger one. The cavity and gap formed by these two elements enable the molecule to support magnetic resonances in the visible and near infrared spectral region, while electric resonances are much weaker to be detected. We show that by changing the relative orientation angle of these two meta-atoms, the resonance wavelength can be changed from the visible to near infrared without modification of the size of the molecule. Anti-crossings and crossings of resonance energy levels, which stem from the coupling effect, are analyzed. When resonating magnetically, the local electric field enhancement at the crescent tip can reach up to hundreds of times with high spatial confinement, which renders the molecule promising applications in many fields.
© 2015 Optical Society of America
1. Introduction
Due to the strong effect of the electric field in the regular media which is magnetic responseless, the electric field modulation is the common approach to manipulate light response. Recently, nanoscale optical antennas based only on electric resonance has been applied to enhance the local electric field [1, 2], to guide the light wave [3–5] and to modulate the wavefront [6,7]. Meanwhile, phenomena related to the coupling between electric resonances were shown to modulate the spectral response effectively, such as Fano resonance [8–10], Plasmon Induced Transparency (PIT) [10], etc. However, in order to realize special phenomenon in electromagnetic wave propagating, magnetic resonant response must be taken into account. Pendry firstly introduced the Split Ring Resonator (SRR) [11] in microwave region, giving rise to the generation of completely novel properties, such as negative refraction [12–14], negative permeability [15, 16], which cannot be found in natural media. Magnetic resonance in SRR was achieved by introducing a gap, through which conduction current was formed, generating the strong corresponding magnetic field. Besides of the proposed SRR [13, 17–19], parallel nanorods [12], nano-crescent [14], and rings of nanoparticles [20, 21] were also designed to support the magnetic resonance. These designs employ straightforward structures as building blocks to develop a composite configuration, in which significant magnetic response was obtained in both visible and infrared region [12–17]. All structures mentioned above reveal both electric and magnetic responses [14, 22–24]. For example, in the crescent, the electric resonance could be the electric dipole moment that is typically located around tips, while the magnetic resonant response is generated in the cavity [14]. Moreover, the scale or shape of the electric or magnetic structures will need to be modified in order to tune the resonance [23–26].
In this paper, we present an energy-level-controllable plasmonic magnetic molecule (PMM), which is a coupled nano-resonator, composed of two different-sized eccentric nano-crescents. We discuss the general energy levels of the PMM and analyze the coupling effect. Benefitting from the pure magnetic resonances in the PMM, the relative azimuth angle between the two crescents, which is analogous to the bonding angle in a real molecule, can be applied to tune the resonant frequency from visible to near-infrared without any scale modification. This features the PMM a new degree of freedom to modulate the frequency response with nano-electromechanical technology. It is also found that the magnetic resonances can induce extremely confined and enhanced electric field on the physical tips, expanding its potential applications to both magnetic and electric.
2. General modes and coupling effect
In Ref. 11, Pendry demonstrated that the magnetic effect (microwave region) of a closed metallic cylinder could be extended by introducing in capacitative elements [11]. He put a gap into the cylinder and duplicating a smaller one in the cavity to develop a “split ring”. The capacitance between the two rings enables current to flow. A LC type resonant circuit model was considered along with the induced inductance of the metallic cylinder and the capacitance at the gaps. The resonant conduction-current-loop generates an enhanced magnetic field. The gap size is a key point for the resonance. Changing the gap size would be a way to manipulate the resonance. At the optical wavelengths, the situation is similar except that the conduction current is weakened, however, the displacement current is enhanced. Materials with positive permittivity (such as air) exhibits capacitive impedance for the displacement current, and that metallic media whose permittivity (real part) is negative, acts as inductive impedance [27, 28]. The PMM we present is similar to the “split ring”, which has plentiful gap configurations. The air gaps and metallic components can form nanoscale LC type circuits to develop current loops. Thus strong magnetic resonances can be produced in the PMM.
The proposed structure is featured in Fig. 1(a) and (b); composed of two nested eccentric nano-crescent disks [Fig. 1(a)], where the radius of the outer crescent is R and the offset of the cavity from the outer crescent center is D with the cavity radius of r. Apexes are rounded for fabrication consideration and the distance between them is w1. The inner crescent is achieved by scaling down the outer one and centered at the cavity of the outer crescent with the tip to tip distance as w2. The gap between the two crescents is d. The angle, θ, between the directions of the two crescent center-openings, defines the relationship between the two crescents (bonding angle). We employ the commercial software Comsol Multiphysics (COMSOL Inc.) to simulate the electromagnetic response of PMM. The parameters in the simulation are defined as, R = 60nm, r = 40nm, w1 = 20nm, w2 = 30nm, D = 19nm, and d = 4nm. Considering the standard nanofabrication, the thickness of the PMM is set to 20nm silver, whose dispersion relation is taken from the literature [29]. Figure 1(b) features the magnetic field of the excited resonance at a wavelength of 1263 nm at θ of 40 deg and thickness of 20 nm; its displacement current distribution is shown in Fig. 1(g). It reveals two clearly closed loops of the displacement currents, the sign of magnetic resonant responses. Other resonances at θ = 40 deg are also presented in Fig. 1(c) with their displacement current distributions in Fig. 1(d)-1(f). The fact that all the displacement currents at the spectral peaks are closed loops, suggests that the proposed structure shown in Fig. 1(a) presents a “pure” magnetic resonant effect. The distributions of the real part magnetic field component Re(Hz) shown in Fig. 1(c) also present the evidences of strong magnetic resonance. The complete magnetic response of the structure lands it as “plasmonic magnetic molecule”, highlighting its nature of two-component structure.
Fig. 1 (a). Schematic and geometrical parameters of a PMM. (b). A 3D overview of the excited magnetic mode related to the wavelength of 1263nm with a bonding angle of 40 deg. The red arrows represent the magnetic field vectors. (c). Spectrum of the maxima of the magnetic enhance factor (MEF) of a PMM (3D) with a bonding angle of 40 deg. (d-g). Normalized distributions of the real part of magnetic field Re(Hz)/H0 corresponding to each peak in c. (d). λ = 694 nm; (e). λ = 833; (f). λ = 1062; (g). λ = 1263 nm. The black arrows represent the displacement current density vectors. Both of the magnetic field and the displacement current loops are the evidences of strong magnetic resonance.
3D and 2D simulations with θ = 140 deg are conducted. In the 3D simulation, the illumination is from the top (normally) and polarized along y-direction, while in the 2D case, the illumination is from left with the same polarization [Fig. 1(a)]. The conducted comparison (not shown here) reveals that the resonance wavelengths of lowest four magnetic modes, blue shift 102, 192, 165, and 116 nm respectively, when switching to 2D simulation. Although the resonance wavelengths are different, the magnetic field and displacement current distributions almost keep the same. The MEFs for the four resonances change from 12 to 25, 11 to 9, 6 to 5 and 50 to 18 respectively. In spite of above shifts, the relative strength of the magnetic enhancement among the four resonances has no big change. That is to say, both the illumination direction and the simulation dimension do not affect the excitation of magnetic modes of the PMM. Because of the time consuming of the full 3D simulations, the spectral responses are performed by 2D counterpart with infinite thickness. For simplicity, the surround medium is selected as vacuum. The PMM is illuminated from left as shown in Fig. 1(a).
Maintaining all parameters but the bonding angle θ, we calculate the resonant variation, defined as the maxima of the magnetic enhancement factor (MEF, |H|/|H0|; |H| and |H0| are the local and incident magnetic field amplitudes, respectively). The resonant modes of the PMM are calculated versus the parameter shown in Fig. 2(a). Figure 2(a) indicates that the resonance in proposed PMM could be tuned from visible light to near infrared ranges. The tone of the color in Fig. 2(a) shows the strength variation of the excited resonances, or the variation of the field enhance factor. It implies that the near field distributions shift when the inner crescent rotates. In Fig. 2(b), the resonances clearly show different patterns. For some molecules, the enhanced magnetic field is in the inner cavity, while for others it’s mainly in the gaps between the crescents. This means that there exist at least two completely different resonance modes in PMM structure (taking the case of θ = 140 deg, or along the white dashed line in Fig. 2(a)). Figure 3(a)-3(d) show the magnetic field distributions of resonant PMMs at these resonant positions, where the displacement current densities are drawn as black arrows. The resonance at 2206nm exhibits the circulation of the displacement current around the inner cavity [Fig. 3(a)], revealing its nature as a magnetic dipole (referred as the cavity dipole pointing out of the plane, or positive value). At 1181, 926 and 754 nm, shown in Fig. 3(b)-3(d), the pronounced current loops are around the gaps, with the maximal magnetic field enhancement inside. The corresponding modes are referred as gap modes. They are actually Fabry–Perot type resonance related to the gap plasmon [30, 31]. The resonance condition can be roughly written as:
where βgp is the real part of propagating constant of the gap plasmon, L is the length of the gap, φ is phase shift related to the reflection of gap plasmon at the gap terminations and n is an integer referring to the order of the resonance. The gap length is defined as the length of the center arc between two gap terminations. As shown in Fig. 1(a), points A and B represent the terminations of the upper gap, and its length is represented by the red dashed line. According to Eq. (1), for a specific order of resonance (fixed n), larger gap length qualitatively corresponds to smaller βgp. Furthermore, the dispersion relation indicates that gap plasmon with smaller βgp is roughly related to smaller frequency, or larger wavelength in the free space [31]. Consequently, the resonance wavelength of the gap mode becomes larger as the gap grows longer. Generally speaking, both cavity and gap modes change their energy levels along with varying bonding angle. In a PMM, the cavity and gap modes are inseparable, or say, they are coupled. The magnetic resonances at 1181 nm [Fig. 3(b)] and 926 nm [Fig. 3(c)] reveal a strong positive magnetic dipole resonance in the gap and a weak negative magnetic dipole in the cavity. The resonance at 754nm [Fig. 3(d)] exhibits that both positiveand negative magnetic dipole moments are in the gap. The coexistent two anti-paralleled magnetic dipole moments could be viewed as a magnetic quadruple. Figure 3(e-h) illustrate the electric field vector and amplitude (normalized by incident field |E0|) distributions. From the electric perspective, the distributions cannot be characterized by any simple electric modes. Actually, the electric field patterns in Fig. 3(f)-3(h) are similar to those from gap plasmons that supported by double strips, which has been suggested as magnetic resonances [30]. This confirms that the PMM is a pure magnetic structure.Fig. 2 (a). MFE map against bonding angle and wavelength (normalized by amplitude of incident magnetic field). (b). Overview of magnetic field distribution (Re(Hz)/H0) of PMM modes on every line of energy levels.
Fig. 3 (a-d). Distributions of Re(Hz)/H0 and displacement current densities corresponding to these peaks along the white dashed line in Fig. 2(a). These four maps indicate that the resonant magnetic fields are either in the inner cavity or in the gap between these two crescents, meaning that two kinds of resonances are excited. (e-f). Distributions of electric field amplitude (|E|/|E0|) related to (a-d) respectively.
The coupling strength between the cavity and gap modes or the two gap modes affect the resonant spectral positions when the bonding angle θ evolves [Fig. 2(b)]. The coupling strength is qualitatively determined by the overlapping between resonant fields. From Fig. 3(a)-3(c), the magnetic field and displacement current distributions show strong coupling between the negative cavity mode and positive gap mode [Fig. 3(b) and 3(c)]. The mode at 2206 nm [Fig. 3(a)] results from the strong coupling between a intense cavity mode and a weak gap mode; the mode at 754 nm is totally opposite: a intense gap mode (2nd order) and a weak cavity mode. The coupling results in the anti-crossings in the energy level. However, the coupling between the two arc-like gaps in the PMM, is weak due to the spatial separation and small field overlapping. The coupling between two modes can be understood in terms of a 2 × 2 Hamiltonian matrix [32]:
where E1 and E2 are complex energy levels of the uncoupled modes. The imaginary part represents the loss rate or the broadening of the resonance. W and V are complex coupling constants. The coupled eigenstates can be obtained by diagonalization of the matrix. The eigenvalues are as follows:The most effective coupling happens when Re(E1(θ)) = Re(E2(θ)), where the difference between the coupled energy levels is:For simplicity, WV is assumed to be real which is true for internal coupling [32]. This assumption does not affect the validity of the analysis. When 4WV>>|Im(E1)-Im(E2)|2, i.e., there is strong coupling between the two modes, the energy level is effectively forced apart and an anti-crossing is observed in the resonance energy levels. However, When 4WV≈|Im(E1)-Im(E2)|2, the difference between the coupled energy levels becomes quite small. If it is smaller than the total broadening |Im(E+(θ)) + Im(E-(θ))|, there will be no observable energy level split.As discussed above, the strong coupling between the cavity and the gap mode produces the energy level anti-crossings. There are two anti-crossings at the wavelength range of 1351-1667 nm, resembling a “band gap” in which no resonance can be excited at all bonding angles [Fig. 2(b)]. Figure 4(a) further illustrates the “band gap”. The two energy level branches represent two modes coupled from cavity and gap mode. The magnetic field distributions for these two modes are plotted in Fig. 4(b)-4(g) for θ = 40, 70 and 110 deg. It is shown that the maximum field for the upper branch moves from the inner cavity to the gap while passing through the anti-crossing [Fig. 4(b)-4(d)]. The bottom branch undergoes the opposite field exchange [Fig. 4(e)-4(g)]. The field exchange indicates the two mode are strongly coupled [32, 33]. The strong coupling stems from the field overlap between the cavity and gap mode [Fig. 4(c) and 4(f)].
Fig. 4 (a). Energy levels near the “band gap”. (b-g). Distributions of Re(Hz)/H0 related to energy levels marked with black dots and corresponding letter labels in (a). The directions of the field in the inner cavity and the gap are marked with cross in circle (point in) and dot in circle (point out).
In contrast to the anti-crossing, energy levels shown in Fig. 2(a) reveal another feature of resonance crossing. Figure 5(a) shows the energy levels near the crossing at λ = 1056 nm, θ = 180 deg. The corresponding magnetic field distributions are shown in Fig. 5(b)-5(d). These two resonant PMMs before the crossing, exhibit the feature of one negative cavity mode and one positive gap mode [Fig. 5(b)-5(c)]. When these four resonances meet at the crossing, the fields in the cavity interact with each other in phase, leading to an enhanced field in the cavity [Fig. 5(d)]. There is no obvious coupling between the two gap modes because of the spatial separation in the field distribution [Fig. 5(d)]. The weak coupling cannot cause visible split of energy level, which results in the crossing phenomenon between two frequency branches.
Fig. 5 (a). Energy levels near a crossing at bonding angle of 180 deg. (b-d). Distributions of Re(Hz)/H0 related to energy levels marked in (a).
3. Electric field enhancement effect
Electric field enhancement at the sharp tip is the signature of the single crescent [23, 26, 34], where electric dipole resonator is excited. In PMM, or two coupled crescents, electric resonances can be barely assigned. Is there still tip enhanced electric field effect at the magnetic resonances in PMM? With the purpose of answering the question, we calculate the local electric field enhance factor (EEF, |E|/|E0|) near the upper tip of the outer crescent in PMM by illuminating the PMM at the same configuration shown in Fig. 1. The parametric electric field enhancement factor is shown in Fig. 6. Comparing to Fig. 2(a), where the magnetic resonances are displayed, the electric enhancement factors shown in Fig. 6 follow the exact traces in Fig. 2(a) apart from a few differences. Figure 2(a) is symmetric along 180 deg, which reflects the symmetry nature of the PMM at 180 deg bonding angle. Figure 6 does not hold the symmetry, in which higher angles (larger than 180 deg) show higher field enhancement. It reflects the non-symmetry of the field enhancement at upper and lower outer tips in PMM. The similarity between the magnetic resonance in PMM [Fig. 2] and the enhanced electric field at the outer tip [Fig. 6] not only proves the pure magnetic resonant response in PMM, but also indicates that magnetic resonance can be employed to generate the highly localized electric field.
Fig. 6 The spectra of EEF near the upper tip of outer crescent as a function of bonding angle and wavelength. The resonant peaks coincide these in Fig. 2(a), which means the electric field enhancement is due to the magnetic resonances.
From Fig. 6, it can be seen that the electric fields at the tip are enhanced with varied strengths at different magnetic resonances. Figure 7 shows the magnetic (upper row) and electric (bottom row) distributions at positions labeled in Fig. 6. The magnetic field plotted in Fig. 7(a) corresponds to position 1 in Fig. 6 (λ = 2308nm, θ = 180 deg), exhibiting a strong cavity mode with particularly high magnetic enhancement. The corresponding electric field shown in Fig. 7(d) indicates that the EEF reaches 223 near the upper tip of the outer crescent. Figure 7(b), corresponding to position 2, is the PMM mode with a positive cavity and a negative gap mode. The EEF near the upper tip of the outer crescent reaches 615 [Fig. 7(e)] although the magnetic resonant enhancement is smaller than that in position 1. The magnetic resonance is more concentrated on the upper tip. Figure 7(c) corresponds to a strong gap mode, where the resonance in the upper gap also brings about an EEF of 216 near the upper tip [Fig. 7(f)].
Fig. 7 (a-f). Real part of magnetic component (Re(Hz)/H0, upper row) and EEF (bottom row) distributions related to positions marked with numbers in Fig. 6. The corresponding wavelengths and bonding angles are shown below each group of pictures.
The electric enhancement in PMM benefits from the sharp tip configuration; but it is different from the single crescent. In PMM, the displacement current corresponding to the magnetic resonance, pushes the charges onto the sharp tip, leading to high concentration of polarization charge density, which results in high electric field enhancement. The concentration of polarization charge near the upper tip is influenced by the overlapping area between the tip and outer surface of the inner crescent in the field distribution. The displacement current of a magnetic resonance flows through the overlapping area into the inner crescent. Smaller overlapping area restricts the current more tightly and produces higher polarization charge density, resulting in stronger electric enhancement. This could be the main reason why the electric enhancement at position 2 is much larger than those in other positions.
To define the confinement of the electric field at the tip, the EEF distribution of the PMM with λ = 1119nm, θ = 320 deg is redrawn in Fig. 8(a). The spatial confinement of the field is shown in the inset, indicating that the energy is highly confined near these two adjacent tips. The higher peak is corresponding to the outer tip while the shorter peak is for the inner tip. The EEF across the terminal of the upper tip [along the white dashed line in Fig. 8(a)] is plotted in Fig. 8(b). The sharp peak has a maximum enhancement factor of 557. The inset reveals a Full Width at Half Maximum (FWHM) of about 1.2nm. The sharpness of the tips can significantly affect the electric field enhancement near the tip of the outer crescent. The maximum EEF in the EEF spectra (shown Fig. 6) as a function of the tip radius is investigated by changing the tip radius. As shown in Fig. 9, the field enhancement dramatically reduces from 434 to 215 when the outer tip radius increases from 1 to 3 nm. There is no significant change when the sharpness of the tip further increases. The reason behind the EEF dependence on the sharpness of the tip is straightforward. The sharper the tip, the denser the charge distribution is, and the higher the induced local field will be.
Fig. 8 (a). Local electric field distribution for enhance factor peak indicated with purple arrow in a, (θ = 320°, λ = 1119nm). Inset shows the spatial distribution of time average total energy density within the purple box, revealing high localization of energy. (b). EEF distribution along with cross line indicated in (a) with white dashed line. Inset shows a Full Width at Half Maximum (FWHM) of about 1.2nm.
Fig. 9 The maximum EEF near the upper tip of the outer crescent (at proper bonding angle and wavelength) as a function of the outer tip radius. The tip radius of inner crescent is fixed at 3 nm.
The gap size is decisive to the achievable maximal EEF in the PMM. Considering the deviation that may happen in fabrication process, the impact of larger gap size on the spectra and maximum EEF are investigated. We study the larger gap size with two methods: magnifying the outer crescent or diminishing the inner crescent. It is revealed that, in both cases, the overall energy levels maintain the same shape, but blue shift while the maximum EEF becomes smaller at the same time [Fig. 10]. The shift is larger by using the latter method than the former [Fig. 10(b) and 10(c)]. In addition, the maximum EEF moves to a bonding angle of 285 deg and 290 deg when the gap size is changed from 4 nm to 7 nm and 10 nm with the latter method. The corresponding maximum EEF varies from 557 to 425 and further to 511 [Fig. 10(a), 10(b) and 10(d)]. Nevertheless, the angle of maximum EEF remains at 320 deg with a maximum EEF of 456 if the former method is employed [Fig. 10(c)].
Fig. 10 The EEF spectrum as a function of bonding angle for gap size d of 4 nm (a), 7 nm (b, c) and 10 nm (d). The larger gap is obtained by decreasing the size of the inner crescent in (b) and (d), while magnifying the outer crescent for (c). The blue shifts of the overall energy levels are 539 nm (b), 429 nm (c) and 761 nm (d). The maximum EEF are marked with purple arrows and corresponding wavelengths are shown. The maximum EEF moves to a bonding angle of 285 deg and 290 deg in (b) and (d) respectively, however, remains at 320 deg in (c).
4. Conclusion
The magnetic response of a PMM composed of two different sized eccentric nano-crescent disks is demonstrated. Owing to the special configuration of the PMM, two kinds of resonances are excited. One is a cavity dipole featured by a displacement loop around the inner cavity and strong magnetic enhancement in the cavity. The other mode is located in the gap between these two crescents. The resonances in the PMM stem from the combination of these cavity dipole and gap modes. The coupling among the resonances gives rise to split of resonance frequency (energy level), which appears as anti-crossings, or the “band gap”. Spectral crossing, where two spectral lines (modes) overlap without the spectral shift, is witnessed in PMM because there is no actual field overlapping when two PMM resonances meet. Another effect brought by the magnetic molecule is the enhancement of local electric field. The electric field of great enhancement with strong spatial confinement, can be achieved at specific wavelength and bonding angle.
The proposed PMM holds great potential in many fields. The magnetic resonant response of the proposed device can be utilized to build a magnetic plasmonic waveguide. In addition, the PMM could be fabricated small enough to be injected in living cells. The enhanced electric field would greatly enhance the Raman signals from the molecules in the cell, which could help researchers to study the activities in living cells. Moreover, pure magnetic resonances of the proposed PMM combined with other electrical resonant nanostructures can be used to build novel metasurface to manipulate optical phase, whose property can be adjusted by changing the bonding angle of PMM with nano-electromechanical technology.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (No. 61377054), Collaborative Innovation Foundation of Sichuan University (No. XTCX 2013002).
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