1. Introduction

Plasmon resonances are associated with electromagnetic enhancement, which is useful in a variety of applications. Design of these resonances, via changes to geometry and materials, is subject to some important constraints. Resonance requires negative permittivity as typically achieved below the plasma frequency of a metal. These materials damp resonances due to frequency-dependent absorption loss [1]. Unfortunately materials with the lowest losses are free-electron-like and tend to be chemically reactive, leaving only a small number of practical choices [2]. There is some prospect of diversifying options through metallic compounds [3] but this has yet to be verified experimentally. As an alternative, here we explore the use of negative permittivity materials in discrete core-shell geometries.

Binary metal core-shell particles have been previously synthesized, targeting a wide variety of applications that can benefit from manipulating chemical, magnetic, and optical properties, including combinations of many different metals [4]. The optical properties of Au-Ag systems [5] have been a particular focus, due to ease of fabrication, stability and strong optical resonances.

The mode positions of spherical Ag-in-Au core-shells have been described as being continuously varying [6] or lying close to the sphere mode of the dominant constituent [7]. Mode sharpness for Ag-in-Au and Au-in-Ag have been measured [8] showing reasonable agreement with theory [9]. Field enhancement for both combinations have also been considered [10] with enhancement increasing significantly as shell thickness increases. Interesting results have also been obtained for Au-in-Ag multishells with dielectric cores [11], including a broad but strong resonance in the interband region of Au.

Other rounded geometries have been investigated, including cylinders [12] and rods [13] – they offer similar tunability to Ag + Au spherical shells. Some investigations of non-spherical shells have been performed, including tuning of stars [14] and cuboids [15], and some experimental evidence suggests that nearly-spherical irregular shapes have modes that are essentially determined by the fill-factor [16].

Only a few investigations have attempted to systematically investigate bimetallic systems. A homogenization approach has been applied to estimate the effective permittivity of dual-Drude spherical core-shells [17], however this omitted some modes of the system. A general eigenmode approach has been used for investigating the complicated behaviour of externally-coupled nanorods [18]. In this article we also use an eigenmode framework, but emphasize using the fill-factor to tune the resonance in arbitrary core-shell particles.

While it is possible to calculate fully-retarded results, to clarify the physics we focus on small particles where electrostatic limits are reasonable approximations. In this regime absorption dominates over scattering and the position of the resonance depends only on geometric ratios, not absolute size. However, the particles cannot be too small because electron-confinement leads to additional size-dependent damping. This effect damps resonances of thin shells on dielectric cores [19], but it is not clear how metal-metal boundaries would be affected. Here we omit these size effects to concentrate on the more fundamental relationship between resonance, permittivity and shape. We develop a dipole model for the behavior of binary core-shells, which we verify using a combination of a known analytical expression for confocal ellipsoids and the boundary element method [20, 21] which is suitable for arbitrary shapes.

We first manipulate analytical expressions for ellipsoid modes to understand the fundamentals of binary core-shells (Sec. 2), show how to generalize the problem by separation into geometric and material factors (Sec. 3), and investigate the modes of spheres (Sec. 4), ellipsoids (Sec. 5), and cuboids (Sec. 6) to test how the ellipsoid model responds to shape perturbation.

2. Ellipsoid core-shell theory

It is well known that spherical core-shell systems have dipole-dipole hybridization of interface modes [22] making them an interesting system for study. We previously derived [1] an approximation for resonances of spherical dielectric-metal core-shells, and we can now generalize this to ellipsoids with metal in both core and shell. The polarizability $\alpha$of electrostatic dipole modes of confocal ellipsoidal core-shells can be written analytically [23]:

We now analyze the resonances arising from Eq. (1). In unary metal systems, mode positions and strengths can be elucidated by exploiting a link between metal permittivity at resonance and the geometric factor L. Although this approach gives insight into this [24] and other [25, 26] geometries, direct extension to bimetallic systems is not straightforward. We still take a low-loss limit with vanishing imaginary parts ${\epsilon }_c\text{'}\text{'}\to \text{0}$ ${\epsilon }_s\text{'}\text{'}\to \text{0}$, but find it most convenient to express the resonance condition in terms of optimum fill factor. We have previously applied this approach to spherical metal shells with dielectric cores [1], but here we show the bimetallic confocal ellipsoid result for the first time:

Equation (2) determines the relationship between geometry and permittivity, and the metal permittivity pairs relate this to the excitation frequency. The physical bounds $0\le f\le 1$ dictate the limits of the regions of resonant permittivities, and since there are two modes each bound results in two bounds on permittivity. The resulting limits are summarized in Table 1. The strength of the resonance can be determined by considering Eq. (3). The polarizability falls to zero on two of the bounds in Table 1. Further, each material contributes to damping of the resonance through the imaginary part of the permittivity, however these contributions are modulated by the environment. On two of the bounds only one material contributes to damping through its imaginary permittivity.

Table 1. Limiting cases for confocal core-shell resonances

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The combinations of resonant permittivities can be further elaborated by plotting the bounds from Table 1 in core & shell permittivity space; the specific result for spheres (L = 1/3) is shown in Fig. 1. There are five resonant regions, which can be accessed through different combinations of materials. Dielectrics have relatively constant permittivity$\epsilon >1$. Metals typically have highly dispersive permittivity$\epsilon$: at low frequencies (where metals are conductors) $\epsilon <<0$, but at higher frequencies $\epsilon ~0$ (characteristic of plasma). The lower right region of Fig. 1 corresponds to a metal sphere embedded in a dielectric shell and is therefore of limited interest here. The upper left region contains metal shells with dielectric cores as previously studied (would be represented horizontal lines on Fig. 1). In addition, the middle-left region contains combinations that would be best described as conductor shell, plasma core. Conversely, the lower-center region contains plasma shell, conductor core.

 

Fig. 1 Optimum fill factor for spherical core-shells (cyan f = 0, magenta f = 1) predicted by Eq. (2) as a function of the core and shell permittivities. Symmetric modes are labelled + + , all others are antisymmetric. Labels CSzd are bounds listed in Table 1. The black lines are K-Al core-shell (solid), and Al-K core-shell (dotted). The black dots are those predicted by Eq. (2) when f = 0, 1/4, 1/2, 3/4, 1.

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Notably, materials with substantially different plasma frequencies trace lines with high or low slope on Fig. 1, and have maximum overlap with the interesting regions where one material is in the conducting regime and the other is near the plasma frequency. In general, the combination with the higher plasma frequency in the shell (low-slope, upper-left Fig. 1) has a charge-symmetric resonance which avoids the zero-strength resonance limits indicated in Table 1, and allows the widest frequency range. This resonance runs from the sphere limit of the core (thin shell “C”) to the sphere limit of the shell (thick shell “S”). There is also a high frequency mode that is anti-symmetric. The opposite combination of low-plasma frequency in the shell (high-slope, lower right in Fig. 1) has two modes that are both anti-symmetric, and are affected by shielding effects near “z” and “d”.

3. Eigenmode method

We now turn our attention to the more general problem of core-shell resonances of arbitrary shapes. The electrostatic response can be written as a generic matrix equation

Eigenmode decomposition of Eq. (4) is a useful tool for understanding how modes of the system contribute to observed behavior such as absorption. In unary metal systems the material factor $\gamma$ is spatially invariant which enables trivial separation of the inversion (for example in multilayered spherical shells [24]), however in binary metal systems this is not true. Instead, it is helpful to partition the matrix equation to highlight the different interfaces between materials, each with an associated value of the material factor which we define as

The utility of the decomposition outlined in Eqs. (6) and (7) is that it represents the interaction in terms of separate interface modes, which depend only on geometry and are independent of material. The diagonal blocks of the new $\Omega$are the uncoupled eigenvalues $\Gamma$which are directly connected to depolarization factors often used to conveniently represent the effect of geometry. The off-diagonal blocks of the new $\Omega$ represent interface mode coupling which is also geometry dependent, and is sensitive to the overall separation between interfaces. We can gain further insight if we consider coupling between sole modes on each of the interfaces. This approach has been used previously to derive an implicit expression for the resonant permittivity of spheroidal shells [29]: here we additionally develop explicit expressions for the mode strengths and in later sections explore the implications of metallic cores and other shapes (cuboids). This dipole-dipole hybridization approach is particularly useful because it is the most fundamental interaction in core-shell systems and dominates many experimental observations. Consideration of Eqs. (4)–(7) shows that singly hybridized modes are given by:

Equation (8) has complex poles that can be used to analyze mode position and absorption strength in the limit of low material losses in$\gamma$. Mode position is determined by the characteristic equation in the denominator, and absorption strength is given by the residue which can be approximated via the real part of the numerator divided by the imaginary part of the denominator. In general the mode parameters cannot be determined analytically, however the known analytical result for ellipsoids, Eq. (1), yields the following terms:

4. Sphere resonances

To illustrate a combination of real materials with suitably different plasma frequencies, combinations of K (low plasma frequency 3.9eV) and Al (high plasma frequency 15eV) have been overlaid on Fig. 1. Although it might be argued that K is difficult to work with, we use it as an example to demonstrate the exceptional tuning range that is possible with widely differing plasma frequencies, and to clearly distinguish resonances. We interpolated the dielectric function of these materials from published experimental Tables [30]. The maximum resonance is also plotted in terms of frequency below in Fig. 2, together with the electric field in Fig. 3. Figure 2 confirms that Eq. (3) is accurate for these materials because it closely matches individual spectral for discrete fill-factors that were calculated using Eq. (1). We verified that these spectra are consistent with the limiting cases of solid spheres (see Table 1), and agree with the numerical method [20, 21] that we use later. K has exceptionally low loss and the combined K-Al resonance is blue-shifted relative to the main resonance of K-only shells, so this combination achieves stronger resonances than most single-metal shells (except Na). As noted above, the combination with K on the inside results in the broadest resonance.

 

Fig. 2 Absorption maxima of (a) K-Al and (b) Al-K spherical core-shells in free-space. The colored spectra are Eq. (1), at fill factors equally spaced from f = 0 (cyan) to f = 1 (magenta), the dots are the corresponding discrete peaks from Eqs. (2) and (3), and the heavy black line is the path of peaks predicted by Eq. (3). Labels CSzd are bounds listed in Table 1, and the paths are the same as those in Fig. 1.

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Fig. 3 Resonant electric dipole fields of (a) K-Al and (b) Al-K core-shells. The fields are calculated at resonance on surfaces, and all are directed normal to the surfaces and parallel to the direction of the excitation field. This is the maximum field on most surfaces except for inside the K-Al shell. The external surface (black), inner and outer shell surfaces (dotted), and core fields are shown. These are consistent with equivalent surface charges, which are symmetric in (a) and antisymmetric in (b). The anti-symmetry is stronger in the lower frequency mode.

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Figure 3 shows the resonant dipole fields on various surfaces - these fields were verified using full vector spherical harmonics [23]. Figure 3(a) confirms that in this case absorption is dominated by the core at low frequency (strong electric field), and the shell at high frequency (outer and largest volume). Note the similarity between the external field and the absorption.

5. Spheroid resonances

Ellipsoidal core-shells offer further tuning possibilities, which has been previously demonstrated for dielectric cores [31], and here we show how our theory applies to some spheroids with metal cores. Specifically, we compare the resonances of K-Al and Al-K core-shells where the outer boundary is the prolate spheroid with aspect ratio of 2 and the excitation is perpendicular to the axis of rotation. Discrete spectra were calculated using a numerical method (boundary elements [20, 21]). Figure 4 shows the confocal result (which was verified against Eq. (1)), and Fig. 5 shows the result when the core has the same shape as the outer surface. In both cases the predicted maximum has been overlaid assuming that the core depolarization factor is equal to the outer (Eqs. (2) and (3)).

 

Fig. 4 Resonances of prolate spheroid core-shells, outer aspect ratio 2 with confocal core, excited perpendicular to the rotation axis. The colored lines were calculated at fill factors f = 0, ¼, ½, ¾, 1 as indicated by the overlaid sketches. These are the full numerical results and have been verified against Eq. (1). The resonance predicted by earlier equations is overlaid in black, using depolarization fixed to that of the outer. The black dots are the discrete maxima predicted by Eqs. (2) and (3) using the same fill-factors as the colored lines. The black line is the path of the maxima predicted by Eq. (3). (a) is K in Al, (b) is Al in K.

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Fig. 5 Resonances of prolate spheroid core-shells of aspect ratio 2 excited perpendicular to the rotation axis, at various fill factors. The colored lines are full numerical results for the sequence f = 0, 1/4, 1/2, 3/4, 1, as indicated by the overlaid sketches. The black dots are the corresponding peaks predicted by Eqs. (2) and (3). The black line is the path of the peaks predicted by Eq. (3). (a) is K in Al, (b) is Al in K.

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4.1 Confocal spheroids

There is relatively good agreement for the results presented here because the confocal core does not change depolarization much compared to the shell. This means that the confocal core can be adequately approximated by assuming that the depolarization of the core is fixed to the outer, as demonstrated by the good agreement between exact and model results (Fig. 4). In general, for other orientations and shapes, mismatch between confocal and fixed depolarization is quite substantial, especially near f = 0, however modes are not uniformly affected. Confocal resonances that are spectrally distant from the deviating resonance of the core-shell interface (the “d”-bound in Table 1) are less affected: these include prolate parallel and oblate perpendicular K in Al, and oblate parallel low frequency Al in K.

4.2 Self-similar spheroids

When the core is the same shape as the outer surface, correct prediction of uncoupled dipoles is simplified due to fixed core, but there is some mode splitting because the dipole fields of the core and shell are no longer confocal. However, in the prolate perpendicular case shown in Fig. 5, the deviation is minimal due to the nearly confocal core.

Sensitivity to mode mixing is dependent on shape and polarization, in a similar way to the sensitivity of confocal modes to core deviation, with the exception of improvement in the high-frequency Al-in-K. This mode (which lies between “C” and “d” resonances) is dominated by the core, so it is insensitive to mode-mixing and its prediction is significantly improved by fixing the core. In contrast, low frequency Al-in-K prolate parallel (Fig. 6(b)) and oblate perpendicular (Fig. 6(c)) show particularly strong mixing due to proximity of a multipole mode. The maps presented in Fig. 6 assist predictions about the likely accuracy of the dipole-dipole model for other material combinations. Dipole-dipole resonances that have both spatial and spectral overlap with multipoles are likely to be affected by splitting. The maps indicate that there is also strong mixing for zero-zero core-shell permittivity for prolate perpendicular / oblate modes.

 

Fig. 6 Resonance maps in dual permittivity space (imaginary permittivities fixed at 0.1), for self-similar spheroid core-shells with fill factor f = 0.5, aspect ratio 2 and 0.5, and polarization perpendicular and parallel to the axis of revolution: (a) prolate perpendicular (b) prolate parallel (c) oblate perpendicular (d) oblate parallel. Bright colors indicate resonance, blue lines are the dipole-dipole prediction, solid black line is K-in-Al and dashed black line is Al-in-K.

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In summary, the model works well for spheroids when the core depolarization does not change much with fill factor or the core shape is nearly confocal. In other cases, agreement varies depending on remoteness from the confocal core-shell interface resonance or same-shape multipole modes.

6. Cuboid resonances

Cuboids are known to exhibit multiple modes, making them a useful test-case for the ideas presented in this article. Previously we have found that for relatively rounded (unary) cuboids the dipole dominates, with the quadrupole and face modes appearing only weakly [26]. We will compare how the internal and external coupling parameters of a cuboid vary with fill-factor compared to ellipsoids (Fig. 7), and test how well a dipole-dipole hybridization model can predict absorption in a K-Al system (Fig. 8).

 

Fig. 7 Resonance parameters of cuboids with roundness exponent p = 0.5. Internal interaction parameters are in (a) and external coupling strengths are in (b). Blue and green are the core and shell respectively, red is the core-shell coupling product, and cyan is the special cross-coupling term given by Eq. (11). Markers are the numerical result; the lines are a generalization of the ellipsoid model where the relevant parameters of the outer shape have been substituted.

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Fig. 8 Resonances of cuboids with roundness exponent p = 0.5. The analytic ellipsoid model with numerical coupling factors is overlaid in black. (a) is K in Al, (b) is Al in K.

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First looking at the coupling parameters, we generalize the ellipsoid model by substituting relevant parameters from the outer cuboid (i.e. using the numerical cuboid value of L, and replacing V with C_OO). Figure 7 shows that there is very good agreement except for minor deviation in the cross-coupling terms. This is perhaps to be expected because the near-field of the dipole of a cuboid is unlikely to scale confocally. Despite this minor error, we now use the model dipole-dipole parameters as inputs to the ellipsoidal maximum model for comparison with numerical resonance results (Fig. 8). Multimode splitting is clearly evident, which notably spreads the low energy anti-symmetric Al-in-K resonances in Fig. 8(b). However, there is reasonable agreement with the symmetric K-in-Al resonance in Fig. 8(a).

Further evidence for the effect of mode-mixing is presented in Fig. 9, which shows that the region with the shell more negative (i.e. K-in-Al) is least affected. In general we would expect the accuracy of the ellipsoidal model to decline further as the cuboid sharpens (p→0) and higher order modes become dominant: further calculations show that in fact the conclusions drawn from Fig. 8 also apply to p = 0.2 and p = 0.1 (which is a relatively hard-edged cuboid), as the differences observed are only moderately enhanced.

 

Fig. 9 Resonance map in dual permittivity space (imaginary permittivities fixed at 0.1), for moderately rounded cuboid core-shell with fill-factor f = 0.5 and roundness p = 0.5. Bright colors indicate resonance, blue lines are the dipole-dipole prediction, solid black line is K-in-Al and dashed black line is Al-in-K.

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7. Conclusion

In conclusion we have shown how to predict the ideal limit of resonances of bimetallic ellipsoidal core-shells, and used this to investigate some of the possibilities of this geometry. In particular we found that some resonances of shapes that are nearly ellipsoidal can be reliably approximated by an ellipsoid model parameterized via the fill-factor: these resonances include spheres (exact), prolate spheroids excited perpendicular to the rotation axis, and well-rounded cuboids. We found that operating near the plasma frequency of the core enables continuous tuning over a range that is not accessible if the core is dielectric. The permittivities and charge symmetry of this mode mean that damping is surprisingly weak and thus resonance strength is maintained relatively well. Ideally the materials should be both free-electron-like and have very different plasma frequencies to allow a wide tuning range. Achieving this type of combination will require some care, but the general idea of using plasma regime of a metal to tune resonances allows additional choice, and other multi-metal geometries should be investigated further.