1. Introduction

Optical nanoresonators have emerged as a promising avenue for a range of applications including metasurfaces [1], sensing [2,3], extreme light confinement [4], and quantum information processing [5]. One crucial factor in the design and performance of optical nanoresonators is the interplay between the shape and material of the device [6,7]. For example, even minute changes of both the shape of an optical nanoresonator and/or its constituent materials can be monitored by tracing the evolution of its spectral features, forming the basis of optical, label-free sensing [3].

Nanoscale optical resonators can be made of metals. Owing to their conduction electrons, metal nanoparticles (NPs) support a strong localized surface plasmon resonance (LSPR) response even for quasistatic particles and are able to confine light in ultra-small volumes [8]. These properties are easily tuned across a broad spectrum [9] and, for example, can lead to deterministic deposition of energy through absorption in appropriately shaped/sized nanorods [10] or offer large field enhancement in gold nanocups for sensitive surface enhanced Raman scattering measurements [11].

The alternative material for making nanoresonators are high index dielectrics which confine the electromagnetic field inside the resonator. While their field enhancements are usually smaller than for plasmonic particles, they offer the benefit of supporting both electric and magnetic modes [12]. Indeed, silicon was for long the material of choice to obtain strong magnetic resonances in the visible and NIR [13]. The shape and size of the dielectric particle determine their optical response [14,15], forming the basis of the ability to spectrally tune the relative resonance frequency of the magnetic and electric dipoles to direct scattering [16] or to use pores or openings to access the enhanced internal fields to manipulate spontaneous emission of magnetic dipoles [17,18].

Instead of limiting oneself to only one of the material types, it is possible to take the best of both worlds and utilize the beneficial aspects of both. For example, dual-resonator assemblies made of gold and silicon or dual-material metasurfaces have been proposed for discrimination of enantiomers [19]. Alternatively, particle-on-mirror setups can couple a low-loss dielectric resonator to its mirror image for extreme light confinement and Purcell enhancement [4].

Metals and dielectrics can be also combined at the subwavelength level into metal-dielectric multilayers [20] or nanorods in a host dielectric matrix [2,21]. When described by an effective medium model, such materials are characterized by a diagonal permittivity tensor with one of the principal components being of opposite sign than the other two [22], resulting in unbounded, hyperbolic isofrequency surfaces. Such dispersion is also found in natural materials, notably in layered van der Waals (vdW) ones [23–25] with new being actively searched for [26]. Due to their high-k modes and thus large sensitivity to small perturbations, hyperbolic materials have found a number of applications. Their use encompasses most of nanophotonics, including highly sensitive nanorod-based materials for biosensing [2,21], modification of linear [27] and nonlinear emission dynamics [28] and control of absorption and scattering channels [29,30].

Motivated by the various advantages and potential applications of hyperbolic materials, we investigate how the interplay of both shape and material anisotropy influences the optical response and mode coupling characteristics of hyperbolic optical nanoresonators. Specifically, our goal is to elucidate how the various multipoles present in hyperbolic nanoantennas combine to yield the observed optical spectra as the nanoparticle NP shape evolves from prolate to oblate. Such understanding is fundamental to deterministic design of efficient optical devices based on hyperbolic materials. In the following we utilize a number numerical and theoretical techniques such as $T$-matrix, finite-difference time-domain (FDTD) and quasistatic approximation to underpin the material origin and shape-dependence of the optical resonances of hyperbolic nanoparticles (HNPs).

2. Methods

The presented study makes use of a type-II hyperbolic material, however, the results and conclusions are independent of the exact way the permittivity is obtained – whether through an artificial metal-dielectric structure [2,20] or a natural hyperbolic material [23–25]. To illustrate the spectral properties of hyperbolic nanoellipsoids we arbitrarily construct the permittivity tensor $\boldsymbol {\epsilon }$ as a silver/silica multilayer, schematically depicted in Figure 1(a). We take the optical axis of the hyperbolic material, which in this case is the dielectric one $\epsilon _{zz}\equiv \epsilon _{\parallel }$, to be parallel to the unique axis of the prolate/oblate NPs. The metallic components $\epsilon _{xx}$ and $\epsilon _{yy}$ are perpendicular ($\epsilon _{xx}=\epsilon _{yy}\equiv \epsilon _{\perp }$) to the optical axis. The tensor elements [31] are

 

Fig. 1. Scheme of studied system and used material models. (a) The optical axis of the hyperbolic uniaxial material is parallel to the shorter/longer axis of an oblate/prolate ellipsoid. (b) The real $\varepsilon '$ and imaginary $\varepsilon ''$ parts of permittivity of Ag which forms the basis of the effective permittivity of the hyperbolic material with the dielectric being $n=1.5$. (c) The effective permittivity as function of the metal fill factor in a Ag/dielectric multilayer.

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In the above we assume that the permittivity thin layers remains at its bulk value. This may not hold for nanometer sized metal particles for which quantum size effects and surface scattering play a role [32]. Hence, mesoscopic electrodynamics at metal surfaces becomes important and requires use of quantum-corrected hydrodynamics or detailed analysis of surface-response [33]. However, for the initial analysis presented in this work we omit material-related effects reported in the literature including modification of permittivity of thin layers based on their thickness [34], nonlocality [35], quantum Landau damping [36], or surface roughness [37]. This is motivated by the objective of exploring mode coupling in particles with shape and hyperbolic-material anisotropy, not necessarily on how a particular dispersion is obtained.

The use of a local, bulk permittivity means that the supporting FDTD calculations used to verify the $T$-matrix ones are expected to yield results quantitatively comparative to ones with an effective medium. We use a nonuniform mesh with a spatial discretization step of 4 Å to accurately represent the fields in 4 nm dielectric and metal layers ($f_m=0.5$) and a 4 nm mesh elsewhere with a graded transition between the meshes. The 4 nm thickness is chosen arbitrarily to ensure good matching to $T$-matrix calculations and is not based on fabrication concerns. A linearly polarized plane wave is introduced via a total-field/scattered-field approach with a frequency span from 0.5 to 4 eV and we use symmetric and/or asymmetric boundary conditions and a perfectly matched layer absorbing boundary conditions to truncate the simulation volume.

2.1 Calculation of the $T$-matrix of hyperbolic nanoparticles

The $T$-matrix method [38] is a useful tool for solving the scattering problem by expanding the incident ($\boldsymbol {a}$) and scattered ($\boldsymbol {b}$) fields in a particular basis, specifically here using the vector spherical wave function (VSWF) basis set as the definition of the multipolar fields. This multipole expansion is referred to in recent literature as an exact multipole decomposition [39], in which the $T$-matrix relates the expansion coefficients in terms of VSWFs as $\boldsymbol {b}=T\boldsymbol {a}$. Here, the $T$-matrix of an object is obtained by the null-field method with discrete sources which is an efficient method of evaluating single-particle scattering properties [40]. The benefit of the $T$-matrix approach is that it directly calculates the relevant multipolar contributions to a particle’s optical spectrum and all coupling elements between them, with the complete spectrum being a sum of these contributions under particular illumination.

The solution of the boundary problem, which ensures continuance of the incident and surface fields of a particle, is formulated via the so-called $Q$-integrals [38]. Briefly, each multipole is characterized by two numbers $(m,l)$ with $l$ being the order (1 – dipole, 2 – quadrupole, etc.) and $m$ the azimuthal mode number. The superscript of $Q$ denotes regular and radiating VSWFs typicaly used for incident and scattered/radiating fields for, respectively, 1 and 3. The $T$-matrix for each multipole pair is obtained by solving a $2\times 2$ block

The $(1,1)$ element of the $Q^3$ matrix is an integral of the form

Here, $\boldsymbol {M}$ and $\boldsymbol {N}$ are the VSWFs, $m_r$ is the relative refractive index, $k$ is the wavenumber, $\boldsymbol {\hat {n}}$ is the unit vector normal to the particle surface, $\boldsymbol {X}^e$ and $\boldsymbol {X}^h$ (as well as $\boldsymbol {Y}^e$ and $\boldsymbol {Y}^h$) are quasi-spherical wave functions (QSWFs) that correspond to the internal modes of the hyperbolic nanoparticle. To obtain other $Q^3$ matrix elements one swaps the functions/modes as follows: an exchange of $\boldsymbol {M}$ for $\boldsymbol {N}$ (and vice versa) follows from changing the first index, while changing the second index necessitates an exchange of the $\boldsymbol {X}$ and $\boldsymbol {Y}$ internal modes as discussed in [38]. Expressions for $Q^1$ are similar, but with $(\boldsymbol {M}^3,\,\boldsymbol {N}^3)$ replaced by $(\boldsymbol {M}^1,\,\boldsymbol {N}^1)$. The QSWFs are described in terms of plane waves parameterized by angles $(\alpha,\beta )$ which are determined by the wave vector in spherical coordinates [38]. In general, solutions of both internal modes and $Q$-integrals require numerical integration and matrix inversion techniques.

Among the main advantages of the $T$-matrix is the fact that it takes an especially simple form for nanoparticles with a symmetry axis. In such a case, the surface equation does not depend on $\phi$, what facilitates analytical integration of the Q-integrals over the azimuthal coordinate. Thus, the numerical calculation of Q-integrals reduces to a one-dimensional integral over $\theta$. This simplification has been widely used for isotropic particles. Below, we show that it can be exploited equally well for uniaxial nanoparticles if the particle symmetry axis and the optical axis are parallel. We use the simplified version of the Q-integrals to evaluate the optical properties of hyperbolic nanoparticles at a computational cost similar to that for isotropic nanoparticles. These computations are performed using the $T$-matrix code as implemented in the SMUTHI package [41] with a custom extension that enables calculating the optical properties of uniaxial nanoparticles with an axis of symmetry or using an analytical approach based upon the quasistatic (QS) approximation.

To facilitate analytical integration over $\phi$, we examine the dependence of VSWFs and QSWFs on $\phi$ and define their $\phi$-independent counterparts by factoring out the $\phi$-dependent part. VSWFs have a simple dependence on $\phi$, which enables us to write

Thus, we define $\phi$-independent QSWF-analogues of $\boldsymbol {\mathcal {M}}_{l,m}$ and $\boldsymbol {\mathcal {N}}_{l,m}$ as

3. Results and discussion

We begin by evaluating the spectral differences of the two extreme cases in the chosen parameter range: a prolate and an oblate ellipsoid with aspect ratios of 1/3 and 3, respectively, assuming an identical volume (equivalent to that of a sphere 50 nm in radius). We define the aspect ratio (AR), $\mathcal {A}=r_x/r_z$, as the ratio of the length of the two identical semiaxes, $r_x$ and $r_y$, to the length of the third semiaxis, $r_z$, where the latter is parallel to the optical axis of the hyperbolic material. The two HNPs are illuminated by a plane wave with three unique polarizations as sketched in the insets of the spectral plots in Figure 2. Here, we assume the permittivity plotted in Figure 1(b) (metal fill factor $f_m=0.5$), whose hyperbolic dispersion region is below ca. 3.5 eV.

 

Fig. 2. Extinction spectra of (a-c) prolate and (d-f) oblate hyperbolic ellipsoids for three unique illumination conditions. The individual multipoles are obtained using the $T$-matrix approach with using an effective permittivity. The total $T$-matrix extinction is a sum of the individual multipoles and quantitatively agrees with total extinction calculated using FDTD for an explicit Ag/dielectric multilayer. The right column shows the corresponding electric fields calculated using FDTD at selected resonances. The volume of nanoparticles is equal to that of a sphere with 50 nm radius. The axes for the prolate NP are 34.7 and 104 nm and for the oblate NP are 72 and 24 nm.

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In Figure 2 we plot the optical extinction cross-section spectra obtained with $T$-matrix calculations and the first four multipoles [magnetic dipole (MD), electric dipole (ED), magnetic quadrupole (MQ), electric quadrupole (EQ)]. This extinction is compared to a reference FDTD calculation with an explicitly defined metal-dielectric multilayer. Extinction of hyperbolic nanostructures plotted in Figure 2 shows a rich spectrum of electric and magnetic resonances [23,29,30]. Figure 2(a) (prolate HNP, TEM illumination) has a three-peaked spectrum in the hyperbolic range ($\lesssim 3.5$) eV. At 2.8 eV one finds a peak dominated by an ED contribution coupled out-of-phase to the MQ. Below in energy at 2.5 eV a strongly mixed and constructively coupled MD–EQ peak is present followed by a mixed ED–MQ coupled in-phase. This spectrum is qualitatively similar to that of a hyperbolic nanosphere of the same volume (50 nm radius, see [30] for details) in that both have a strong ED resonance to the blue of the coupled MD–EQ peak (although both appear at different energies due to a different shape). An additional difference is the appearance of a low-energy ED peak.

When the prolate HNP is illuminated normally to the optical axis, these three peaks are selectively excited. For TE illumination the spectra in Figure 2(b) show only the two ED–weak-MQ resonances, although the phase of the MQ contribution is reversed. The coupled MD–EQ mode is, however, not excited. The situation is reversed for TM illumination, cf. Figure 2(c), as of the original three resonances only the MD–EQ mode is active. The other, an admittedly stronger mode, is an almost pure ED that coincides with the large permittivity of the $\epsilon _{zz}$ component and is a lossy dielectric-type resonance.

In Figure 2(d-f) we plot the corresponding spectra for the oblate hyperbolic nanoellipsoid. Due to the markedly changed shape, it is expected that the resonances shift in energy toward the red. The dominant resonance with the electric field polarized along the longer semiaxis (TEM and TE in, respectively, Figure 2(d and e)) is the ED with a weakly coupled MQ. This resonance is absent for the TM case (Figure 2(f)), however, a weak mode is seen beyond the hyperbolic range at ca. 3.8 eV. In the TE case a higher energy EQ is also seen. The other important mode is the coupled MD–EQ resonance at ca. 0.9 eV, although its amplitude is small due to the oblate shape.

An interesting observation in the multipolar contributions to extinction are both positive and negative contributions depending on the order and illumination. Such is the case at ca. 2.5 eV for the MD–EQ resonance and at 2.8 eV for the ED–MQ peak. Negative multipolar contributions were discussed previously in silicon nanodisks [15] or hyperbolic nanospheroids [30]. Their specific properties stem from the symmetry of the particle in question and illumination conditions [30,42] and we shall return to discuss them in subsection 3.2.

In addition to the extinction cross sections, in Figure 2 we also present the induced field enhancement at the resonances. These show highly localized fields at the resonant energies with the largest amplitude enhancement reaching up to a factor in the 10–20-fold range. Such large enhancements are typical for plasmonic resonances of metallic nanoparticles. However, here such amplification of the electric field is also observed for the magnetic modes. It is the strongest for prolate particles with an amplitude exceeding 10-fold enhancement, demonstrating strong interaction of the magnetic mode with light, what is typically not seen in dielectric particles.

3.1 Shape-dependence of the coupled ED–MQ and MD–EQ modes

While the origin of the unusual magnetic dipolar mode in small hyperbolic nanoparticles has been investigated for spherical particles [30], it is our intent to derive the dependence of this mode on the shape. In this analysis we limit ourselves to the quasistatic regime [43] for particles small compared to the wavelength. The key element of this approach is based on the expansion of the internal modes into plane waves following Kiselev [44] (as discussed in the Methods section) to solve the $Q$-integrals as written in Eq. (3). In a general case this problem requires numerical integration and matrix inversion, however, the complexity of this solution may be eased by limiting the number of plane waves used to describe the modes of our anisotropic particles.

First of all, we note that the analysed hyperbolic nanoellipsoid belongs to the $D_{\infty h}$ point group, which is characterized by continuous rotation symmetry and a horizontal plane of reflection [42]. Hence, the $Q$-integral over the azimuthal angle can be performed analytically. Second, the polar integration can be replaced by a sum over two angles while still retaining sufficient accuracy [30]. Employing the Gauss-Legendre quadrature integration scheme we retain reasonable accuracy by choosing the polar angles as $\beta _{1}$ = $\frac {\pi }{4.75}$, $\beta _{2}$ = $\pi - \frac {\pi }{4.75}$ and corresponding weights ${w}_{1}$ = ${w}_{2}$ = $\frac {\pi }{2}$. A final simplification makes use of the horizontal plane of symmetry and only a single term has to be evaluated

When the spheroid’s modes are approximated further [45] by employing the Taylor expansion of the Bessel function, the integration over the spheroid’s surface can be computed analytically. The surface element of the spheroid, $dS$, is

In the small particle limit the electric dipole can be solved for independently of other multipoles [46]. The $T$-matrix components necessary for a description of the ED are close analogues of the QS Mie approximation. The exact QS solution ($T$) for the ED of a hyperbolic nanoparticle (assuming in vacuum) depends on the particle’s aspect ratio $\mathcal {A}$

The above results for the $T^{1,1}_{1,1,1,1}$ ED together with those for ARs in the range $\mathcal {A}\in \langle 0.33,3\rangle$ are plotted in Figure 3(a), where we compare the accuracy of our approximation for the extinction cross section (red circles) and the shape-dependent pole of the denominator (red diamonds) against the full $T$-matrix calculation (red dot) and Rayleigh approximation (black line). These results confirm the very good accuracy of the QS approach and support the plasmonic, material-dependent character of the ED resonance for the hyperbolic nanoellipsoid under appropriate excitation. In contrast, the $T^{1,1}_{0,1,0,1}$ does not have a significant resonance due to $\epsilon _{\parallel }>0$ in the hyperbolic region.

 

Fig. 3. (a) Shape-dependence of resonant energy of the magnetic and electric dipole of hyperbolic nanoellipsoids. The ED resonance is the typical LSPR, the magnetic one is the coupled MD-EQ resonance. The exact $T$-matrix results calculated for a sub-20 nm nanoellipsoid match the quasistatic results. The black line marks the absorption maximum of the Rayleigh polarizability for a spheroidal particle with an isotropic permittivity equal to $\epsilon _\perp$. (b) ED and MD resonance scaling vs. particle aspect ratio. Left $y$-axis: comparison of the QS-estimated scaling parameter (circles) of the ED resonance and the Rayleigh polarizability free term in the denominator (solid line) assuming vacuum as the surrounding material. Right $y$-axis: calculated ratio of the QS-derived $-\epsilon _{xx}/\epsilon _{zz}$ (diamonds) MD resonance condition and fitted quadratic (dotted line) trend line.

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A corresponding solution for the MD mode is more involved. This complexity stems from the fact that the materials employed here are nonmagnetic and do not support a QS magnetic response. Indeed, for $\mu =1$ a large dielectric particle supporting a geometrical resonance [13] or a complex metal structure such as a split ring resonator or parallel wire pair is required [47]. To overcome the above-mentioned limitation of a simple QS approach, one needs to treat the coupled MD–EQ peak, as dictated by the simultaneous (coupled) contribution of both multipoles to the resonance in Figure 2(a,c) at 2.5 eV. Indeed, the quasistatic limit of the $T$-matrix approach is a viable in yielding general understanding [45].

Expanding the $Q$-integrals of the coupled MD–EQ mode yields the following general expression for the magnetic dipole in the QS approximation

The evolution of the MD with the aspect ratio, including the above equation, is plotted in Figure 3(a) for the full $T$-matrix calculation (blue dots), the full QS approximation (blue circles) and the material condition in Eq. (12). The agreement between the three approaches is very good, confirming the validity of our simple approach to elucidate the origin and shape-evolution of the coupled MD-EQ in hyperbolic nanoellipsoids. Specifically, the ratio of the ordinary and extraordinary permittivities need to be of opposite signs and the resonance shifts to the red for an increasing ratio $\epsilon _\perp /\epsilon _\parallel$ as the HNP changes from prolate to oblate.

Interestingly, the red shift of the the MD mode with the changing aspect ratio is much larger than for the ED. For the ED this shift is given by the typical expressions for the principal dipolar polarizabilities $\alpha _i$ for the longitudinal and transverse electric fields being proportional [46] to

The corresponding term which is responsible for the MD resonance shape-sensitivity is different in that it is a ratio between the dielectric and metallic permittivities of the hyperbolic tensor. The obtained values of $g^{0,0}(\mathcal {A})$ are plotted as diamonds in Figure 3(b). Interestingly, they show a quadratic dependence (dotted line) on the aspect ratio, indicating a stronger response to the shape change when compared to the ED. Specifically, the red shift for the coupled MD–EQ resonance for a shape change from a prolate ($\mathcal {A}=0.33$) to an oblate ($\mathcal {A}=3$) particle is 1.8 eV, while for the ED only 0.7 eV.

3.2 Probing mode coupling and evolution with aspect ratio

Figure 4 presents the total absorption and scattering cross sections of the studied hyperbolic ellipsoids accompanied by a decomposition into electric and magnetic dipole and quadrupole contributions under TEM, TE, and TM illumination. The numbers below the descriptions of the particular plotted quantity refer to the maximum value in that plot, i.e. the larger the multiplicative factor, the stronger the spectrum. From the plots one can denote the two, mostly distinct, groups of modes – the MD–EQ and the ED–MQ – and their red shift with the aspect ratio. The presence of the two coupling “groups” originates from the symmetry of the particles which can be obtained by utilizing a group theory based approach [49]. These hyperbolic nanoellipsoids belong to the $D_{\infty h}$ point group [42], which means that coupling between electric and magnetic multipoles happens only if one order is odd and the other is even, while coupling for the same type requires skipping every other order [30].

 

Fig. 4. Absorption and scattering cross sections of hyperbolic ellipsoids with volume equivalent to a 50 nm sphere for different plane wave illumination. Left two columns: TEM illumination with the wave vector $k$ parallel to the optical axis. Middle two columns: TE illumination with both $k$ and electric field perpendicular to optical axis. Right two columns: TM with $k$ and magnetic field perpendicular to optical axis.

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The MD–EQ modes for the TEM and TM polarizations are clearly coupled, as both follow the exact same dependence vs. aspect ratio. Moreover, the sign of the coupling for TEM and TM is due to the symmetries of excitation and coupling [30]. The TE case is different, since in this case the illumination conditions dictate that the particle behaves like a plasmonic one with a strong EQ (and ED, see below).

The ED–MQ excitation is qualitatively similar for TEM and TM illuminations. A strong ED, which red shifts from ca. 3 to ca. 2 eV, is accompanied by a coupled MQ whose sign switches when passing the aspect ratio of 1 (sphere) for a given polarization. Such coupling behavior is typical for nonspherical particles even with isotropic permittivity due to symmetry breaking of the system (restriction to the $D_{\infty h}$ point group). However, this coupling is qualitatively different from that of the material-dependent MD–EQ mode, in which the coupling sign is constant regardless of the aspect ratio.

In addition to the main ED–MQ mode with its AR-dependent coupling sign, one can clearly see another, much narrower, coupled ED–MQ mode to the red. In the TEM case the MQ contribution to absorption is positive, while for TE illumination it is negative. In both cases this additional mode maintains the sign of its MQ contribution in the whole AR range. Based on the similarity to the MD–EQ mode, we hypothesise that this is also a material-dependent mode which appears in hyperbolic nanoparticles.

The origin of the negative contribution to absorption of certain multipoles (TM for EQ; TEM and TE for MQ) is consistent with that of extinction shown in Figure 2. At the mathematical level it stems from the solutions of the $T$-matrix of the particles and the projection of a given incident plane wave onto these multipoles [30,38]. Qualitatively, this negative contribution to both extinction and/or absorption can be understood in terms of internal coupling between the multipoles of the particle and the phase of incident light when projected onto the multipole. In any isolated system total extinction/absorption is positive (barring gain). This applies equally to a single particle with many multipoles or to an array of coupled particles. In the presence of coupling between multipoles in a single particle, some discrete modes may receive more energy than directly provided to them by the external source. This underpins the basis of using optical antennas to enhance physical processes. If the coupling rate is large enough and out-of-phase with the incident field, the mode in question may exhibit negative extinction/absorption. In essence, it will effectively return more energy to the electromagnetic field than receive directly from the source [50]. This explains negative extinction in an element, however, absorption in an individual particle is still positive, even for a coupled one. Simultaneously, all scattering multipolar components are positive [38]. Conversely, extinction does not obey this condition, since extinction of individual multipoles is expressed by expansion coefficients of both incident and scattered fields, implying that it may be larger or smaller than scattering of the same multipole, or even negative [38]. Consequently, absorption for a particular multipole does not have to be always positive even in the absence of gain. Fundamentally, negative contributions of individual dipoles is enabled by the out-of-phase coupling between interacting multipoles [30].

3.3 Impact of metal fill factor on the optical multipoles of hyperbolic nanoellipsoids

The shape-dependent anisotropic response of a HNP can be tuned by other various parameters, such as the type of metal (its plasma frequency), the permittivity of the dielectric layers, the volume ratio of the two materials. As an example, we vary the metal fill factor to tune the plasma frequency of the metallic part of the hyperbolic dispersion (and simultaneously also $\varepsilon _\perp$). This can be tailored at fabrication by choosing appropriate layer thickness, however, an alternative approach is to use anisotropic and/or hyperbolic vdW materials and modify their properties after fabrication by electrostatic gating [51] or strain [52].

Figure 5 plots the spectral evolution of the four first multipoles (absolute values of extinction) for a variable metal fill factor for a range of ellipsoids from an aspect ratio of 0.33 (prolate, gray) to 3 (oblate, red). The red shift of all the modes with the shape change is clear (following the colors from gray to red), in line with the results in Figure 4. Simultaneously, for each AR two types of behavior can be identified in the hyperbolic dispersion region when analyzing the metal fill factor dependence. The plasmonic modes (like for isotropic metal nanoparticles) exhibit a continuous blue shift with an increasing $f_m$. This occurs for TEM polarization for the ED (and the coupled MQ) and for the TE polarization for the ED and the EQ. This results from an increase of the effective plasma frequency of the particle with an increasing $f_m$.

 

Fig. 5. Evolution of the extinction cross section and its constituent multipoles versus the metal fill factor and aspect ratio (different colors). Absolute values of total extinction (ext), ED and MQ (for TEM and TE) are plotted for AR of 0.33, 1, and 3 for clarity; MD, EQ, and MQ for TM are plotted for the whole listed AR range. The particle volume is equivalent to a 50 nm radius sphere. The black lines denote the zeroing of the $\epsilon _\perp$ and $\epsilon _\parallel$, as illustrated by the inset.

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In contrast, the modes originating from the hyperbolic dispersion – the coupled MD–EQ show a crescent-like dependence, that is also subject to a considerable red-shift when going from a prolate to an oblate ellipsoid. By changing $f_m$ these material modes can be tuned, but the furthest shift toward the blue is obtained around $f_m\approx 0.5$. A QS analysis based on Eq. (11) yields a dependence of the magnetic dipole resonance that is symmetric with respect to $f_m=0.5$ ($\omega ^\mathrm {MD}_{res}\propto f_m(1-f_m)$) [30]. However, it is clear from Figure 5 (beyond the QS approximation) that the crescent-like traces of the MD resonance are not perfectly symmetric. This is seen for AR of 0.33 for which the furthest blue shift is for $f_m\approx 0.6$. For larger aspect ratios this maximum blue shift tends towards $f_m=0.5$, although the amplitude of the modes is larger for $f_m>0.5$. Finally, a similar behavior is observed for the hyperbolic MQ modes (crescent-shaped) for both TEM and TM polarization, demonstrating the qualitatively different behavior of the plasmonic and hyperbolic modes of the HNP.

Qualitative understanding of the crescent-shapes lies in that the metal fill factor determines both $\varepsilon _\parallel$ and $\varepsilon _\perp$. According to Eq. (12) the MD resonance is set by the ratio of $\varepsilon _\parallel$ and $\varepsilon _\perp$. Since for small $f_m$ the in-plane metallic permittivity changes more quickly than the out-of-plane dielectric one, the MD exhibits a blue shift. Once $f_m>0.5$, $\varepsilon _\perp$ grows more quickly than $\varepsilon _\parallel$ causing a red shift as the in-plane component needs to become more negative to fulfill Eq. (12) for a quickly growing $\varepsilon _\perp$. It is interesting to note here, that if it would be possible to increase the plasma frequency of $\varepsilon _\parallel$ independently of $\varepsilon _\perp$, then for a fixed value of the out-of-plane permittivity the MD resonance would also shift to the blue. Hence, HNPs make it possible to tailor the absolute and/or relative positions of the ED and MD modes by material and shape changes which enable independent red and blue shifts of these modes.

4. Conclusions

In summary, we have analysed the material- and shape-dependent optical modes of hyperbolic spheroidal nano-resonators. Due to having both metallic and dielectric principal components of its permittivity tensor, a HNP presents both a typical plasmonic-like response as well as a quasistatic strong magnetic resonance. The former is dominated by an electric dipole, namely a material-dependent LSPR. The quasistatic magnetic resonance is unique to the hyperbolic nanoparticle, being absent in both purely dielectric and metallic nanoellipsoids. This resonance is inherently formed by a coupled magnetic dipole and an electric quadrupole and the necessary condition for its existence is a different sign of the diagonal permittivity tensor elements. Hence, such a resonance does not exist in a spheroidal metallic or dielectric resonator. Another significant difference, which sets this mode apart from those of pure dielectric or magnetic resonators, is how the phase of the coupling depends on the aspect ratio. In an isotropic dielectric or metallic particle, whose shape deviates from spherical, the dipolar multipoles of one type couple to the quadrupole of the other type. For constant illumination the phase of this coupling changes sign when the shape changes from prolate to oblate, what is clearly evident in absorption and extinction spectra as shape-dependent properties. This is also the case for the LSPR/ED of a HNP. However, for the unique MD–EQ mode of hyperbolic nanoellipsoids this phase is constant under shape changes, what is a significant distinction from the properties of isotropic particles. The MD–EQ mode of HNP is not the only such material-dependent one. Indeed, for particles significantly deviating from spherical similar modes of hyperbolic origin exist as coupled ED–MQ solutions.

These MD–EQ modes, which are unique to hyperbolic antennas, can be subsequently modified by the shape factor. By changing from a prolate to an oblate ellipsoid the mode exhibits a very significant shift to the red. This shift is 2.6-times larger (1.8 eV) than of the metallic LSPR (0.7 eV) under the same shape change, showing much greater sensitivity to the resonance conditions. As such, it might be possible to leverage the material-dependent properties to work with or against the shape-dependent ones to tailor the optical response to a very large degree. This could be possible in actively or at-fabrication tunable hyperbolic resonators, where the in-plane conductivity would determine the base spectral separation of the ED (only blue shift with increasing conductivity or fill factor) and MD modes (variable shift depending on conductivity or fill factor), while the shape would tune both modes to the red. Active tunability can, for example, be done in vdW materials whose electronic properties can be modified by external gating or strain [51,52]. Of course, for many natural hyperbolic vdW materials the shape-dependence may be the only recourse to tune their optical response. Alternatively, the hyperbolic dispersion may be defined during fabrication. Deposition of thin, 10 nm material layers can be accomplished with physical vapor deposition and wetting layers [34] and this thickness could be reduced by employing atomic layer deposition [53], creating a multilayer which can be etched to create hyperbolic nanostructures [54]. Even better quality few-nanometer layers are expected to be created by stacking appropriately selected vdW materials [55] or through the use of chemical vapor deposition [56].

In summary, we expect that the understanding of the modal structure of hyperbolic nanoparticles will be relevant in designing future optical devices and will further understanding of already presented examples found in literature. Here, the different character of the modes should prove useful in being able to independently tune the electric and magnetic resonances by shape, material content or even electrostatic gating of vdW materials, as well as provide potential means to introduce two types of anisotropy (shape and material) into the optical system to augment or suppress certain properties.