1. Introduction

Akin to their photonic counterparts in cavity quantum electrodynamics (cQED) [1], plasmonic nanocavities offering unprecedented optical field localization to deeply sub-wavelength regions (typically $10^{-5}\lambda ^3$) and large field enhancements can tailor the local electromagnetic environment of individual quantum emitters, facilitating research in quantum nanophotonics [2,3]. These properties have already enabled a variety of applications including surface-enhanced fluorescence [4–7], Raman [8,9] and circular dichroism [10–12] spectroscopy, single-molecule microscopy [13,14] and Purcell-enhanced single-photon sources [15–18], all of which benefit from the enhanced extraction of optical signals achievable in the so-called weak coupling regime. More recently however, staggering progress in cavity fabrication and single-emitter manipulation with nanoscale precision [19–21] have finally culminated in the experimental observation of strong coupling with individual organic molecules and quantum dots under ambient conditions [20,22–24]. Indeed, the extreme field confinement and enhancement afforded by, for example, bowtie [7,22,25] and nanoparticle-on-mirror (NPoM) [20,26–28] nanogaps or scanning nanoresonator tips [23,29] have been shown to compensate for the rapid Ohmic dissipation inherent to their metallic constituents, enabling the formation of plasmon-exciton polaritons (or plexcitons) that dramatically alter the spectral response of the hybrid system [2,24,30,31]. Access to the strong coupling regime in the single-emitter limit represents an especially key resource for emerging quantum technologies [3], facilitating coherent energy exchange between light and matter on a sub-picosecond scale together with the emergence of single-photon nonlinearities. As such, plasmonic nanocavities constitute compelling architectures for quintessential components like quantum logic gates [32] and reconfigurable quantum light sources [33], as well as strong-coupling-enabled sensing [34,35] and spectroscopy [23,29], benefiting from nanoscale compactness, femtosecond-scale dynamics and room-temperature viability.

To date, tailoring plasmonic nanocavity modes to excitonic resonances in molecules and two-dimensional (2D) materials has relied primarily on varying the morphology and physical dimensions of the cavity. In particular, for NPoM cavities, the gap modes depend exquisitely on the shape of the nanoparticle facets, providing an obvious mechanism of tunability. However, the facet shape also controls the efficiency with which light can be coupled into and out of such extreme nanogaps [28], and may complicate the experimental characterization of plexcitonic systems due to mode splitting effects that arise from natural asymmetric faceting [36]. Moreover, although 2D semiconductors offer excellent potential for harnessing room-temperature excitonic photoluminescence, nonlinearities and plexcitonic strong coupling [37–44], their interaction with NPoM devices is rendered inefficient due to the misalignment of their in-plane excitonic dipole moments with respect to the cavity field [39,45,46]. As a result, one may be compelled to compromise the field enhancement of the cavity by increasing the gap size [39,41,47], such that an in-plane electric field component is allowed, or to harness the interlayer excitons supported by van der Waals heterostructures [48–50].

In a recent study [25], we showed that a rational selection of the substrate can facilitate facile manipulation of the spatiospectral and polarization properties of nanogap modes. In particular, by placing a canonical bowtie nanoantenna on gold or silicon substrates, the plasmonic hotspot confined to the nanogap can be elevated from the bowtie-substrate interface to the upper surface of the device. Such substrate-engineered antenna modes provide easier hotspot access for external quantum matter, such as single molecules and quantum dots. Moreover, they possess a dominantly in-plane polarized near-field, which constitutes a decisive advantage for achieving efficient interaction with excitons and defects in 2D materials such as transition-metal dichalcogenides (TMDCs) and hexagonal boron nitride (hBN) [51–53]. This feature is especially appealing in view of their potential applications in quantum information science [54], where high-fidelity, single-qubit coherent control and multipartite entanglement, previously studied for isolated emitters [32,55–58], have yet to be properly explored. Additionally, we note that in contrast to many popular nanocavities relying on bottom-up wet-chemical synthesis and assembly, the substrate engineering strategy we propose is fully compatible with top-down lithographic nanofabrication, allowing a more deterministic control of the cavity topology and potentially enabling mass-production of plexcitonic devices for integrated quantum photonics.

In this work, we present a general exploration of substrate-engineered plasmonic antenna modes. Adopting once more the exemplary bowtie nanocavity, and defining the complex refractive index ($n+ik$) of the substrate, we vary $n$ and $k$ independently, and systematically investigate their impact on the spectral characteristics of the antenna mode, as well as on the spatial distribution and attainable field enhancements of its associated near-field hotspots. Our studies evidence that the substrate material provides additional design flexibility in tailoring the spatiospectral properties of plasmonic nanocavity modes, beyond the widely practiced synthetic control over the cavity topology or physical dimensions. As such, they bear strong relevance in promoting the versatility and performance of plasmonic nanocavities across the gamut of their emerging applications, from spatially- and temporally-resolved, near-field spectroscopy to plexcitonic quantum information processing.

2. Hotspot engineering via substrates

As shown in Fig. 1(A), we consider a gold (Au) [59] bowtie that is placed on a substrate characterized by the complex refractive index $n+ik$, and exposed to air ($n_\mathrm {air}=1$). Unless otherwise stated, we fix the geometrical parameters of the bowtie: height $h=30$ nm, gap distance $d=2$ nm, width $w=100$ nm, apex angle $\alpha =60^\circ$, and radius of curvature at the corners $r=5$ nm. We also neglect any dispersive character of the substrate to better isolate the physical consequences of the $n$ and $k$ magnitudes (i.e., $n$ and $k$ are assumed to be independent of frequency). Numerical simulations are performed using the finite-element method implemented in COMSOL Multiphysics, treating the nanobowtie as a scatterer placed on an infinite substrate [25,60]. The absorption cross-sections of the Au bowtie and the electric near-field distributions of the bowtie-on-substrate cavities are extracted to analyze the optical response. We focus particularly on the electric field enhancement $|E|=|\textbf {E}|/|\textbf {E}_0|$, where $\textbf {E}$ is the total electric field in the presence of the scatterer (i.e., the Au bowtie) and $\textbf {E}_0$ is the background electric field injected onto the substrate in the absence of the scatterer. Our numerical simulations show that the bowtie-on-substrate cavity can be engineered to support an antenna mode with controllable spatiospectral hotspot characteristics through the choice of substrate material. Indeed, as exemplified in Fig. 1(B), two distinct schemes can be devised to construct such an antenna mode by using either refractive or absorptive substrates. Here, the refractive substrate corresponds to a lossless material with large $n$ (i.e., $k=0$), while the absorptive one is a substrate with purely large $k$ (i.e., $n=1$). As can be seen in Fig. 1(C), both refractive and absorptive substrates give rise to a plasmonic, antenna-mode resonance featuring a hotspot that is both tightly confined to the bowtie gap and elevated to the upper surface of the device. Such hotspots allow easier access for externally introduced quantum matter such as molecular and quantum-dot emitters as well as biochemical analytes, and are especially appealing for coupling sub-wavelength optical fields with excitons and luminescent defects in 2D materials such as TMDCs, graphene and hBN. Our proposed substrate-mediated cavity engineering provides a simple approach to tailor plasmonic nanogap hotspots for enhancing light-matter interaction at the nanoscale, with potential benefits across a myriad of contemporary research directives in nanophotonics, extending from single-molecule spectroscopy to optical quantum information processing technologies.

 

Fig. 1. Antenna modes of the bowtie-on-substrate cavity. (A) Schematic of a gold bowtie placed on an arbitrary substrate characterized by the complex refractive index $n+ik$. (B) The absorption spectra arising from the antenna modes constructed in two distinct schemes, using highly refractive and highly absorptive substrates. (C) The electric near-field profiles of the bowtie in the two representative cases, showing the elevation of the antenna-mode hotspot to the top of the bowtie nanogap.

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In the following, we demonstrate how to construct such antenna modes via appropriate selection of the substrate material, by separately analyzing the effects of the substrate $n$ and $k$ on the modes of the corresponding bowtie-on-substrate nanocavity. Notably, our reported conclusions regarding substrate-dependent effects in this system remain robust with respect to variations in the bowtie geometry, which modify only the spectral mode characteristics without affecting the disposition of the hotspots in space.

3. Effects of the real refractive index n

We first study the effects of $n$ and fix $k=0$, corresponding to a purely dielectric substrate without any absorption. A range of $0.01 \leq n \leq 5$ is chosen to encompass the most commonly used dielectric materials in photonic applications. Note that a finite but small value of $n=0.01$ is used to represent the case of a near-zero-index substrate, thus avoiding convergence issues during the numerical simulations.

3.1 Spectral response and intracavity hotspot location

Figure 2(A) presents the evolution in the absorption spectrum of the Au bowtie as $n$ is varied, where more and more spectral peaks can be observed as the latter is increased (see also Fig. S1A in Supplementary Material for additional detail). To elucidate the corresponding near-field properties of these spectral features, we show in Fig. 2(B) the respective field enhancements $|E|$ at the “Top” and “Bottom” (denoted in the inset) of the bowtie nanogap as functions of wavelength. By comparing the field enhancements $|E|_\mathrm {top}$ at location Top and $|E|_\mathrm {bot}$ at location Bottom, we can quickly discern whether the resonance of interest has an exposed hotspot ($|E|_\mathrm {top}>|E|_\mathrm {bot}$) or not, and then analyze its radiative character to confirm antenna-mode functionality. It is worth noting that the field enhancement $|E|$ is taken at the center of the bowtie gap, rather than in the vicinity of the nanoprism apexes at which the maximum field enhancement should be located. In other words, the field enhancement spectra shown here serve to provide a qualitative insight into the nature of the plasmonic modes, while a comprehensive understanding should of course rely on an analysis of the 3D near-field profiles for each case (see later in Fig. 3).

 

Fig. 2. Spatiospectral hotspot manipulation via the real part of the refractive index $n$ of the substrate (fixing $k=0$). (A) Absorption spectra of the Au bowtie as the substrate $n$ is increased from 0.01 to 5. Green symbols designate the different types of mode discussed in the main text. (B) Field enhancement spectra $|E|$ at the respective locations of the bowtie nanogap. Inset: “Top” and “Bottom” refer to the upper and lower extremities at the center of the bowtie gap.

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Fig. 3. Plasmonic mode analysis for substrates with different $n$ (fixing $k=0$). (A) Scalar electric field enhancement $|E|$ across the bowtie surface (color maps) and vectorial electric field distributions ${\bf E}$ inside the bowtie (arrows) for four representative modes, with $n$ values indicated in bold. The arrow direction indicates that of the electric field, while the arrow length is proportional to its magnitude in a logarithmic scale. Note that the resonant wavelengths of the modes change as $n$ varies. (B) Radiative decay rates $\Gamma _\mathrm {rad}$ for a classical dipole aligned along the $x$-axis and placed at Top, normalized with respect to a dipole in free space. The spectral locations of the four modes are designated by their corresponding symbols, indicating that the antenna mode has the best out-coupling to free space. Note that the data shown in the pink box are scaled by a factor of 2. (C) Electric field intensities $|E|^2$ for far-field radiation directed upwards into air (left panel) and downwards into the substrate (right panel) for (i) the bonding dimer mode, (ii) HOM$_1$, (iii) HOM$_2$, and (iv) the antenna mode.

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For the cases of $n=0.01$ and 1, the absorption spectrum of the bowtie-on-substrate cavity presents only a single resonant peak (green circles in Fig. 2(A)), undergoing a progressive red-shift as $n$ increases (see Fig. S1A for $0.01 \leq n \leq 1$). The field enhancements $|E|$ in Fig. 2(B) also exhibit a peak at the corresponding wavelengths, with $|E|_\mathrm {top}$ decreasing and finally becoming equal to $|E|_\mathrm {bot}$ as $n$ increases to 1. The case of $n=1$ is special as it represents a Au bowtie suspended in an isotropic air environment; the symmetry between Top and Bottom locations ensures identical field enhancements $|E|_\mathrm {top}=|E|_\mathrm {bot}$. However, for $n = 0.01$, $|E|_\mathrm {top}$ dominates because of the higher-refractive-index medium towards the top of the bowtie gap where $n_\mathrm {air}=1$. Note that despite the asymmetric distribution of field enhancement between the Top and Bottom locations for $n = 0.01$, the values of $|E|_\mathrm {top}$ and $|E|_\mathrm {bot}$ are both sizeable, which contrasts with the $n\geq 4$ scenarios where $|E|_\mathrm {bot}$ becomes mostly negligible. Our results thus point to a previously unrecognized and potentially useful application of near-zero-index materials [61]. Indeed, such media are known to facilitate enhanced light-matter interaction and control of emission properties when acting as cavities or hosts for embedded quantum emitters [32,62], but may also provide some flexibility as functional substrates for hotspot nanoengineering.

For the case of $n=2$, the bowtie-on-substrate cavity develops multiple absorption features, and shows a non-trivial mode evolution with an increasing number of peaks despite only small, incremental changes in $n$ (see Fig. S1A for $1.5 \leq n \leq 2.5$). It can be seen that the mode designated with a circle undergoes a significant red-shift, and finally departs the spectral range of interest (beyond 1000 nm) for $n>2$. Meanwhile, a higher-order mode (HOM$_1$, triangle symbol in Fig. 2(A)) begins to play a role, and red-shifts as well when $n$ rises (Fig. S1A). As for the field enhancement, $|E|_\mathrm {bot}$ generally dominates over the spectral range of interest as shown in Fig. 2(B). Whilst the individual peaks are discernible for $n=2$, the spectral $|E|_\mathrm {bot}$ for $n=2.5$ (see Fig. S1B in Supplementary Material) presents peaks that are sufficiently broad and/or numerous so as to give the appearance of a broadband field enhancement, extending from 700 nm to 1000 nm. In either case, the modes designated by triangles evidence a peak $|E|$ shifted to longer wavelengths as $n$ becomes larger.

Upon increasing to $n=3$, the absorption cross-section retains a multi-peak structure, where another higher-order mode (HOM$_2$, diamond symbol in Fig. 2(A)) appears accompanied with a general red-shift of the spectral features (see also Fig. S1A for $n=3.5$). The field enhancement spectrum now presents an interesting behavior: while $|E|_\mathrm {bot}$ continues to dominate in the near-infrared region ($\sim 950$ nm), $|E|_\mathrm {top}$ dominates in the visible range ($\sim 650$ nm), and grows as $n$ increases (see Fig. S1B for $n=3.5$). This suggests that the same nanocavity design may be capable of supporting plasmonic mode coupling to matter vibrational transitions (in the infrared range) or electronic transitions (in the visible range) according to the wavelength of the exciting light, representing a significant degree of device flexibility for a hybrid cavity-emitter system. In particular, the value $n=3.5$ represents the commonly used semiconductor silicon, and we note that such a bowtie-on-silicon cavity was explored very recently [25] as a candidate for achieving enhanced light-matter interaction with low-dimensional quantum matter via its elevated hotspots.

When $n=4$ and 5, $|E|_\mathrm {top}$ becomes distinguished while $|E|_\mathrm {bot}$ is negligible, as shown in Fig. 2(B). Although the absorption cross-sections still have multiple peaks, $|E|_\mathrm {top}$ exhibits a single dominant one (denoted by green squares) and maintains its maximum at $\sim 650$ nm regardless of the substrate $n$ (see also Fig. S1B for $n=4.5$). This is precisely the antenna mode of interest in the present article, and we will further explore its spatial near-field characteristics in the following.

3.2 Near- and far-field properties

According to Fig. 2, we can identify several representative resonances at which either $|E|_\mathrm {top}$ or $|E|_\mathrm {bot}$ has a relatively large value. We have indicated these peaks with green symbols in Fig. 2(A), and will elucidate their nature by analyzing the corresponding spatial electric near-field distributions and far-field properties. Specifically, we extract the profiles of field enhancement $|E|$ for the entire Au bowtie surface in order to identify the location of the plasmonic hotspots, as well as the vectorial properties of the electric field inside the bowtie structure (see Fig. 3 and Fig. S2 in Supplementary Material) to aid in differentiating those modes with similar hotspot locations. In this way, the formation mechanism of the modes can also be better understood.

Figure 3(A)(i) exemplifies (via the case $n = 1$) the near-field properties of the mode indicated by circle symbols. Combined with Fig. S2A, we observe that the location of the hotspot depends purely on the contrast between $n$ and $n_\mathrm {air}$. The vectorial field distribution indicating the charge flow inside the Au bowtie (arrows) suggests that this mode originates purely from the coupling between adjacent prisms, in correspondence to the well-studied bonding dimer plasmon mode [25,63]. The mode thus appears as the longest wavelength, lowest-order bright mode of the Au bowtie nanoantenna, with negligible contribution from the substrate.

At shorter wavelengths, a higher-order mode HOM$_1$ (triangle symbol) appears, whose near-field characteristics are exemplified in Fig. 3(A)(ii) (for $n = 2$). Similar to the bonding dimer mode, the corresponding cavity hotspot becomes located at position Bottom since $n > n_\mathrm {air}$, and the intracavity field becomes evermore concentrated at the lower end of the gap as the disparity between $n$ and $n_\mathrm {air}$ grows (see also Fig. S2B). Distinct from the bonding dimer mode, the field vectors pertaining to this mode are directed towards the substrate in the gap region, such that the cavity field is no longer purely longitudinal in nature. The field penetration into the substrate observed at the bottom of the gap also suggests that this mode does not arise solely from the coupling between adjacent prisms, but bears a contribution from the coupling between the bowtie and substrate. As corroborated by the spectral decay rates in Fig. 3(B), HOM$_1$ displays an essentially non-radiative character with a negligible $\Gamma _\mathrm {rad}$, unlike the bonding dimer mode.

As $n$ continues rising, another higher-order mode HOM$_2$ (diamond symbol) appears within the spectral range 800 – 1000 nm, for which a representative ($n = 3$) is shown in Fig. 3(A)(iii). The vectorial near-field distributions pertaining to this mode for different $n$ values (see Fig. S2C) are manifestly more complex than those for HOM$_1$, but the two modes nevertheless bear a key common feature, namely an accumulation of electric field strength at the bottom of the gap. Beyond this however, there are notable quantitative differences. For example, for this mode, the maximum field enhancement attained across the bowtie lower surface is generally smaller ($|E|_\mathrm {bot}=189$ for HOM$_2$ versus $|E|_\mathrm {bot}=644$ for HOM$_1$ in Fig. 2(B)), and a localized field enhancement by a factor of $\sim 20$ can now be observed around the outer corners of the device. As confirmed by the radiative decay rates in Fig. 3(B), HOM$_2$ is essentially non-radiative in character, akin to HOM$_1$. Whilst not of immediate interest in this study, examination of the plasmon-induced surface charge polarities for the modes HOM$_1$ and HOM$_2$ (see Fig. S3 in the Supplementary Material) suggests that they are both anti-symmetrically coupled (i.e., bonding) in character, arising principally from the hybridization of multipolar modes supported by the individual nanoprisms.

Finally, when $n\geq 3.5$, a peak around 650 nm signifies the antenna mode (square symbol). As shown in Fig. 3(A)(iv) (for $n = 4$) and Fig. S2D (for larger $n$), the hotspot is now elevated to the location Top, in stark contrast to the other three modes. The electric field distribution in the device is now rather distinct from those in Figs. 3(A)(i-iii); around the bowtie gap, the field vectors display a circulating pattern in which they emerge vertically from the substrate into the bowtie, show a strong directional preference from one prism tip to the other at the top of the gap, and then vertically descend into the substrate once more. The existence of this mode must therefore be attributable to the coupling between the bowtie and substrate, and the implied charge carrier motion is a free migration between the two components. As seen in Fig. 3(A)(iv), the field enhancement is rather small at the Au surface ($\sim 20$), but can be up to 300 fold at the bowtie gap (see $|E|_\mathrm {top}$ for $n=5$ in Fig. 2(B)). Aside from the elevated hotspot, this mode bears several further striking features that differentiate it from the other modes, including its strongly radiative nature (see Fig. 3(B)) and the much weaker sensitivity of the resonant wavelength to variations in $n$ (see Fig. S2E). This radiative character may stem from the dominant $E_\mathrm {x}$ component that is symmetrically distributed in space, giving rise to a non-zero contribution in the far field.

Complementary to the spatial near-field distributions and calculation of $\Gamma _\mathrm {rad}$ reported here, we also discuss the far-field emission profiles of these modes in the presence of an intracavity dipole source. A brief summary of the calculation procedure can be found in Sec. S4 of the Supplementary Material. As shown in Fig. 3(C), all of the emission profiles display the same reflection symmetry characteristics as the coupled cavity-dipole system itself (namely, invariance under reflection about both the $x$- and $y$-axes). Naturally, symmetry is lost between the upward and downward directions in all cases, particularly in the presence of an inhomogeneous environment. For the bonding dimer mode, the radiation field magnitudes in the upward and downward directions are most comparable, as would be expected given the symmetric environment of the bowtie device in air. For the other modes however, the downward emission is considerably stronger than the upward one, following the refractive index contrast $n > n_\mathrm {air}$. For the radiation directed upwards, the bonding dimer mode and the higher-order modes HOM$_1$ and HOM$_2$ share qualitatively similar patterns, namely a spot-shaped emission peaked at a polar angle of $\theta = 0^{\circ }$ with some elliptical distortion either along the bowtie axis (iii) or perpendicular to it (i-ii). The emission in this direction appears somewhat more diffuse for the higher-order modes than for the bonding dimer one. In contrast, for the radiation directed downwards, the bonding dimer mode continues to display a spot-shaped profile (i), while the higher-order modes present two dominant emission lobes oriented perpendicular to the bowtie axis (ii-iii). Notably, the emission lobes for HOM$_1$ show a larger angular divergence (with peaks at $\theta \approx 30^{\circ }$) than those of HOM$_2$ (with peaks at $\theta \approx 20^{\circ }$). The antenna mode (iv) exhibits radiation patterns that are rather distinct from all other modes for both upward- and downward-directed emission. In particular, we find two dominant emission lobes oriented along the bowtie axis with peaks at $\theta \approx 35^{\circ }$ in the upward direction, and $\theta \approx 25^{\circ }$ in the downward one. Moreover, and in line with the radiative decay rates in Fig. 3(B), the antenna mode provides the strongest emission, especially in the downward direction (i.e., into the substrate).

Our findings reveal that placing a Au bowtie on a high-$n$ substrate can render the plasmonic hotspot totally exposed at the upper surface of the device and easily coupled with free space, making it readily exploitable for applications in surface-enhanced spectroscopy or nanoplasmonic cQED with single molecules, excitonic 2D materials or their quantum defects. Before concluding this subsection, we wish to highlight some real materials that could be employed as the substrates discussed above. For example, crystals like quartz and sapphire have refractive indices of $n < 2$, while inorganic nitrides like gallium nitride and silicon nitride exhibit values of $n$ in the range $2 < n < 3$, with their transparency window extending to the infrared [64,65]. The semiconductors silicon and gallium phosphide possess high refractive indices of $3 < n < 4$, and are largely lossless throughout the visible and near-infrared ranges [66]. Materials such as germanium [67], silicon-germanium alloys [68], and aluminum antimonide [69] may provide even higher index contrast with $n > 4$, though they are not strictly lossless in the visible range.

4. Effects of the imaginary refractive index k

In this section, we discuss the impact of the imaginary part $k$ of the substrate refractive index on the optical response of the bowtie-on-substrate cavity. To exclude any effect of the substrate $n$, we fix $n=1$ and tune the value of $k$ from 0 to 8. Naturally, the case of $k=0$ is a special one, corresponding to a Au bowtie suspended in air (i.e., no substrate), while the case of $k=8$ corresponds to a substrate that is highly absorbing (more lossy than the common noble metals). Similar to our earlier discussions surrounding the role of $n$, we will analyze in detail the spectral absorption cross-sections of the Au bowtie, the spectral field enhancements $|E|$ at locations Top and Bottom, as well as the spatial near-field properties of the key plasmonic modes.

As shown in Fig. 4(A), the absorption spectrum of the Au bowtie undergoes a relatively simple evolution as the substrate $k$ is tuned. Following our previous analyses for a bowtie suspended in air (i.e., $n=1$ in Figs. 2–3), the single peak at $\sim 745$ nm (gray dashed line) for $k=0$ signifies the bonding dimer mode of the bowtie, thus the field enhancement satisfies $|E|_\mathrm {top}=|E|_\mathrm {bot}$, as highlighted in the green box in Fig. 4(B). Apart from the case of $k=0$, the spectra in Fig. 4(A) all exhibit a broad continuum below $\sim 650$ nm (see also Fig. S4A in the Supplementary Material), with increasing absorption towards 400 nm that is likely attributable to the intrinsic interband transitions of Au constituting the bowtie [70]. In addition to this spectral background, as $k$ increases a single peak emerges at $\sim 720$ nm, accompanied with a slight blue-shift. Although this resonance occurs at a similar wavelength to that for $k=0$, they have distinct modal properties as we will show in the following. In terms of the field enhancement spectra shown in Fig. 4(B) and Fig. S4B, $|E|_\mathrm {top}$ dominates over the entire spectral range in all cases with $k > 0$ studied here, with a single peak in good agreement with that appearing in each absorption spectrum for $k\gtrsim 4$. Meanwhile, $|E|_\mathrm {bot}$ becomes progressively weaker as $k$ increases from zero and is finally negligible when $k\gtrsim 4$. Notably, another peak at $\sim 600$ nm becomes manifest in the $|E|$ spectra for small $k$, but is hardly seen in the absorption spectrum (being obscured by the pronounced spectral continuum).

 

Fig. 4. Spatiospectral hotspot manipulation via the imaginary part of the refractive index $k$ of the substrate (fixing $n=1$). (A) Absorption spectra of the Au bowtie as the substrate $k$ is increased from 0 to 8. The gray line indicates the wavelength of the bonding dimer plasmon mode for $k=0$. (B) Field enhancement spectra $|E|$ at the “Top” and “Bottom” locations of the bowtie nanogap. The green box highlights the special case with no substrate, which is intrinsically different from the others. (C) Electric near-field profiles for the higher-order mode at $\sim 600$ nm (star symbol) and antenna mode at $\sim 700$ nm (square symbol), with superimposed vectorial electric field distributions (arrows) inside the bowtie. The arrow direction indicates that of the electric field, while the arrow length is proportional to its magnitude in a logarithmic scale. (D) Radiative decay rates $\Gamma _\mathrm {rad}$ for a classical dipole aligned along the $x$-axis and placed at the top of the gap center, normalized with respect to a dipole in free space. The data for large $k$ (red lines) show that the antenna mode has a good out-coupling to free space.

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To gain more insight into the two peaks at $\sim 600$ nm and $\sim 720$ nm, we choose two representatives (green symbols in Fig. 4(A)) and investigate their near-field modal properties. As shown in Fig. 4(C), the shorter-wavelength mode (star symbol) presents multiple hotspots at all corners of the bowtie upper surface, with $|E|$ at the outer corners growing stronger as $k$ increases (see also Fig. S4C). The vectorial field distribution suggests a clear explanation for this disposition of hotspots in terms of the classical motion of charge carriers, which would be driven primarily towards the upper extremities of the constituent prisms (particularly the opposing upper corners), producing a corresponding accumulation of surface charge. In contrast, the longer-wavelength mode (square symbol) presents a hotspot only at Top, with some spatial extension of the field enhancement down the opposing prism surfaces. The corresponding vectorial electric field distribution bears a notable qualitative similarity to that for a highly refractive substrate in Fig. 3(D). Indeed, just as for a high-$n$ substrate, large values of $k$ appear to admit a circulating field distribution spanning the bowtie-substrate system, in which the field vectors are directed vertically into and out of the substrate, and the intracavity field at the upper end of the gap has a dominantly longitudinal (along $x$-axis) character. Such a field pattern suggests a vertical migration of carriers between the substrate and bowtie, producing surface charge accumulation at the opposing, upper corners of the prisms.

The calculated radiative decay rates for different substrate $k$ values are shown in Fig. 4(D). For relatively weakly absorbing substrates with $k=2$, the decay rate spectra are rather broad (similar to the corresponding field enhancement spectra in Fig. 4(B)), and show some discernible peak structure in the range 600 – 800 nm followed by a featureless, monotonically decreasing behaviour extending to the longest wavelengths considered. In particular, a shoulder emerges at $\sim 600$ nm, consistent with the resonance designated by a star symbol in Fig. 4(A). This feature persists for all values of $k$ considered here, although the radiative character of the mode clearly diminishes as $k$ increases and is ultimately negligible for $k = 8$. In contrast, a peak that initially appears rather poorly defined at $\sim 750$ nm (for $k=2$) grows larger and sharper as $k$ increases, while also exhibiting non-trivial spectral shifts (see also Fig. S4D). When $k\geq 4$, this peak dominates the radiative decay spectrum, and its spectral position around 710 nm matches well with the resonance indicated by a square symbol in Fig. 4(A). Furthermore, as $k$ increases, this mode attains greater wavelength stability, undergoing only a minor blue-shift. The calculated decay rates suggest that the latter mode offers a good free-space out-coupling efficiency, meriting its designation as an “antenna mode” and enabling the use of standard far-field measurement techniques in plasmonic cQED applications. On the other hand, by virtue of its shorter resonant wavelength and multiple elevated hotspots, the mode at $\sim 600$ nm (star symbol) may be regarded as a higher-order, non-radiative variant of the antenna mode. It is worth underlining that as $k$ increases, the antenna-mode peak becomes narrower (see Figs. 4(A), 4(B) and 4(D)), presumably contrary to what would happen upon rendering the nanoresonator (e.g., bowtie) material itself more lossy. Larger $k$ thus appears to result in a stronger coupling between the bowtie and substrate, which in turn sharpens the spectral response and improves the quality factor of the antenna mode.

Regarding some real-world materials that could function as the lossy substrates discussed in this subsection, it is natural to suggest the conventional plasmonic metals like Au, Ag and Cu. Note that although we fix $n=1$ here to exclude any non-trivial influence of $n$ on the hotspot location or the device optical response, it is not a necessary condition for construction of the antenna mode. In fact, the aforementioned metals have $n < 1$ throughout much of the visible region, which in conjunction with their relatively high $k$ values, will further promote the hotspot elevation effect as discussed in Section 3.

5. Discussion

Before concluding, it is prudent to give a brief, qualitative summary of the effects of the substrate on the spatiospectral mode characteristics of the bowtie nanocavity. As discussed in connection with Figs. 2–3, the role of increasing $n$ is to red-shift the plasmonic resonances of the bowtie and facilitate additional, higher-order modes. For relatively small $n$ ($n\leq 3$), the dominant modes within the spectral window studied here are the bonding dimer mode and its higher-order variants, whose hotspot locations mainly depend on the environmental index contrast at locations Top and Bottom. In particular, epsilon-near-zero materials may serve as substrates that mediate an elevated hotspot. For relatively large $n$ ($n\geq 4$), the dominant mode within the same spectral window becomes the strongly radiative antenna one, which features a hotspot elevated to Top and a dominantly in-plane polarized near field. This mode originates from the coupling between bowtie and substrate, where a circulating field pattern in the gap region suggests the vertical migration of carriers within the coupled system. In the intermediate range ($3<n<4$), multiple types of mode can be observed in the same spectral window.

In contrast, the evolution of the spatiospectral response with rising $k$ is dominated by the emergence of a single antenna mode with the same elevated hotspot and in-plane field polarization characteristics. Indeed, although distinct from $n$ in its physical meaning, large values of $k$ (i.e., more lossy substrates) appear to achieve a qualitatively similar effect to that of high $n$, where values of $k \geq 4$ admit an electric field circulation spanning the bowtie-substrate system, enabling the vertical migration of carriers in the vicinity of the bowtie gap and facilitating a hotspot positioned at location Top. It is perhaps counter-intuitive that a lossy substrate can offer some practical advantages. In plasmonics, increasing the amount of dissipation is the price we pay for the benefits of sub-wavelength field confinement and enhancement. Here, we take a step further: introducing even more dissipation into the system (via the substrate, and beyond the material constitution of the bowtie itself) opens a coupling channel between the bowtie and substrate, resulting in the beneficial effects of hotspot elevation and in-plane field polarization that are desirable for coupling with external quantum matter.

An interesting and complementary element to the lossy substrate is one with gain (see also Fig. S5). As a very simple but nonetheless insightful approximation, we treat this case by means of a fixed, negative value of $k$, namely $k=-4$, and neglect any dynamic gain depletion in the system. We find that the resonant wavelength and the hotspot location remain largely unchanged, compared to the case of $k=4$. However, the gain in the substrate supplies a pumping to the antenna mode [71], resulting in a greatly enhanced local field at the hotspot. These features are of interest for applications in nanolasing and non-Hermitian systems with PT-symmetry.

6. Conclusions

In conclusion, we have presented a systematic exploration of substrate engineering as a means for spatiospectral manipulation of plasmonic nanocavity antenna modes. Adopting the canonical bowtie nanocavity as an example, we have shown that both the real and imaginary parts of the substrate refractive index provide significant flexibility in tailoring the properties of the cavity modes, including their resonance wavelengths, hotspot locations, intracavity field polarization and radiative decay rates. Our study has uncovered the particularly important result that highly refractive ($n \geq 4$) and highly absorptive ($k \geq 4$) substrates provide two complementary approaches to engineering nanocavity modes with a unique combination of features, namely an elevated and thus more exposed hotspot with a dominantly in-plane polarized near field, together with a strongly radiative character. The feasibility of creating such modes in a simple, substrate-mediated fashion without the need for intricate modification of the cavity topology itself is highly appealing, and is expected to promote the versatility and performance of plasmonic nanocavities across a broad range of their emerging applications involving active quantum matter, from spatially- and/or temporally-resolved, near-field microscopy and spectroscopy to room-temperature, plexcitonic quantum information processing. Perhaps most notable of all, substrate engineering of nanocavity antenna modes may support the development of plasmonic cQED strategies for two-dimensional, excitonic quantum matter, opening the possibility of plasmon-mediated control of excitonic photoluminescence and optical nonlinearities in TMDCs or doped graphene, as well as the near-field manipulation of luminescent quantum defects in hBN.