Abstract
Phonon polaritons, hybrid lightmatter quasiparticles resulting from strong coupling of the electromagnetic field with the lattice vibrations of polar crystals are a promising platform for midinfrared photonics but for the moment there has been no proposal allowing for their electrical pumping. Electrical currents in fact mainly generate longitudinal optical phonons, while only transverse ones participate in the creation of phonon polaritons. We demonstrate how to exploit longcell polytypes of silicon carbide to achieve strong coupling between transverse phonon polaritons and zonefolded longitudinal optical phonons. We develop a microscopic theory predicting the existence of the resulting hybrid longitudinaltransverse excitations. We then provide an experimental observation by tuning the resonance of a nanopillar array through the folded longitudinal optical mode, obtaining a clear spectral anticrossing. The hybridisation of phonon polaritons with longitudinal phonons could represent an important step toward the development of phonon polaritonbased electrically pumped midinfrared emitters.
Introduction
Phonon polaritons are mixed lightmatter excitations arising from hybridisation between photons and transverse optical phonons in polar dielectrics. In the crystal’s Reststrahlen band, between the transverse optical (TO) and longitudinal optical (LO) phonon frequencies, the real part of the dielectric function is negative. In this spectral window these resonances result in electric fields which are strongly localised at the crystal surface, leading to the appearance of localised modes termed surface phonon polaritons (SPhPs).
Following initial studies on surfaces^{1,2} or in waveguides^{3}, SPhPs in user defined nanostructures were demonstrated^{4,5}. Such localised fields allow for energy confinement on lengthscales orders of magnitude shorter than that of the free photon wavelength^{6}, without the strong optical losses associated with plasmonic systems^{7,8}. These modes are also extremely tunable, thanks to their morphologic nature^{9}, their dependence on carrier density^{10,11} and their ability to hybridise with propagating^{12} or epsilonnearzero modes^{13}. Recent investigations have demonstrated the potential of localised phonon polaritons for sensing^{14}, nonlinear optics^{15,16,17,18}, waveguiding^{19}, nanophotonic circuitry^{20} and rewritable nanooptics^{21,22}.
The tunable, narrowband nature of SPhP resonances makes them a good candidate for realisation of integrated midinfrared emitters. To this end phonon polaritonbased thermal emitters have been demonstrated^{2,23,24}, but thermal pumping is intrinsically inefficient and does not allow for an increase in the degree of temporal coherence. In polaritonic systems based on electronic excitations, electrical injection of polaritonic modes has been demonstrated^{25,26}, but similar schemes with phonon polaritons are difficult to implement as their energies typically lie an order of magnitude below the electrical bandgap. Electrical currents do however couple efficiently to crystal lattice vibrations. In fact one of the main sources of Ohmic loss in polar dielectrics is through LO phonon emission via the Fröhlich interaction^{27}. The use of such an interaction to pump SPhPs is however problematic as the LO phonon frequency defines the upper edge of the Reststrahlen band, which limits the spectral overlap and resonant transitions between the LO phonon and SPhPs. More fundamentally the photonic field, due to its transverse nature, does not couple with longitudinal excitations. We are thus at an impasse: we can efficiently create large populations of LO phonons via electrical currents, but only TO phonons can form SPhPs and emit light in the farfield.
In this paper we theoretically predict and experimentally demonstrate an approach to strongly couple LO phonons with SPhPs, exploiting silicon carbide (SiC) polytypes whose unit cells are elongated along the caxis. This extension of the atomic lattice inserts an additional Bragg plane in the direction of the caxis, folding the phonon dispersion back to the Γ point. These zonefolded LO phonons (ZFLOs) can then become resonant with the SPhPs at optical wavelengths, as illustrated in Fig. 1a. The mechanical boundary conditions for the nuclear displacement at the crystal interface can mix the two excitations, leading to novel hybrid modes arising from the strong coupling of SPhPs and ZFLO phonons. The resulting quasiparticles, which we name LongitudinalTransverse Phonon Polaritons (LTPP) possess both a longitudinal character, which could potentially allow resonant generation by Ohmic losses^{27}, and a transverse one, making farfield emission possible^{23}.
The ZFLO modes we exploit, typically termed weak phonons, are highwavevector states accessible near the Γ point due to Bragg scattering induced by the periodicity of the crystal lattice. They manifest as a dip in planar reflectance and are usually phenomenologically described by adding oscillators to the material's transverse dielectric function^{28,29}. The negative dispersion of the LO phonon ensures that these weak phonon modes exist within the Reststrahlen band, coexisting in frequency with propagating or localised SPhPs^{17}. This is illustrated in Fig. 1a for 4HSiC, the material studied in this Letter, whose weak phonon lies at around 837.5/cm. Around 250 unique polytypes of SiC exist, each with different weak phonon frequencies, allowing the weak LO phonon to be tuned throughout the Reststrahlen region. The weak phonons of 15R and 6HSiC for example lie near 860 and 885/cm, respectively.
Longitudinaltransverse hybridisation has been also theoretically predicted in polar quantum wells and superlattices^{30,31}, and realised in plasmonic systems, where nanoscale confinement of transverse plasmonic modes makes very large wavevectors accessible, intersecting the negative dispersion of the longitudinal oscillation of the electron gas^{32,33}. This results in red shift of the modal frequency, which can become nonnegligible when an appreciable portion of the plasmonic field exists at large wavevectors. Contrastingly in the systems under investigation Bragg folding ensures that the longitudinal mode is accessible for all values of the wavevector, which as we will show leads to a strong hybridisation even in optically large resonators.
In the following we will initially develop a microscopic theory of lightmatter coupling in polar dielectric systems including spatial dispersion. This theory will be then used to theoretically investigate the reflectance of a 4HSiC surface. Finally, we will present experimental results demonstrating strong coupling, and thus the existence of LTPP, using arrays of 4HSiC nanopillars.
Results
Theory
Our starting point in order to microscopically model the hybridisation of phonon polaritons with ZFLOs is to expand the theory describing ionic motion in a polar dielectric^{34,35,36} to the retarded regime. In frequencyspace the material displacement X obeys the equation
where ϕ(A), is the electromagnetic scalar (vector) potential, the material highfrequency dielectric constant is ε_{∞}, the transverse (longitudinal) optical phonon frequency at the Γ point is ω_{T}(ω_{L}), the material density is given by ρ, the phonon damping rate by γ, the transverse (longitudinal) phonon velocities in the limit of quadratic dispersion by β_{T}(β_{L}) and the polarizability α. In this we assume that the only effect of the anisotropy is the Bragg folding along the caxis.
In section 1 of the Supplementary Note we solve Eq. (1), in conjunction with the Maxwell equations by the introduction of auxiliary scalar and vector potentials Y = ∇ ⋅ X, Θ = ∇ × X, allowing us to write the ionic displacement as a hybrid, containing both transverse and longitudinal components whose mixing will be instigated by application of the appropriate mechanical and Maxwell boundary conditions
in which ϕ_{H} is the homogeneous electric scalar potential, solution to the Laplace equation, and ε(ω, 0) is the lattice dielectric function in the absence of spatial dispersion. Including spatial dispersion it is easy to show that the full dielectric function is given by
for the transverse fields.
Application to SPhPs
In order to clearly demonstrate how our theory leads to the appearance of hybrid LTPP modes, here we apply it to the analytically solvable case of an acut uniaxial polar dielectric halfspace in vacuum, shown in the inset of Fig. 2a. We choose an acut crystal because in this system the ZFLO phonon manifests as a dip in the planar reflectance, permitting us to compare our analytical solution with experimental observation of ZFLOs previously reported in the literature^{4,28,29,37}. In section 2 of the Supplementary Note we apply the formalism outlined in the previous section to this geometry, applying the appropriate electrical and mechanical boundary conditions and calculate the Fresnel coefficient for TM polarised light incident along the caxis, including spatial dispersion, to be
where k_{zB} (k_{zT}) are the outofplane wavevector components of the transverse mode in the vacuum (dielectric) defined as
the vector k_{T} = (k_{x}, 0, k_{zT}), c is the speed of light in vacuum, v_{T} is a characteristic velocity describing the TO phonon dispersion and Ω encodes the mechanical boundary condition \(\bar \sigma \cdot {\mathbf{z}}_{z = 0} = 0\), where \(\bar \sigma\) is the stress tensor. It is important to note that Eq. (4) cannot be reduced to a simple equation where the dispersionless dielectric function ε(ω) is replaced by the spatially dispersive ε(ω, k) derivable from Eq. (1). This is because additionally to the effect of spatial dispersion calculating the reflectance now involves application of both mechanical and Maxwell boundary conditions, resulting in a mixing dependent on the wavevectors on each side of the boundary. We can apply this result to the description of the ZFLO modes observed in planar reflectance measurements of acut SiC polytypes by shifting the inplane wavevector of the longitudinal component k_{xL} inside Ω
while leaving the wavevectors of the transverse components unchanged. Here a is the length of the unit cell along the caxis and k_{x} is the inplane wavevector of the incident photons. The result for a 4HSiC substrate is shown in Fig. 2a. The anisotropy of such a polytype can be taken into account by noting that the leading terms in the numerator and denominator of Eq. (4) when neglecting the ZFLO mode (Ω = 0) are just those from the Fresnel coefficient of an isotropic halfspace. We can then replace the term for the lower halfspace with that derived for an uniaxial halfspace in the local approximation by considering the transverse wavevector to be that of the extraordinary wave in the crystal, yielding a characteristic dip in the reflectance at the ZFLO frequency 837.5/cm, consistent with previously reported experimental data^{29,37}.
This result also allows for investigation of the guided modes of the planar structure, satisfying
whose dispersion is seen in Fig. 2b, where the imaginary component of the reflectance coefficient Eq. (4) is plotted utilising standard parameters for the 4HSiC dielectric function with damping rate γ = 4/cm. The clearly visible spectral anticrossing between the dispersion of SPhPs supported on the planar interface and the comparatively dispersionless ZFLO phonon is the hallmark of strong coupling. It demonstrates that, close to resonance, the bare modes (black dashed lines in Fig. 2b) hybridise, creating two novel spectrally resolved hybrid longitudinaltransverse quasiparticle branches which we named LTPP (green solid lines).
Experimental results
In order to verify the existence of LTPP, we consider square arrays of cylindrical 4HSiC resonators on a samematerial ccut substrate^{4,12,38}. Such systems, sketched in Fig. 1b, support a variety of transverse SPhP modes, with highly tuneable frequencies dependent on the geometrical parameters^{9,12}. The monopole mode in particular, polarised outof the substrate plane, is highly sensitive to the interpillar spacing (pitch) due to the repulsion of like charges on adjacent pillars and can effectively be tuned throughout the Reststrahlen band as has been shown in previous studies^{9,38}. This mode is often referred to as longitudinal in the literature but this naming convention only refers to the electric field orientation with respect to the pillar long axis, the mode is nonetheless electromagnetically transverse, with nonvanishing curl. Apart from its technological relevance thanks to small mode volumes and narrow and tuneable resonances, this system presents a key advantage over the planar system discussed in the previous section. The SPhPs here exist within the lightline, and it is thus possible to spectroscopically probe the anticrossing without the need for complex prism coupling setups^{39}.
The underlying mechanism that gives rise to the longitudinaltransverse hybrid mode in the theoretical treatment of the polar dielectric halfspace and in the micropillar arrays we experimentally probed is the same. As such we expect to observe a spectral anticrossing in the micropillar array resonances, essentially analogous to the one shown in Fig. 2b, but entirely contained within the lightline. However, the nanostructured array complicates the theoretical analysis since there is no analytical solution to Eq. (1) for this system. The experimental practicality of the nanopillar array comes in fact at the cost of heavy numerical complications, which dramatically increase the computational power required to solve the problem. In order to tackle the problem numerically the electromagnetic fields, described by Maxwell’s equations, must be coupled through the mechanical boundary conditions to the ionic equation of motion in Eq. (1). In contrast to the the planar case considered in the previous section where solutions were Bloch modes for which the inplane wavevector was a good quantum number, the micropillar modes contain many wavevectors whose values will be affected by their coupling to the longitudinal degrees of freedom as in plasmonic nonlocality^{33}. Given the importance of mechanical boundary conditions for the longitudinaltransverse hybridisation, prohibitively expensive full 3D simulations of the coupled electromagnetic and material displacement fields are thus necessary to describe LTPP in the resonator array on substrate system.
We exploit the broad tunability of the monopole mode by fabricating samples with interpillar spacings in the range 700–2000 nm, over which the resonance is expected to tune from the high to the low energy side of the ZFLO phonon at 837.5/cm. Pillars were fabricated with a uniform height of 950 nm and diameters of 300 and 500 nm. Nanopillar arrays were fabricated from semiinsulating ccut 4HSiC substrates by reactive ion etching^{4}. We probe the planar reflectance utilising Fourier transform infrared spectroscopy.
Results are shown in Fig. 3 for pillars of nominal diameters 300 nm (a) and 500 nm (b) for the full range of interpillar spacings explored. A larger diameter results in a blue shift of the monopolar mode^{9}, while increasing the interpillar spacing causes the monopolar mode to red shift as a result of decreased coupling between resonators. When the monopolar mode approaches the 4HSiC ZFLO, illustrated by the horizontal dashed line in Fig. 3, a second branch appears in the reflectance on the low energy side of the ZFLO mode. Rather than continuing to red shift through the ZFLO for large interpillar spacings, the monopolar mode remains on it’s high energy side and the new branch red shifts. This anticrossing behaviour, previously illustrated in Fig. 1, is a hallmark of strong coupling and demonstrates that the monopolar phonon polariton mode of the pillar array is hybridised with the ZFLO phonon^{12,13}.
Further evidence for the hybrid nature of the observed LTPP resonances can be acquired from the magnitude of the recorded reflectance dips, calculated by subtracting the reflectance of the pillar array from that of the planar substrate, as shown in the lower panel of Fig. 3. In this panel the blue (red) circles correspond to the upper (lower) branches in the upper panel. For small or large interpillar spacings, where the detuning between monopolar and ZFLO modes is large we see that, as expected, the upper and lower branches have characteristics of the bare modes. The monopolar mode of the resonator array couples well to the impinging light resulting in a deep reflectance dip, while the ZFLO couples weakly. In the intermediate region instead, the modes are linear combinations of the bare monopolar and ZFLO modes, leading to the crossing for the reflectance dip which demonstrates their hybrid nature. In the lower panel of Fig. 3 the shaded regions indicate where the upper (lower) branch is more ZFLO (monopolar) in character. Particularly the equalisation of the branches reflectance is a signal of the anticrossing point, indicating that both branches are composed of equal parts monopolar and ZFLO modes. This can be seen by following the vertical lines up onto the reflectance maps, corresponding to the avoided crossing of the LTPP branches.
Additional samples were fabricated with the same nominal array parameters. Wide intersample variability ensures that these arrays have different monopolar frequencies. All samples reproduce the anticrossing at the ZFLO frequency, this data is available in section 3 of the Supplementary Note.
Discussion
Our results illustrate the hybridisation of longitudinal and transverse modes in polar dielectric structures, thus providing the first clear experimental evidence of the LTPP modes theoretically predicted by our theory. This hybridisation, mediated by the mechanical boundary conditions at the crystal surface, cannot be achieved in bulk as the longitudinal mode cannot be matched without an interface. This kind of surfaceinduced hybridisation is well understood in plasmonic systems, where spatial dispersion arises as a result of electron pressure^{33}. In plasmonic systems however these effects are only accessible where the electric field is confined on the nanoscale, meaning that resonances are comprised of sufficiently highwavevector Fourier components to experience the dispersion^{32}. In the polar dielectric systems discussed here these large wavevector Fourier components are instead accessible in optically large resonators meaning that the hybridisation is essentially accessible in any appropriately tuned polar dielectric resonator.
Fabrication of resonators whose eigenmodes are linear superpositions of transverse and longitudinal waves has important technological implications. Such modes could be directly pumped electrically through the Fröhlich interaction, providing way toward the realisation of phonon polaritonbased midinfrared emitters. Furthermore our simulations show that for hybridised LTPPs nonradiative and radiative losses are of the same order, leading to an appreciable radiative efficiency and potentially allowing for the creation of efficient electroluminescent devices operating throughout the SiC Reststrahlen band. An efficient injection scheme could also potentially lead to the development of coherent phonon polaritonbased light sources, an idea which has received some attention in recent literature^{40,41}. Further flexibility can be found by applying these results to superlattice systems in which the Brillouin folding can be finely tuned^{42} and the hybrid material dielectric function can be controlled^{43}, potentially allowing for the creation of electroluminescent devices operating across the midinfrared spectral region.
Methods
Nanofabrication of SiC nanopillar arrays
Pillar arrays were fabricated by etching into deep semiinsulating 4HSiC substrates. The pillar geometry was defined using Al/Cr hard masks deposited via electronbeam lithography, liftoff and evaporation. The exposure time of a following reactive ion etch determined the pillar height. The masked substrate was exposed for 38 min at 150 W utilising SF6 and Ar in equal partial pressures, followed by a chemical wet etch. To remove any residual fluorine, a commercial PlasmaSolv treatment was performed.
Fourier transform infrared spectroscopy
Infrared measurements were performed in the reflectance mode of a Thermo Scientific, Nicolet FTIR Continuum microscope. A 15×, 0.58 NA reverse Cassegrain objective provided illumination at angles of 10–35° offnormal, with weighted average of 25°. Spectra were taken as an average of 32 scans with 0.5 cm^{−1} resolution acquired from a 50 μm^{2} area
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
S.D.L. is a Royal Society Research Fellow. S.D.L and C.R.G. acknowledge support from the Innovation Fund of the EPSRC Programme EP/M009122/1. C.T.E. and J.G.T. acknowledge support from the Office of Naval Research. M.A.M. and C.T.E. acknowledge support from the National Research Council Research Associateship program. R.B. acknowledges the Capes Foundation for a Science Without Borders fellowship (Bolsista da Capes, Proc. No. BEX 13.298/135). S.A.M. acknowledges the DFG Cluster of Excellence Nanoinitiative Munich (NIM) and the Bavarian Solar Technologies Go Hybrid (Soltech) programme.
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S.D.L. conceived and led the project. S.D.L. and C.R.G. developed the theory, analysed the data, and wrote the paper. J.G.T., S.A.M., J.D.C. and S.D.L. conceived and designed the experiments. A.J.G., V.D.W. and J.D.C. designed and fabricated the nanostructures. R.B., M.A.M., C.T.E. and J.D.C. all performed FTIR measurements of the nanostructure arrays. All authors discussed the results and commented on the manuscript.
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Gubbin, C.R., Berte, R., Meeker, M.A. et al. Hybrid longitudinaltransverse phonon polaritons. Nat Commun 10, 1682 (2019). https://doi.org/10.1038/s41467019094144
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