Ultra-confined surface phonon polaritons in molecular layers of van der Waals dielectrics

Improvements in device density in photonic circuits can only be achieved with interconnects exploiting highly confined states of light. Recently this has brought interest to highly confined plasmon and phonon polaritons. While plasmonic structures have been extensively studied, the ultimate limits of phonon polariton squeezing, in particular enabling the confinement (the ratio between the excitation and polariton wavelengths) exceeding 102, is yet to be explored. Here, exploiting unique structure of 2D materials, we report for the first time that atomically thin van der Waals dielectrics (e.g., transition-metal dichalcogenides) on silicon carbide substrate demonstrate experimentally record-breaking propagating phonon polaritons confinement resulting in 190-times squeezed surface waves. The strongly dispersive confinement can be potentially tuned to greater than 103 near the phonon resonance of the substrate, and it scales with number of van der Waals layers. We argue that our findings are a substantial step towards infrared ultra-compact phonon polaritonic circuits and resonators, and would stimulate further investigations on nanophotonics in non-plasmonic atomically thin interface platforms.


Supplementary Note 1. Derivation of the polariton confinement factor expression.
A dispersion of TM-polarized surface wave at the three-layer interface is given by the following inexplicit relation 1  where we introduced dimensionless complex confinement factor as the ratio between polariton and free space wavevectors ( / ), and set & , ' and , -1; depicts the wavelength in free space; the minus sign before * + term originates from the choice of ≃ representing physically propagating mode. We also confirmed, that for confinement factors (|Re | 0 10) and the two-dimensional material (MoS 2 ) discussed in the main text, direct numerical solutions of Supplementary Equation (1) are almost indistinguishable from results given by simplified Supplementary Equation (3); examples of the approximated versus exact confinement plots are shown in Supplementary Fig. 1a. The figure of merit for polaritons propagation is more sensitive to the approximation procedure, however the difference in exact and approximated solutions is negligible even for several tens layers of MoS 2 ( Supplementary Fig. 1b).

Supplementary Note 2. Silicon carbide permittivity.
The dielectric function of 6H-SiC is calculated using a single harmonic oscillator model: In addition, we report highly confined SPhPs modes in deeply sub-diffractional TMD crystals on silicon carbide substrate. As an example, a tri-layer nanoribbon with a width of approximately 100 nm (which is less than /100), a length of ~1 µm (< /10) and MoS 2 thickness of 2.1 nm is shown in Supplementary Fig. 7. The structure was fabricated in the process of the layered material exfoliation. The mode inside the ribbon is characterized by deeply subwavelength confinement, and is featured by a period ~100 nm. We believe these experimental data may be of interest for potential future applications such as nanoscale resonators or waveguides.

Supplementary Note 4. Estimate for MoS 2 -SiC phonon polaritons propagation loss.
The measured fringes are fitted into a model developed in literature 3 : where fitting parameters <, @, are complex numbers and E, F , F , Greal; ? -distance from s-SNOM tip to MoS 2 edge. The first term represents tip-launched surface waves. These waves travel a total distance of 2? (from tip to edge, and back to tip) and experience a geometry decay of √? when the wavefront expands circularly. The second term represents all possible channels of launching/detection of the surface wave that travels only a single edge-tip distance ?. The ? A term in the denominator accounts for the geometrical decay of such waves and left as a fitting parameter. The third term represents a background signal, and the fourthpossible fast SPhP modes coexisting on SiC (later can be approximated by an exponential decaying function since the wavelength of the fast modes is much larger than the range of ultra-confined (slow) modes mapping). The first period of the fringe is excluded from the fitting optimization as it would overcomplicate the model (it is known that complex tip-edge coupling near an edge discontinuity cannot be modelled straightforwardly 3 without providing additional assumptions). Example of the modelling is shown in Supplementary Fig. 8; corresponding estimate for the SPhPs propagation loss figure of merit is ~13. In addition, we note that the weight of edge-launched versus tip-launched contribution may depend on particular geometry/shape of the crystal edge termination, and the tip.

Supplementary
Supplementary Figure 9. SPhPs confinement factor, |Re(β)|, calculated for MoS 2 -SiC interfaces according to the equation (1) of the article main text as a function of the twodimensional material thickness (plotted in number of van der Waals layers) and the excitation frequency.

Supplementary Note 5. Analysis of the phonon polariton wavelength based on singlefringe features near MoS 2 crystal edge.
It is known 3 that complex tip-edge coupling near an edge discontinuity may result in that the first fringe (nearest to the edge) is related to the polariton wavelength (or confinement) in more sophisticated way compared to all sequent fringes further inside the crystal (later fringes can be well modelled by Supplementary Equation (4), and provide most accurate information on ). From the other hand, as the first fringe scales in size with the laser excitation frequency, it may straightforwardly provide an estimate for (or Re ) if: imaging is spectroscopically carried out at several frequencies, and the wavelength value is priory-known accurately at least at one of these frequencies. We analyse the distance (O) between first minimum of the scattering signal inside MoS 2 crystal edge and the neighbouring maximum further inside the crystal (Supplementary Fig. 10; O represents a semi-width of the first interferometric fringe, clearly defined on the experimental data). Between frequencies of 930, 924.5 and 897 cm -1 O scales as 104, 78 and 22.1 nm correspondingly. The values of Re at 930 and 924.5 cm -1 are priory known (40 and 54, correspondingly) from the analysis of inner fringes depicted with red-colour arrows in Supplementary Fig. 10a,b. Assuming that O linearly depends on we observe a good agreement between priory-known values and re-calculated ones using scaling of the parameter delta (i.e. |Re | P N.J |Re | P Q R STU R S:V.W 53.3 ≈ 54), which demonstrates suitability of this method (in addition we confirmed that the other data, e.g. for 7-layer MoS 2 , also follow this trend). Applying the same calculation technique to the bi-layer MoS 2 data at 897 cm -1 we obtain an estimate for the confinement factor, which is |Re |~190.