Guiding light with surface exciton–polaritons in atomically thin superlattices

: Two-dimensional materials give access to the ultimate physical limits of photonics with appealing properties for ultracompact optical components such as waveguides and modulators. Speciﬁcally, in monolayer semiconductors, a strong excitonic resonance leads to a sharp oscillation in permittivity from positive to even negative values. This extreme optical response enables surface exciton–polaritons to guide visible light bound to an atomically thin layer. However, such ultrathin waveguides support a transverse electric (TE) mode with low conﬁnement and a transverse magnetic (TM) mode with short propagation. Here, we propose that realistic semiconductor–insulator–semiconductor superlattices comprising monolayer WS 2 and hexagonal boron nitride (hBN) can improve the properties of both TE and TM modes. Compared to a single monolayer, a heterostructure with a 1-nm hBN spacer separating two monolayers enhances the conﬁne-ment of the TE mode from 1.2 to around 0.5 μ m, while the out-of-plane extension of the TM mode increases from 25 to 50 nm. We propose two simple additivity rules for mode conﬁnement valid in the ultrathin ﬁlm approximation for heterostructures with increasing spacer thickness. Stack-*

propagation lengths exceeding 100 m. 26Although the near-zero thickness of the monolayer can support waveguide modes, they are loosely confined to the TMD monolayer and require a symmetric refractive index medium.One possibility to increase confinement is patterning the monolayer into a photonic crystal, which has been demonstrated for suspended structures. 27For unpatterned monolayers, however, the proximity of the guided mode to the light line complicates experimental detection due to the requirement for a perfectly symmetric optical environment with low scattering. 28Furthermore, detection relies critically on achieving narrow excitonic linewidths, which can require cryogenic temperatures. 29re, we address the fundamental challenge of guiding light bound to atomically thin semiconductors.We propose van der Waals superlattices based on semiconductor-insulatorsemiconductor heterostructures to improve the propagation characteristics of surface exciton-polaritons (Figure 1a).We show the existence of both TE and TM guided modes and compare their dispersion relations in monolayers, heterostructures, and superlattices made of monolayer WS 2 and hexagonal boron nitride.Compared to negligible confinement in a monolayer, we demonstrate increased confinement of the TE mode in heterostructures.Then, we clarify the impact of the thickness of the spacer layer on the guided modes.In the ultrathin film approximation, we find that the decay constants of the TE and TM modes supported by heterostructures follow simple additivity rules for their constituent layers.Additionally, we investigate the electrostatic tuning of the modes.To guide experimental realizations under different excitation conditions, we investigate the differences between two approaches for solving the modes of the superlattices using either a complex in-plane wave vector,  or a complex frequency, .Our study thus produces specific directions to tailor and tune guided modes in semiconductor monolayer superlattices as a platform for nanoscale photonic and optoelectronic devices.

Strong exciton oscillator strength and permittivity
We use WS 2 monolayers due to their strong exciton oscillator strength and narrow linewidth, which are better than in other semiconductors at room temperature and result in a record absorption coefficient.
To retrieve the permittivity of a realistic, high-quality monolayer, we deposit a mechanically exfoliated WS 2 monolayer on polydimethylsiloxane (PDMS) on a glass substrate.Using PDMS as a substrate facilitates a narrow and strong exciton peak while preserving the quantum efficiency of the monolayer emission. 30Transmittance spectroscopy shows a strong excitonic resonance with approximately 17% transmittance contrast and a narrow exciton linewidth,  A = 22.7 meV (Supplementary Section S1).We fit the measured transmission spectrum using transfer-matrix analysis and model the in-plane permittivity of monolayer WS 2 with 4 Lorentzian oscillators 16,31,32 as where is the dielectric constant in the absence of excitons, the index i represents the excitonic resonances.The spectrum features peaks associated with the A and B exciton ground states, as well as the first excited state (n=2) of the A exciton.The peak of the C exciton at higher energies is also included in the fitting to reproduce the overall shape of the spectrum., , and, are the peak energy, oscillator strength, and linewidth of each exciton.depending on the permittivity in the yellow and green areas, respectively.c, Electric and magnetic field profiles for the modes guided by a monolayer (black) and a heterostructure with a spacer thickness of 1 nm (red) at energies of 2 and 2.0223 eV.The field is confined in the out-of-plane direction, while the wave propagates in the plane.Heterostructures contribute to increased confinement of the TE mode and reduced confinement of the TM mode.
The permittivity oscillation around the exciton energy in Figure 1b is so pronounced that the real part of the permittivity, Re(), goes from positive to negative across the excitonic resonance.
Effectively, the material behaves optically like a high-refractive-index dielectric when Re() >0 or a reflective metal when Re() < 0. These permittivities facilitate two regimes for guiding SEP waves: a TE mode can be supported in the range of positive and high real permittivity (above 612.5 nm, orange area), while a TM mode can be sustained where the condition Re(())+ Re( medium ) < 0 is met (from 606.5 to 612.5 nm, green area).

Surface exciton-polaritons in monolayers and heterostructures
We consider a semiconductor monolayer as a thin film of thickness t with permittivity ε m clad between two homogenous media with refractive indices n 1 and n 2 .Such a layered medium can support TE and TM modes.To support a guided mode in monolayer WS 2 , however, the environment refractive index must be nearly symmetric with n 1 ~ n 2 .Otherwise, a cut-off appears in the minimum required TMD thickness (Supplementary Section S2).To study the mode propagation characteristics, we base our calculations on the transfer-matrix method (see Methods). 33,34Specifically, the matrix element M 22 must be zero for a guided mode.We solve the equations numerically in the complex- plane to obtain the real in-plane wave vector β of the supported guided mode (Supplementary Section S3) and evaluate its effective width W eff = 1/Re(q) and effective SEP wavelength λ SEP = 2π/Re(β), where the momentum in the out-of-plane direction in a given medium and it is known as the decay constant.
This method is appropriate for guided waves in any layered system, including atomically thin superlattices.
We use this method first to show that a WS 2 monolayer can support SEP modes at energies close to the exciton.SEP waves propagate along the monolayer and decay evanescently in the perpendicular direction (z-axis in Figure 1a).TE and TM modes can be excited in different energy ranges depending on the sign of the monolayer permittivity.The TE mode is only supported when Re(ε m ) > 0, while the TM mode starts to appear as the sign of the permittivity changes to negative (Figure 1b, yellow and green areas).The TE mode of a monolayer is very close to the light line (Figure 2a, black), with an effective refractive index close to the surrounding medium.Close to the exciton energy = 2.017 eV, the SEP wave vector becomes higher than the light line, but this mode is still only loosely confined to the monolayer.On the other hand, the TM mode is tightly confined to the monolayer, owing to the proximity of the propagation constant to the exciton energy line.
To overcome the confinement challenges associated with monolayer WS 2 , we introduce a hexagonal boron nitride (hBN) layer with a refractive index of 2.3 between two WS 2 monolayers.This modification significantly alters the dispersion behavior of the TE and TM modes.In this heterostructure, the bending of the dispersion curve starts further away from the exciton energy compared to the monolayer and evolves more slowly with energy (Figure 2a, red).This enhanced mode confinement facilitates experimental observation because loosely bound guided waves are easily scattered by imperfections.
The mode profiles for a monolayer (Figure 1c, black lines) and a heterostructure (red) for the TE and TM modes at energies of 2 and 2.0223 eV, respectively, illustrate the confinement close to the monolayer.All modes show evanescent behavior outside the waveguide core, with the TM mode being more confined than the TE mode.Using a heterostructure with an hBN spacer thickness of 1 nm provides opportunities to customize the waveguide characteristics.When transitioning from a monolayer to a heterostructure at an energy of 2 eV, the effective width of the TE mode is compressed from 1.2 to approximately 0.5 µm (Figure 2b).For the TM mode at energies above the exciton peak, the mode confinement exhibits the opposite behavior and becomes less tightly confined, with the TMmode width increasing from 47 to 90 nm for a heterostructure at an energy of 2.023 eV.

Contribution of the spacer to confinement
In heterostructures, the confinement of the guided mode depends on the insulator spacer thickness.
Increasing the spacer thickness typically increases the propagation constant of the TE mode.On the other hand, the TM dispersion line moves towards the monolayer curve.To gain insight into these modes, we evaluate the intensity modal profile for heterostructures with varying spacer thickness (Figure 3a).The confinement of the TE mode at E = 2 eV is enhanced by an order of magnitude as the spacer thickness goes from 1 to 100 nm.The TE mode shifts to higher β as the spacer thickness increases, resulting in a more confined SEP width and a shorter SEP wavelength (Figures 3b and 3c).
This apparent confinement is, however, due to the introduction of a material with a higher refractive index than the surrounding medium, which shifts the dispersion curve away from the PDMS light line towards that of the spacer material.Similarly, for the TM mode, increasing the spacer thickness reduces the width, resulting in higher confinement and shortening of the SEP wavelength (Figures 3b and 3c).
Note that the TM-mode intensity profile at E = 2.0223 eV (Figure 3a) corresponds to an antisymmetric electric field distribution (Supplementary Section S4).Next, we compare the behavior of the guided mode in a single monolayer, a heterostructure, and the hBN spacer alone, all embedded in a symmetric dielectric environment (Figure 4).Using the ultrathin film approximation, we explicitly calculate the dependence of the decay constant of the heterostructure, q hetero , on the constituent layers for both modes.For the TE mode, the heterostructure decay constant follows a simple additivity rule of the decay constants of the individual layers, namely q hBN and q monolayer , given by q TE,hetero = q hBN + 2 q monolayer (proof in Supplementary Section S5).For the TM mode, on the other hand, the heterostructure decay constant is described by q TM,hetero = −2 ε Bg /(h ε hBN + t ε monolayer ), where ε Bg denotes the permittivity of the background medium, h and t represent the thicknesses of the hBN layer and the semiconductor monolayer, respectively, and ε hBN and ε monolayer are their corresponding permittivities (proof in Supplementary Section S5).These two additivity rules for TE and TM modes demonstrate the simple but distinct relations between the permittivities and thicknesses of the constituent layers and confinement in heterostructures.
We analyze first the behavior of the decay constant for the WS 2 monolayer and hBN layers alone.
The decay constant for the monolayer is a horizontal black line in Figures 4a and 4b, as there is no spacer.If we consider an hBN film only, it supports a TE mode with increasing confinement for increasing thickness (gray line in Figure 4a).Conversely, the TM mode is absent for hBN alone at this photon energy due to its positive refractive index (no gray line in Figure 4b).For complete heterostructures containing both WS 2 and hBN, we observe an excellent agreement between the decay constants obtained using the analytical additivity rules (red lines in Figures 4a and 4b) and the numerically simulated decay constants (dark red), particularly for small thicknesses below a few tens of nanometers.

Engineering the guided modes in superlattices
A superlattice geometry − a heterostructure stack − can further improve the mode confinement and make the SEP properties more appealing for nanophotonics.The TE mode moves away from the light line for superlattices, providing higher confinement for an increasing number of monolayers (Figures 5a and 5b).The effective TE-mode width is 1.2, 0.5, and 0.3 m for one, two, and three monolayers, respectively.The TM-mode width at E = 2.023 eV rises to 100 nm with three monolayers, suggesting reduced confinement of the TM mode within the structure.The dispersion line moves away from the exciton peak energy as we go from one to three monolayers (Figures 5a and 5b).Comparison between the decay constant obtained from numerical simulations (dark red) and the theoretically calculated decay constant (red) using the analytical additivity rules for a, TE mode at E = 2 eV, and b, TM mode at E = 2.0223eV.Both show excellent agreement for thin spacers.
To exploit the advantageous tunability of SEPs, we evaluate how the electrical control of the A exciton can allow active tuning of the guided mode.The refractive index of monolayer TMDs can be tuned using electrical gating; carrier injection can tune and broaden the in-plane permittivity around the exciton resonance. 35To incorporate tunability in our simulations, we suppress the excitonic behavior of the WS 2 monolayers by reducing the oscillator strength from 1.6 to 0.1 eV 2 , resulting in a drop of ~ 50 % of its original permittivity near the A exciton (Figure 5c, inset).With this modified permittivity, we can control and potentially modulate the guided modes (Figure 5c).The TE mode confinement is frustrated in the heterostructure after suppressing the exciton by turning it towards the light line.
Simultaneously, electrical tuning eliminates the possibility of sustaining the TM mode altogether because the permittivity is now positive in the energy range where a strong exciton produced a negative permittivity.Therefore, both modes show promise for modulation.The TM mode cannot be supported in the absence of a strong exciton.Inset: in-plane permittivity when the exciton is off.

Approaches to solve the dispersion relation: complex β and complex ω
The conditions under which SEPs can be observed in heterostructures depend on the experimental configuration.The governing equations of the guided modes are defined in the complex plane.
7][38] The complex-β and complex-ω approaches lead to different dispersion relations and describe different experimental conditions for polariton excitation. 373][44][45][46] In all our results so far, we solved the guided modes using the complex-ω approach.Here, we compare the dispersion relations obtained using the complex-β and complex-ω approaches.
For the complex-ω solutions, we observe an asymptote for large values of β for the TE and TM modes (Figure 6a¸ gray lines).Instead, for the complex-β approach at a given real ω (purple lines), the dispersion relation of the TE mode shows a back-bending limiting the maximum value of β.We remove the unphysical branches from the complex- results because physically meaningful solutions should have real and imaginary parts of the wave vector with the same sign.The TM-mode dispersion lines occur within different energy ranges with a shift between complex-β and complex-ω solutions.The reason for this shift is that when we keep ω as a real value and solve the mode equation, the obtained β values possess a significant imaginary part for the TM mode.However, enforcing a real β requires the film permittivity to be strongly negative.Negative permittivity only occurs near the exciton, which shifts the obtained real part of ω closer to the exciton peak.We also compare the spacer thickness dependence of the guided modes using both approaches.While the TE mode can propagate for any spacer thickness, we obtain a cut-off thickness for the TM mode for a 1-nm hBN spacer (Supplementary Section S6).Above this thickness, the effective total permittivity of the stack becomes positive, and no TM mode is supported.
Focusing on specific energies, we observe a similar evolution of the effective mode width for both approaches (Figure 6b).For complex , the TE mode at E = 2 eV is confined to around 0.55 m for the heterostructure (red) compared to 1.15 m for the monolayer (black), whereas the TM mode at E = 2.03 eV expands from 30 nm for the monolayer to 80 nm for the heterostructure.Additionally, the complexβ approach allows us to calculate an additional SEP property: the propagation length, = 1/(2 Im( )).For the monolayer, the TE and TM modes can propagate for approximately 30 m and 2 nm at energies of 2 and 2.03 eV, respectively (Figure 6c).For the heterostructure, the propagation length of the TE mode shortens to 5 m at the same energy and drops rapidly as the energy gets closer to the exciton due to strong absorption related to the imaginary part of the monolayer permittivity.The TM mode propagation remains extremely dampened but improves to 4 nm.a, Mode dispersion in a heterostructure obtained using the complex-β approach (purple) compared to the complex-ω approach (gray), corresponding to different experimental situations.Unphysical branches are not shown in the dispersion diagram.The heterostructure consists of two WS 2 monolayers separated by a 0.3-nm-thick hBN monolayer in a symmetric PDMS environment.b-c, Effective width and propagation length using the complex- approach for both guided modes in the same heterostructure and in a monolayer.
Finally, we demonstrate the effect of the exciton linewidth γ A , which can be controlled by lowering the temperature, 47,48 on the different modes in our heterostructures.We show that decreasing the linewidth (or increasing the oscillator strength) is particularly beneficial for complex-β solutions of both TE and TM modes.We vary the A-exciton linewidth in our 4-Lorentzian permittivity model and calculate the SEP dispersion curve for γ A = 22.7 (experimentally retrieved value at room temperature), 15, 10, and 5 meV (Figure 7).The TE mode has a more pronounced back-bending line and higher confinement using the narrowest linewidth, underscoring the need for high-quality excitons and possibly low temperatures to ease observation in experiments described by the complex-β approach. 29 the complex-ω approach, adjustments to the linewidth do not significantly affect the dispersion.
However, the propagation length exhibits changes because modifying the linewidth causes a shift in the permittivity in the complex plane, bringing it closer to the real axis.

Conclusion
We have investigated surface exciton-polaritons supported by atomically thin semiconductor-insulatorsemiconductor heterostructures and their superlattices.These guided waves rely on having strong exciton resonances with high oscillator strength and narrow linewidth, which are present in WS 2 monolayers even at room temperature.They also require a symmetric optical environment for observation.Both TE and TM modes are possible for high-quality monolayers within spectral ranges with positive and negative permittivities, respectively.Compared to the monolayer modes, the heterostructure architecture modifies the effective width, exciton-polariton wavelength, and propagation length.Increasing the insulator spacer thickness provides higher confinement for the TE and TM modes.Similarly, using heterostructures with more monolayers and ultrathin spacers can further increase the TE mode confinement while decreasing it for the TM mode.We proposed strongly controlling and modulating the guided modes by switching the monolayer excitons on and off.Finally, we have shown that the surface exciton-polariton waves can be predicted with either a complex wave vector or a complex frequency approach.These approaches provide qualitatively different mode dispersions and properties.As they describe different experimental conditions, it is critical to consider the right complex-plane approach to model a specific experiment.
The diverse tuning mechanisms of excitons in monolayer semiconductors provide a control knob for guided waves based on changes to the exciton strength, linewidth, and peak energy.For example, all-optical modulation due to lattice heating has been shown to substantially alter the reflectivity of atomically thin mirrors 19 and could be used to modulate exciton-polaritons in space and time.Based on our results and given the fast pace of developments in this area, atomically thin semiconductors hold great promise for nanoscale tunable Photonics at visible wavelengths.

Transfer-matrix method
Consider two different media separated by a planar interface.The forward and backward wave amplitudes in medium 1 are denoted by A 1 and B 1 , respectively.Similarly, A 2 and B 2 are the waves in medium 2. The interface transfer matrix connects the amplitudes of the waves in the two media through , where .
By applying the electric and magnetic boundary conditions depending on the polarization of the wave (TE or TM), we can evaluate the matrix elements M ij, , which depend on the optical properties of the layered medium.For the TE mode, we obtain For the TM mode, we have , and and β is the in-plane wave vector.
The propagation transfer matrix in a homogeneous medium is , where 0 0 accounts for the propagation phase, and d is the thickness of the layer.The complete transfer matrix is the product of the interface and propagation matrices.For example, the transfer matrix of a film waveguide based on a monolayer is M = M 3←2 P 2 M 2←1 .For a heterostructure consisting of three stacked films, the transfer matrix has the form M= M 5←4 P 4 M 4←3 P 3 M 3←2 P 2 M 2←1 .
We require A 1 = B 2 = 0 to guarantee confinement so that the field vanishes at infinity.In addition, k zi must have an imaginary component (and a vanishing real component if losses are neglected for the sake of determining the dispersion relation) to have decaying fields at the bottom and top layers.For a guided mode, the matrix element M 22 should be zero.Based on this condition, we can obtain the propagation constant of the mode, as well as the field distribution in each layer.
This method remains applicable for both the complex-ω and complex-β approaches.The complexω approach involves finding ω for each real value of β using a permittivity defined in the complex-ω plane.The obtained ω from the mode solution is used to extend the permittivity in the complex-ω plane using the 4-Lorentzian model (Supplementary Sections S1 and S3).Consequently, the permittivity for the complex-ω approach encompasses two branches, one for the TE mode and another for the TM mode, each requiring the determination of complex ω independently.On the other hand, the complex- approach relies on finding the real and imaginary parts of  with a permittivity defined at each real ω.
In medium I and III, as shown in Supplementary Figure S6, we have: where q is the out-of-plane decay constant, β is the propagation constant (or in-plane wave vector), k 0 is the free-space wavenumber, and k z is the component of the wave vector in the out-of-plane direction (perpendicular to the propagation direction).
In medium II, we have: where m is a parameter related to the decay constants.
In the infinitesimally thin monolayer approximation, the boundary conditions at the interfaces are At the upper monolayer (z = a), we obtain: We can then write all the boundary conditions as a matrix equation: For the mode to exist, the matrix determinant should be zero: The lowest-order solution, taking the positive sign, is: Using the small thickness approximation, Equation S7 is satisfied when m ~ R:  Heterostructure with two WS 2 monolayers and hBN as a spacer: Replacing Equation S6 in Equation S5, we obtain

TM mode
For medium n: We use the same approach as in the TE mode: For a mode to exist, the matrix determinant should be zero: Let us also set = , = , and (1 + ) = .
 Only monolayer (no hBN): this is equivalent to setting a=0: Taking the positive sign solution:

Figure 1 |
Figure 1 | Waveguiding in WS 2 monolayers around a permittivity oscillation due to high exciton

Figure 2 |
Figure 2 | Guided modes for a WS 2 monolayer and a WS 2 -hBN-WS 2 heterostructure.Stack with

Figure 3 |
Figure 3 | Guided mode properties in a heterostructure as a function of spacer thickness.a,

Figure 4 |
Figure 4 | Additivity rules for the decay constants of the guided modes in a heterostructure.

Figure 5 |
Figure 5 | Engineering the dispersion of guided modes in superlattices.a, Superlattices of WS 2

Figure 6 |
Figure 6 | Dispersion relations in the complex-wave-vector and complex-frequency approaches.


investigate the decay constant for a monolayer, an hBN film, and a heterostucture:  Only monolayer (no hBN): this is equivalent to setting = 0: Only hBN (no monolayer): this is equivalent to setting = 0: = ± .S7 all the boundary conditions as a matrix equation:

.
) = 0 with = 0 not possible, We take the positive sign and work out the → 0 limit: