Monolayer-thick two-dimensional (2D) materials have become objects of intense studies due to their potential applications in electronics and optics [1,2]. Despite their thickness in angstroms, 2D materials can support substantial electric currents, as well as absorb and emit measurable amounts of light. Among the optical properties, the ability to support a guided mode is among the most fundamental and consequential ones. Graphene, the original 2D material, is known to support a 2D surface-plasmon polariton (SPP), largely confined to the graphene sheet and having a large propagation constant and slow group velocity [3,4]. Graphene plasmonics is a thriving field where numerous potential applications are being investigated. While plasmons are transverse magnetic waves existing in media with negative permittivity, it had been shown in [5] that in the narrow interval where the permittivity of graphene turns positive, yet below the absorption edge, a weakly confined transverse electric (TE) can also exist. However, guided waves in graphene require high doping and are restricted to a frequency region below the absorption edge occurring at twice the Fermi energy. It follows that graphene photonics is practicable mostly in the range from THz to perhaps mid-IR [6], and if one is to expand this range to more interesting visible or UV ranges, a monolayer material with preferably a wide direct bandgap should be considered.

Over the past few years, a family of such materials, transition metal dichalcogenides (TMDCs), has been identified [7,8]. High-quality monolayers of TMDCs, ${\mathrm{MoS}}_2$ and ${\mathrm{WSe}}_2$ being the most prominent representatives, have been fabricated and explored, revealing strong excitonic features in the visible range [9–12]. Due to a large effective mass and a small dielectric constant of the surrounding material, peak excitonic absorption in TMDCs reaches 15%–20% per monolayer and the exciton binding energy is measured in hundreds of meV, making the excitonic feature stable at elevated temperatures. Strong exciton–photon coupling has been observed in microcavities [13]. It is then reasonable to expect that a 2D exciton can couple with a propagating photon and engender a 2D exciton–polariton—a confined quasi-particle propagating in a single atomic layer of TMDCs. In this work, we show the conditions under which a guided-mode 2D exciton–polariton in a TMDC monolayer exists and estimate its most relative characteristics, dispersion, confinement factor, and propagation length.

The TMDC exciton–polariton, shown in Fig. 1, is a TE wave of frequency $\omega$ propagating along a TMDC monolayer placed inside a dielectric cladding with index $n$, with the relative fields described as

 

Fig. 1. (a) Fields in an exciton–polariton supported by a TMDC monolayer and (b) absorption and reflection by a TMDC monolayer.

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The 2D susceptibility in Eq. (3) is directly related to the absorption by the TMDC monolayer of the harmonic wave propagating in the direction normal to it [Fig. 1(b)]. Using the boundary condition for the tangential electric and magnetic fields yields

If we now approximate the excitonic absorption by a Lorentzian shape with peak absorption at the exciton resonance $\alpha ({\omega }_0)={\alpha }_{\max }$ and FWHM linewidth $\gamma$ (Fig. 2), then the 2D susceptibility can be evaluated as

 

Fig. 2. Effective mode width, $W_{\text{eff}}$, and propagation length, $L_p$, of an exciton–polariton in ${\mathrm{WSe}}_2$, whose normal incidence absorption $\alpha$ is also shown.

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To provide an example, we rely on the results obtained in Ref. [12], where absorption by a monolayer of ${\mathrm{WSe}}_2$ placed on a quartz substrate ($n=1.41$) had been studied. Maximum excitonic absorption of ${\alpha }_{\max }=0.15$ with an FWHM linewidth of $\hslash \gamma =50\mathrm{meV}$ was measured at a resonance energy of $\hslash {\omega }_0=1.6\mathrm{eV}$(A exciton), and the binding energy was estimated to be $E_B=0.37\mathrm{eV}$. Note that our rough estimate of exciton oscillator strength for this example is correct to within 15%, even though the excitonic radius is only 8 Å, indicating that Wannier–Mott exciton model might be not quite applicable here.

Using these values in Eqs. (8) and (9), one can determine the effective width and propagation length as shown in Fig. 2. The minimum effective width occurs at a wavelength of $\lambda =788\mathrm{nm}$ and equals 2.3 μm. The propagation length, however, is rather short, 30 μm. But at a slightly longer wavelength of $\lambda =800\mathrm{nm}$, the propagation length exceeds 100 μm, whereas the effective mode size is only 3 μm. It should be noted that although the effective mode size is larger than that in a conventional “thick” waveguide, a large mode size does have advantages. First of all, the wide mode is less sensitive to small surface perturbations; second, for the wide mode, coupling in and out of an optical fiber becomes easier.

It is instructive to compare these results with conventional “3D” dielectric waveguides. The maximum surface (sheet) susceptibility can be found from Eq. (5) as ${\chi }_{r,\max }^{(s)}={\alpha }_{\max }\lambda n/4\pi \approx 13\mathrm{nm}$. For a dielectric waveguide of thickness $t_g$, the surface susceptibility is simply ${\chi }_{r,3D}^{(s)}=t_g\mathrm{\Delta }{\chi }_r$, where $\mathrm{\Delta }{\chi }_r=n_g^2-n^2$ is the “additional susceptibility” provided by the core with index $n_g$ relative to the cladding with index $n$. If we had considered a waveguide core made, say, from heavy glass with $n_g=1.85$, then the thickness required to provide a similar sheet susceptibility, i.e., the same sheet current as that in the TMDC monolayer, would have to be $t_g=13\mathrm{nm}/(n_g^2-n^2)\approx 9\mathrm{nm}$. Thus, a monolayer of ${\mathrm{WSe}}_2$ has the same confinement ability as a slab of 3D material that is nearly 30 times thicker. Additional comparison can be made with long-range SPP modes [14] in metals, which offer slightly shorter propagation distances for a comparable confinement. Interestingly, since TMDCs can be doped up to ${10}^{13}{\mathrm{cm}}^{-2}$, regular 2D SPP modes, in a way similar to graphene, should also be observable in these materials, but, due to large effective masses, only in the THz range.

Another significant observation is that a confined mode exists only if the refractive indices on both sides of the monolayer are close to each other. If one retraces the derivations above for the case where the indices on two sides differ by a small amount $\mathrm{\Delta }n$, then the confined mode can be shown to exist only for as long as

Our final observation is that if anything, in this simple model, we have underestimated the light guiding performance of the TMDC monolayers because there is an additional contribution to the susceptibility due to the split-off exciton B situated about 100 meV above the main exciton A [6,8]. Furthermore, propagation away from resonance may be even longer than that estimated here because, away from the resonance, absorption decays exponentially [15] (Urbach tail) rather than quadratically as in a Lorentzian line shape. It could be reasonable to expect some improvement of the confinement with bilayers and trilayers of TMDCs, but the improvement will not be proportional to the number of layers as the character of the material would change from a direct to an indirect bandgap [16,17], causing a decrease in the oscillator strength of the exciton; hence, a monolayer probably remains the best option.

A few words can now be said about the experimental techniques that could be employed to excite and observe 2D exciton polaritons. Since the effective index contrast between the surrounding dielectric and the guided mode is small, whereas the mode size is large, butt coupling appears to be feasible, while guiding could be detected by near-field techniques [18,19]. A classical m-line method used so successfully with dielectric waveguides is also an option [20]. There is also an intriguing possibility of “intrinsic excitation” of a polariton by simply trapping the photoluminescence from the low-energy tail of the exciton itself inside the monolayer with subsequent near-field detection.

In conclusion, we have demonstrated theoretically that a single monolayer of TMDCs is capable of supporting confined exciton–polariton modes in the visible and near-IR ranges with an effective width of just a few micrometers and a propagation length in excess of 100 μm. It is reasonable, in our view, to expect that these characteristics will be further improved if and when higher quality materials with narrower excitonic linewidths become available. Besides being scientifically notable, this impressive light guiding ability of TMDC monolayers enhances the chances that TMDCs will find their own practical niche in nanophotonics.