1. Introduction

Atomically thin two-dimensional (2D) semiconductors have recently emerged as a new class of materials widely considered promising candidates for various optoelectronic and photonic applications (Refs. [1–5] and numerous citations therein). Among them, monolayer transition metal dichalcogenides (TMDs) such as MoS₂, MoSe₂, WS₂ have been extensively studied experimentally and found to exhibit a direct bandgap in the visible [6] accompanied by strong optical responses, including substantial excitonic effects [3, 7]. Owing to the 2D confinement and reduced dielectric screening, excitons in such systems possess binding energies on the order of several hundred meV. Large optical response in these materials has also been associated with the phenomenon of band nesting [8, 9]. Moreover, the optical properties of the 2D crystals can be controlled on ultrashort timescales via photoinduced changes [10, 11]. This indicates that 2D semiconductors could be gainfully integrated in ultrafast photonic devices, analogous to the integration explored for plasmonic and graphene materials [12]. The spectral response of 2D semiconductors is, however, very different from the plasmonic and graphene materials [5] leading to qualitatively distinct results.

The strong light-matter interaction exhibited by 2D semiconductors can result in the formation of mixed exciton-polariton states [5,13]. Such states have in fact been experimentally observed in TMDs embedded in optical microcavities [14, 15]. We discussed earlier [16] the intrinsic exciton-polaritons confined in the vicinity of a 2D semiconducting layer at the interface between two dielectric media. In this paper, we suggest that the integration of 2D semiconductors into dielectric waveguides, one of the staple structures of photonic devices [17], is expected to result in interesting new possibilities. We will demonstrate that high in-plane polarizability of 2D layers can substantially alter the dispersion of “proper” waveguide modes, in addition to the appearance of polaritons confined to the layers and coupled through the waveguide medium. At the same time, 2D semiconductors in these structures can absorb the light that propagates along them, unlike the traditional absorption geometry for harvesting light incident on the layers [2]. As an analogy, the absorption of waveguided light in Si nanomembranes has been discussed recently [18] as a possible route to ultrathin Si solar cells.

2. Framework description

While various geometrical and environmental arrangements can be envisioned for the integration, our basic illustrations here are restricted to the standard example [17] of a planar symmetric waveguide, where the dielectric slab of refractive index $n_2=\sqrt{{\epsilon }_2}$ is embedded in the dielectric medium of refractive index $n_1=\sqrt{{\epsilon }_1}<n_2$, as shown schematically in Fig. 1. In structure (a) of that figure, two identical 2D layers are positioned at the interfaces between the media, while in structure (b) a single layer is positioned at the center of the waveguide. Following a macroscopic electrodynamics approach to surface currents [19], we treat 2D layers as infinitesimally thin and characterized by the 2D susceptibility χ [16, 20] that determines the in-plane (parallel to the layers) polarization. As appropriate for TMDs [4], this response is isotropic for in-plane directions. (Generally, the susceptibility of the layer would be a tensor for different anisotropic in-plane and out-of-plane responses. These developments are outside of the scope of this paper.)

 

Fig. 1 (a) and (b) Schematically, two types of structures under consideration. The bare waveguide of thickness d is a dielectric of dielectric constant ε₂ embedded in the medium with constant ε₁. The 2D semiconductor layers are described by the in-plane dielectric susceptibility χ. In structure (a), two such layers are used at the interfaces. In structure (b), the layer is positioned in the middle of the waveguide. (c) The exemplary dispersion ω(k) of the bare waveguide modes in between the two bulk-media light lines: red lines for s-polarized and green lines for p-polarized waves. Their crossing with a dispersionless resonance at frequency ω₀ is depicted.

Download Full Size | PDF

Figure 1(c) exemplifies the well-known [17] dispersion curves ω(k) for several bare waveguide modes that are situated in between the corresponding bulk light lines: ω = ck/n₁ and ω = ck/n₂. We use k to denote the in-plane (≡ k_‖) wave number (also known as the propagation constant β for a guided mode [17]). This dispersion can be readily derived from the corresponding eigenvalue equation:

It is clear from the view of Fig. 1(a) that eigenvalue equation (1) remains the same for that structure but the reflection coefficients would be altered by the in-plane polarizability of the interfacial 2D layers. (It is worth noting that even in a generalized treatment including the out-of-plane polarization, the TE modes would still be affected only by the in-plane polarizability.) Using standard [19] boundary conditions with interfacial currents, they are derived [16] as

The results for the structure in Fig. 1(b) are obtained with the appropriately modified eigenvalue equation:

For our illustrations below, we will utilize response functions $\tilde{\chi }(\omega )$ that represent both a model excitonic response as well as the extraction [21, 22] from our own transmission experiments on an MoS₂ monolayer. In the vicinity of a well-separated dispersionless excitonic transition of frequency ω₀, e.g., the 2D layer susceptibility would acquire a frequently used [20,23] single-oscillator form:

3. Results

In order to provide an unobscured illustration of the underlying effects, we will first address the case of an ideal single-oscillator resonance with vanishing χ₀ and γ. In a bulk 3D medium, the dielectric function ε(ω) containing such resonance contribution would lead to the classical exciton-polariton spectrum with the polaritonic gap [13, 23]. In our system, however, the resonance takes place not in the bulk but at the interfacial boundaries. As the waveguide modes are discussed in relation to the phase shifts [17] experienced upon total internal reflection from the boundary “mirrors” (the regime of real-valued $k_{2_z}$ and imaginary $k_{1_z}$), it is clear from Eqs. (2) and (3) that real-valued $\tilde{\chi }$ can substantially affect those shifts.

We proceed now by the way of numerical examples. Figure 2 illustrates fundamental changes that the ideal excitonic resonance at $\hslash {\omega }_0=2\mathrm{e}\mathrm{V}$ in 2D layers imposes on the spectrum of the normal electromagnetic modes in waveguiding glass-like (n₂ = 1.5) slabs in the air (n₁ = 1). Three columns of this figure correspond to three different values of increasing slab thickness d exemplifying the cases where the bare waveguides would respectively support only 1, 2 or 3 waveguide modes of each polarization in the vicinity of the resonance frequency. Comparing these cases, one can clearly see the trends that are exhibited by the dispersion curves in progression from single-mode to multi-mode waveguides. Rows (a) and (b) of Fig. 2 display results numerically obtained from Eq. (1) for the structure of Fig. 1(a); results in rows (c) and (d) were derived from Eq. (5) for the structure of Fig. 1(b).

 

Fig. 2 Dispersion of the intrinsic eigen modes as a function of the dimensionless in-plane wave number ck/ω for different thicknesses of the glass (index n₂ = 1.5) waveguide in air (n₁ = 1): d = 250 nm (first column), d = 500 nm (second column), and d = 700 nm (third column). Rows (a) and (b) show the behavior for the model configuration of Fig. 1(a), rows (c) and (d) for the configuration of Fig. 1(b). Rows (a) and (c) marked with S are for the s-polarized; rows (b) and (d) marked with P are for the p-polarized modes. Solid red and blue lines display, respectively, the behavior for “+” and “−” modes, Eqs. (1) and (5), in the waveguide with 2D semiconductors, which is compared to the behavior of the modes in the bare waveguide, shown by dashed colored lines. Columns 1, 2, and 3 thereby illustrate the cases of bare wave guides sustaining correspondingly one, two or three eigen modes of each polarization in the vicinity of the resonance frequency $(\hslash {\omega }_0=2\mathrm{e}\mathrm{V})$. The black dashed lines show the position of the ideal excitonic resonance at ω = ω₀, and of the light line in glass, ck/ω = 1.5. See text for more detail.

Download Full Size | PDF

For our illustrative calculations here, we used the strength of resonance parameter A of such magnitude that ${\hslash }^{2}A{\omega }_0/c=0.01\mathrm{e}{\mathrm{V}}^2$. With this strength, the appreciable changes in the dispersion of the waveguide modes take place within few meV from the resonance, and this is the region displayed in Fig. 2. It should be noted that no direct experimental access to the intrinsic resonance parameters in TMD materials is available now as the experimental measurements of excitons are modified by extrinsic effects. This also concerns the environmental screening effects on the resonance parameters for 2D layers on substrates (the most common experimental arrangement) or embedded in media. It is however reasonable to assume that glass-like media would result only in relatively moderate screening effects. We refer the reader to the review [24] of the current state of various microscopic calculations of exciton properties in 2D materials, including environmental screening. Our representative choice of effective resonance strength A here is based on the order-of-magnitude estimates [16, 25] of the exciton dipole transition moments in TMDs and should be considered as such. Importantly, however, in our earlier work [16] we showed that this resonance strength also leads to the intrinsic polariton radiative linewidths ~ 1 meV, which appears to agree well with experimental findings [15, 26] for TMDs on typical substrates. We also note Rabi polariton splitting magnitudes of ~20 – 50 meV reported for TMDs in optical microcavities [14,15].

Figure 2 shows the intricate evolution of the proper waveguide modes “between” the bare waveguide dispersion curves as well as the appearance of the modes where they were not present in the absence of the resonant layers. This, in particular, includes the appearance of the modes beyond the bulk slab material light line, that is, for k > n₂ω/c, where both $k_{1_z}$ and $k_{2_z}$ are purely imaginary. Such states correspond to polaritons “bound” to the 2D layers. In the structure of Fig. 1(a) there are always two eigen modes of this nature describing even and odd combinations of single-layer polaritons coupled through the slab. (A useful analogy for the system under consideration can be drawn from one-dimensional quantum mechanics. The waveguide slab would then correspond to the potential well while the 2D layers to delta-function potentials: either wells or barriers depending on the sign of $\tilde{\chi }$.) The coupling is of course stronger for smaller d resulting in bigger polariton splitting in thinner slabs. In the structure of Fig. 1(b) (see rows (c) and (d) of Fig. 2), on the other hand, there would be only one mode of this nature. It is noteworthy that these modes evolve continuously from proper waveguide modes, with the dispersion curves actually crossing the light line, as indicated by points A and B in the example of Fig. 2(a1). At such points with $k_{2_z}=0$, the electric field exhibits linear rather than sinusoidal spatial variations across the slab; at point A, in particular, the in-plane electric field would be just constant.

With regard to structure of Fig. 1(b), one immediately notices that, even in the presence of the 2D layer, the reflection and transmission coefficients satisfy the following relationships: $r_{22}^{(s)}-t_{22}^{(s)}=-1$ and $r_{22}^{(p)}+t_{22}^{(p)}=1$. This signifies that “−” TE and “+” TM modes would be precisely the same as their bare waveguide counterparts, which is transparent in results of rows (c) and (d) in Fig. 2. This, of course, is a reflection of the fact that the in-plane electric field in modes of those symmetries has nodes in the middle of the waveguide, hence no in-plane layer polarization would be taking place.

As our current model considers only the in-plane layer polarization, it is understandable that its effects on TE modes are generally expected to be more substantial than on TM modes, as is indeed seen in comparison of panels in Fig. 2. The difference of the effects of the polarizable layer on TE and TM modes is however also qualitative. In particular, one should notice that, while subsequent bare waveguide TE and TM modes appear at the same cutoff frequencies (see Fig. 1(c)), the cutoff frequencies for the s- and p-polarizations are clearly different in the vicinity of the resonance in Fig. 2.

It should be stressed now that the presence of dissipation in the resonance, finite γ in Eq. (6), can significantly affect the described picture, just as the dissipation is known to affect 3D and surface polaritons [13, 19, 20]. The dissipation can result from both intrinsic (such as the interaction with phonons) and extrinsic (such as the interaction with defects) processes and is generally temperature-dependent. The dissipation makes the eigen modes decaying so that the in-plane wave vector (number) for a given frequency ω becomes complex-valued: k = k′+ ik″. In this paper we do not discuss strongly decaying modes and modes that would exist only in the presence of dissipation. It is also clear that very large values of γ would lead to a complete washing out of the resonance-induced dispersion signatures. For our computational example in Fig. 3, we utilize two values of γ that are comparable with the energy scale of the resonance-induced features. Panel (a) of Fig. 3 shows a modification of the dispersion curves ω(k′) and panel (b) a relatively weak decay, k″ ≪ k′, that may take place as a result of such moderate dissipation. For the cases illustrated in Fig. 3, the dissipation results in the disappearance of polariton modes with k′ > n₂ω/c. On the other hand, it leads to the appearance of the region of anomalous dispersion and negative group velocity [27] in the vicinity of the resonance frequency. This is similar to the observed effect of damping on the surface plasmon dispersion [19]. Given the relatively weak decay (that can be tuned, e.g., via the waveguide thickness) at the particularly attractive visible spectral region wavelengths, this anomalous region might be of interest for studies in the context of negative-refraction applications [5,27].

 

Fig. 3 The effect of dissipation γ on the dispersion of the s-polarized “+” mode in the waveguide of Fig. 1(a) with thickness d = 200 nm. The red lines show the dispersion in the absence of dissipation: the dashed line for the bare waveguide, the solid lines with the 2D layers. Panel (a) refers to the resulting real part k′ of the in-plane wave number k, panel (b) to its imaginary part k″. The blue lines are for ħ γ = 2 meV, green for ħγ = 4 meV. The black dashed lines show the position of the resonance and of the light line in glass.

Download Full Size | PDF

As the quest for better-quality samples of various 2D semiconductors is currently underway, it is not clear yet how strong and narrow excitonic transitions would become practically achievable. In this regard, it is important to note that sizable effects should be possible to observe already with the available 2D materials that exhibit strong optical responses. We illustrate this in Fig. 4 for the same glass-in-the-air waveguide of Fig. 1(a) by employing the complex-valued 2D response function $\tilde{\chi }(\omega )$ extracted from our own room-temperature transmission measurements of MoS₂ monolayers on sapphire substrates over a broad range of frequencies ω. The real and imaginary parts of the response function are shown in the inset to Fig. 4(d). We discussed and used this function previously to rationalize experimental results [21,22] on energy transfer into such monolayers. The examples of waveguide eigen mode calculations with this function are shown in Fig. 4, both for their dispersion and linear absorption coefficient. As expected, the dispersion of TE modes, panel (a), is affected more strongly, clearly showing the effects from well-known lower-frequency excitons as well as from the higher-frequency absorption feature of MoS₂. We point out that the absorption of light propagating along the 2D semiconductor layers could be exploited in 3D structures for light harvesting applications [2]. An interesting similarity can be found in the recent work to utilize waveguided modes to increase the overall light absorption by ultrathin Si layers [18]. In the illustration of Fig. 4, the resulting absorption lengths are ~ 1 − 10 μm that can be made smaller or larger by waveguide modifications. The tunable absorption lengths may, for instance, be useful to control parasitic losses to nonlinear recombination processes in 2D semiconductor layers [22,28].

 

Fig. 4 Dispersion (panels a,c) and linear absorption coefficient (b,d) for eigen modes in the waveguide of Fig. 1(a) with thickness d = 250 nm that employs monolayer MoS₂ for the 2D semiconductor layers. Red lines are for “+” and blue for “−” modes of both s- (panels a,b) and p- (c,d) polarizations, they are compared to the dispersion of the bare waveguide modes (dashed lines). The inset: the real and imaginary parts of $\tilde{\chi }(\omega )$ used for response of a MoS₂ monolayer.

Download Full Size | PDF

4. Conclusions

In summary, we provided computational illustrations of potentially strong effects and interesting opportunities that may result from integration of 2D semiconductors into dielectric waveguides, in particular due to excitonic resonances in 2D layers, such as occurring in TMD materials. It may be apt to quote here from a recent review [5]: “The field of exciton-polaritons in 2D materials is still at the nascent stage, and we expect exciting future developments”. As the waveguide geometry and environment can be judiciously chosen to modify the electric field profiles, it is conceivable that such integration may open a fertile research area (e.g., to engineer polariton mixing) that might be of interest for various photonic and optoelectronic applications. While our numerical examples in this paper were limited to the visible spectral region, it should be clear that the results have a generic nature and can also be realized in other spectral regions with appropriate changes of waveguide parameters.