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Abstract
Strong optical responses of two-dimensional (2D) semiconductors like transition metal dichalcogenides (TMDs) draw substantial attention for prospective applications in optoelectronics and photonics. Here, we propose a potentially attractive application avenue via embedding patterns of 2D semiconductors (shaped, e.g., as strips or disks) in planar optical microcavities to engineer photonic modes in the dissipation-free spectral range below the optical gap. While the cavity confines electromagnetic fields to its interior, the high in-plane polarizability of 2D materials causes the appearance of the cavity modes that are bound to the patterned pieces in the lateral directions along the cavity. A TMD strip would then act to guide such bound cavity photons, while a pair of neighboring strips could operate similar to coupled photonic waveguides. Our calculations relying on experimentally measured TMD optical suspectibilities, explicitly demonstrate this type of behavior accompanied by photonic binding energies on the order of 10 meV and micron-scale spatial extents. They indicate that patterned 2D semiconductor structures employed within microcavities could represent a new material platform to enable various functionalities of integrated photonics.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Atomically thin two-dimensional (2D) semiconductors have recently emerged as a new class of materials drawing considerable attention for various optoelectronic and photonic applications, such as light-harvesting and photodetection, light-emitting diodes, lasers and nanophotonic devices (see reviews in [1–7] and numerous citations therein). In particular, monolayer transition metal dichalcogenides (TMDs): MoS2, MoSe2, WS2 and WSe2, have been extensively studied experimentally; they exhibit a direct optical gap in the visible and near-IR range accompanied by overall strong optical responses. Widely discussed, for instance, are excitonic resonance features around their optical gap, with exciton binding energies on the order of several hundred meV due to the 2D confinement and reduced dielectric screening [3,8]. Large optical response at higher frequencies has also been associated with the phenomenon of band nesting [9, 10]. Moreover, it has been demonstrated that the optical properties of 2D semiconductors can be controlled on ultrashort timescales via photoinduced changes [11–13]. This suggests that the layered semiconducting materials could be gainfully integrated in (ultrafast) photonic devices.
As an example of an integrated photonic structure, it has been recently discussed [14] how TMD sheets embedded into planar dielectric waveguides can alter electromagnetic modes supported by the system thanks to the high in-plane polarizability of 2D layers. In the vicinity of the excitonic resonance, that alteration concerns both the dispersion of proper waveguide modes as well as the appearance of mixed exciton-polariton states confined to the layers and coupled through the waveguide medium. Exciton-polaritons, a signature of the strong light-matter interaction [5, 15], have in fact been reported [16–20] in experiments with TMDs embedded in optical microcavities, another type of photonic structures. A theoretical analysis of such exciton-polariton modes in a Fabry-Pérot microcavity has been performed in [21]. The polariton formation has also been observed for TMD monolayer excitons interacting with photonic crystal modes [22], with Bloch surface waves near the surface of a dielectric mirror [23] and in Tamm-plasmon photonic microstructures [24, 25].
As the spectral range above the optical gap ordinarily features substantial dissipation in TMDs [26, 27], it may, however, limit polariton propagation lengths. On the other hand, the dissipation can become practically negligible sufficiently below the optical gap potentially rendering 2D sheet bound polaritons in that spectral range nearly intrinsic [28] or long propagating [29]. Importantly, the experimental data [26, 27] clearly shows that the in-plane polarizability of TMDs retains high values below the optical gap, where no actual excitonic resonances take place but the polarizability, by the Kramers-Kronig relation, is contributed to by all electronic transitions [15, 30].
Here, we point out that the high polarizability of TMDs in the dissipation-free range could be exploited for judicious manipulation of electromagnetic modes in appropriately tuned planar optical microcavities. This is achieved via integration into these cavities of patterned 2D semiconductors (e.g., in the shape of strips or disks) and their structures. The cavity itself confines electromagnetic fields to its interior [31, 32], hence “cavity photons”. The high polarizability of 2D materials then leads to the appearance of the new modes that are spatially bound to the patterned pieces in the lateral direction(s) along the cavity. A TMD strip can thus act to guide such bound modes within the cavity, while a pair of neighboring strips could act in a similar way to photonic waveguide couplers [31]. Our calculations in this paper will illustrate the formation of bound cavity modes around such structures as well as around 2D semiconductor disks and rings. One can envision much more involved, including periodic, patterned 2D semiconductor structures employed within planar microcavities as a platform to enable various functionalities of integrated (on-chip) photonics [31, 33, 34].
It should be noted that lateral modulation has been of growing interest to manipulate exciton-polaritons in planar microcavities with traditional (such as GaAs- and CdTe-based) semiconductor quantum wells (see [35, 36] and references therein). Various techniques to experimentally implement such modulated structures are generally demanding and the achieved energy modulation can frequently be less than or comparable with 1 meV [36, 37], although several-meV-deep potential traps have been reported with mesa-patterning of cavity thickness [38,39]. In this sense, it is remarkable that binding energies on the order of 10 meV that we illustrate in this paper can be achieved just with patterns of monolayer TMDs. Combined with the versatility of the layer deposition techniques [40–42], patterned 2D semiconductors thus appear as an attractive material alternative to fabricate integrated photonic structures.
Figure 1 schematically illustrates a planar-mirror cavity of thickness d filled with a dielectric of refractive index n and the effect on the lower-frequency photonic cavity modes that a whole uniform sheet of a 2D material would have in such a structure (see also [21]). As a result of the confinement across the cavity, the main bare cavity (no 2D layer) modes start from the cutoff frequency [31] , where c is the speed of light in vacuum. The cavity is tuned so that lies sufficiently below the lowest exciton resonance frequency ω0 of the 2D layer. When a uniform layer is positioned in the central plane of the cavity, the cutoff frequency experiences a red shift to ωc and the dispersion branches for different light polarizations undergo splitting. For small red shifts, the calculation yields
where χ is the in-plane 2D optical susceptibility of the layer (assumed real-valued in that spectral range). From the experimental data reported in the form of the the in-plane optical dielectric function ε║(ω) for the given sheet thickness ds, the 2D optical susceptibility is evaluated as χ(ω) = (ε║(ω) − 1) ds. Figure 2 displays a compilation of the behavior of the real part Reχ(ω) from the experimental data [26, 27] for a set of TMD layers. It is clear from the data that χ ~ 10 nm would give a good representation in illustrative calculations for the spectral range below the respective optical gaps. As an example, with d = 270 nm, n = 1.5 and χ = 10 nm, the bare cutoff energy would undergo, by Eq. (1), a red shift of about 25 meV, comparable with Rabi polariton splitting magnitudes of ~ 20–50 meV reported for some TMDs in optical microcavities [16, 17], albeit not as high as ~ 100 meV observed for the WS2 A-exciton transition in the metallic Fabry-Pérot cavity [18]. If one were to draw some quantum-mechanical analogies, a red shift would result from a laterally uniform attractive potential. The patterned pieces of 2D materials would then correspond to effective potential wells that may lead to laterally bound states in their vicinity. This is indeed what our calculations will demonstrate along with resulting binding energies and spatial profiles of lowest-frequency bound states.
Fig. 1 Schematically, the dispersion ω(k) of the cavity modes as a function of the in-plane wavenumber k : Black line is for the bare cavity, red and blue lines for the cavity with a uniform polarizable 2D layer in the central plane (as shown in the inset). The layer causes a splitting of the dispersion branches for the s-polarized (or TE) and p-polarized (TM) waves. The dashed black lines indicate the dispersion ω = ck/n of the bulk-medium light line (refractive index n) and the position of the ideal (in this illustration) single dispersionless excitonic resonance at frequency ω0.
Fig. 2 The frequency dependence of the real part of the 2D optical susceptibility χ(ω) compiled for a set of TMDs from the experimental data. From [26]: monolayer MoS2 (red line), WSe2 (cyan), WS2 (green). From [27]: monolayer MoS2 (blue line), trilayer MoS2 (magenta).
2. Theoretical framework
The main results for this paper are obtained via numerical solutions to the wave equation:
for the electric field E in the cavity subject to the appropriate boundary conditions [43]. All time-dependent quantities correspondingly vary with time t as exp(−iωt). The 2D semiconductor patterns will be positioned in the z = 0 plane (see the inset in Fig. 1). Their in-plane polarization is caused by the tangential components of the field: Et = (Ex, Ey), on which we will particularly focus.We restrict our attention to the ideal cavity, where the tangential field components vanish at the cavity walls [21, 43]:
The other conditions to be met are due to the polarizable interface at z = 0. With only the in-plane polarization, the tangential field is continuous across this interface: Et (z = +0) = Et (z = −0). It induces the surface (interface) polarization current: wherever there is the polarizable material (ε0 is the vacuum permittivity). Spatially nonuniform surface current (4) would be accompanied, via the continuity equation, by surface charge density σ = −ε0τ, where and ∇t = (∂/∂x, ∂/∂y) refers to the tangential spatial derivatives. Standard macroscopic electrodynamics [43] dictates that the surface current causes a discontinuity in the tangential magnetic field Ht, while the surface charge density a discontinuity in the normal component of the electric displacement n2Ez. These discontinuities translate into a discontinuity for the normal derivative of the tangential electric field: Note that Et in the right-hand-sides of Eqs. (4), (5) and (6) refers to the field in the z = 0 plane. From a mathematical perspective, the eigenvalue problem for Eq. (2) with boundary conditions (3) and (6) is reminiscent of problems with mixed boundary conditions [43] but is more involved. In addition, the full solution would need to be self-consistent with the spectral variation of the susceptibility χ(ω), which can be achieved, e.g., iteratively.For the spatially uniform χ (the whole 2D material sheet), the eigenvalue problem is straightforwardly analyzed analytically (similar to a discussion in [21]): the resulting dispersion of the lower-frequency branches is illustrated in Fig. 1 and the red shift in Eq. (1). The right-hand-side of Eq. (6) is a sum of two contributions, where the second term originates from the induced surface charge density. Its contribution does not vanish whenever the density is spatially nonuniform. It is this term that leads to the splitting of the branches for the s- and p-polarized modes shown in Fig. 1: TM-waves are affected by the surface charge even in that case of spatially uniform χ. For patterned structures, χ in boundary condition (6) becomes spatially nonuniform, that is, it becomes a function χ(x, y) of the position in the z = 0 plane. This function χ(x, y) equals to the material value χ at those points (x, y), where the polarizable material is present and equals to zero, wherever the 2D material is absent. Spatial non-uniformity of χ(x, y) may lead to extra surface charge effects as per Eqs. (5) and (6).
The screening effect of the cavity medium on the 2D layer polarizability can be easily noticed from Eqs. (1) and (6): χ for the suspended layer is effectively replaced by the combination χ/n2.
3. Computational results
In what follows we describe several representative examples for the cavity with parameters d = 270 nm and n = 1.5 intended to illustrate the salient features of the systems under consideration rather than to provide an exhaustive treatment of various geometries and regimes. Depending on the system parameters, the patterned structures can support a variety of bound modes; here we restrict our attention to relevant modes of the lowest energies.
Our first numerical example concerns a strip of the 2D material that we orient along the x-axis (that is, χ(y) is not zero only within the strip) in the z = 0 plane and look for plane wave solutions that vary with x as exp(ikxx). In the case of a uniform layer with propagation along the x-axis, purely TE-modes would correspond to the electric field component Ex = 0, while purely TM-modes to Ey = 0. From the explicit form of the surface-charge term in Eq. (6):
it is clear, however, that the plane waves along the strip would generally feature a mixture of the Ex and Ey components of the field. One may refer to the wave solutions of different polarizations as s-like (or p-like) when the magnitude of Ey is much larger (or much smaller) than that of Ex. An example of such relationships is shown in Fig. 3 displaying the spatial profiles of the electric field components in the z = 0 plane for waves running along the strip of width w = 1.5 µm.Fig. 3 Spatial profiles of the electric field components in the z = 0 plane for waves of different polarization: (a) s-like, (b) p-like. Red lines are used for the Ey component, blue for Ex. The waves are running along the strip of width w = 1.5 µm, whose edges are indicated by vertical dashed lines, with the wave vector kx = 2 µm−1. Solid lines display the results obtained with χ = 10 nm and dashed lines with χ = 15 nm. The electric field profiles shown here and in other plots have been smoothed over the discrete values calculated on variable spacing grids.
Figure 3 unequivocally demonstrates that the lowest energy normal modes in the cavity with a 2D strip are spatially localized (bound) in the vicinity of the strip; in the example shown the field extends laterally beyond the strip over a micron-scale distance. This spatial extent would be increasing while the binding energy decreasing for narrower strips. It should be noticed that the strip geometry relates to one-dimensional (in the y-direction) quantum-mechanical potential wells that are known to always produce at least one bound state [44]. The running bound waves are thus guided by the strip. The exemplary dispersion ω(kx) of such guided waves as a function of the wave vector kx along the strip is displayed in Fig. 4 and found “in between” the dispersion curves of the bare cavity modes and the cavity modes with the whole 2D material sheet in the z = 0 plane. As mentioned before, the actual dispersion of the waves would depend on the spectral behavior χ(ω) of the susceptibility for a specific material. In lieu of that, Fig. 4 shows results obtained for two different but constant values: χ = 10 nm and χ = 15 nm (see the spectral variations in Fig. 2), that should give one a good idea of the degree of possible modifications of the dispersion curves. At kx = 0, the binding energy of the bound modes is computed at about 16 meV for χ = 10 nm, increasing to about 27 meV for χ = 15 nm. (A slight difference between cutoff frequencies for guided waves of different polarizations is not seen on the scale of Fig. 4.)
Fig. 4 The model dispersion ω(kx) of the strip-guided waves as a function of the wave vector kx along the strip, shown by the thick colored lines: red for the s-like and blue for the p-like polarizations. Also shown are the dispersions of the uniform cavity modes: black line for the bare cavity and thin color-coordinated lines for the cavity with the whole 2D layer in the z = 0 plane. The strip width w = 1.5 µm. The solid lines shows the results for χ = 10 nm and dashed lines for χ = 15 nm. Inset: Energy splitting between symmetric and antisymmetric bound modes around a couple of parallel strips described in text (red symbols for the s-like and blue for the p-like polarizations).
Figure 5 displays the spatial profiles of the electric field components in the bound eigenmodes derived for a simple “phase matched coupled waveguides” [31] structure, composed of two parallel strips in the z = 0 plane, each of width w = 1.5 µm and separated by a distance of 1 µm. These profiles exhibit a typical [44] “double-well behavior” of roughly symmetric and antisymmetric combinations of the fields in the waves guided by individual strips. The inset in Fig. 4 shows the energy splitting Δ between symmetric and antisymmetric modes as a function of the wave vector kx for propagation along the strips, derived here with constant χ = 10 nm. In the illustrated case this splitting Δ ∼ 1 − 2 meV, it can be made larger or smaller by modification of the structural parameters. As coupled waveguides are frequently used in photonics to transfer optical power from one waveguide to another, the relevant transfer distance [31] can be estimated as L0 = πvgħ/Δ. Taking, as an example, the bare cavity group velocity vg at kx = 2 µm−1 and Δ = 1.8 meV (see Fig. 4), L0 would thus be estimated at about 39 µm. It is worth noting that phase mismatched coupled waveguides [31] structures can also be employed just by having a pair of parallel 2D material strips of different widths.
Fig. 5 Spatial profiles of the electric field components in the z = 0 plane for waves of different polarization: (a) s-like, (b) p-like. Red lines are used for the Ey component, blue for Ex. The waves with the wave vector kx = 2 µm−1 are running along a pair of strips, each of width w = 1.5 µm, separated by distance of 1 µm, strips’ edges are indicated by vertical dashed lines. The susceptibility used is χ = 10 nm. Solid lines show the results for symmetric and dashed for antisymmetric modes of the predominant field polarization.
If the case of the photonic lateral confinement by strips of the 2D material in the cavity can be compared to the situation with quantum wires [15], our next example could be related to the situation with quantum dots, where the confinement takes place in all directions. We exemplify this behavior by considering 2D semiconductor patterns in the shape of a disk and a ring. Such geometries are conveniently treated in the cylindrical coordinate system (r, ϕ, z) utilizing the radial Er and azimuthal Eϕ components of the tangential electric field Et instead of Ex and Ey. The patterns we consider are then characterized by r-dependent susceptibility χ(r) in the boundary conditions (6) for the z = 0 plane. Generally, the eigenmodes would be featuring a periodic dependence ∝ exp(imϕ) on the azimuthal angle ϕ with integer m = 0, 1, 2 … [43] and coupling between Er and Eϕ components. The lowest energy solutions, however, are azimuthally symmetric (m = 0) and do not depend on ϕ. This is the case of our interest here, which entails ϕ-independent charge density (5):
and decoupling of Er and Eϕ field components. The explicit form of boundary conditions (6) in this case: shows that only the radial component Er is affected by the radially variable charge density.Figure 6 displays spatial profiles of the electric field components in the azimuthally-symmetric bound photonic modes derived for two 2D material patterns: panel (a), where χ = 10 nm within a disk of radius 2.5 µm, and panel (b), where χ = 10 nm within a ring between 2 and 3.5 µm radii. For each of these configurations, there are two bound modes of different polarizations: one is purely radial with Eϕ = 0 and the other is purely azimuthal with Er = 0. Since the surface charge effects are numerically small for the parameters used, the modes of different polarizations here turn out to be nearly degenerate: the binding energy 15.9 meV of the radial mode in panel (a) is only slightly less than binding energy 16.2 meV of the azimuthal mode; for configuration of panel (b), the numbers are respectively 15.7 meV vs 15.9 meV. The figure clearly shows the spatial confinement of these bound modes, with the electric field extending outside the material patterns over micron-scale distances. This spatial extent and the binding energy of course depend on the radial dimensions of the patterns.
Fig. 6 Spatial profiles of the electric field components in the z = 0 plane for azimuthally-symmetric photonic states bound to 2D material patterns in the shape of: (a) disk of radius 2.5 µm, (b) ring with inner and outer radii of 2 and 3.5 µm, respectively; those boundaries are indicated by vertical dashed lines. Red lines are used for the Er component in the radial mode, blue for Eϕ in the azimuthal mode, they practically coincide in this case and are indistinguishable on the scale of the panels. The susceptibility magnitude here is χ = 10 nm.
4. Conclusions
We have provided explicit theoretical demonstrations of the spatial binding of the photonic cavity modes to the patterned pieces of 2D semiconducting materials embedded within planar optical microcavities. With the frequency tuning to the below-optical-gap spectral range, such modes are expected not to suffer from dissipation losses and thus to be long-lived. With the frequency tuning towards the excitonic resonances in TMDs, one is likely to encounter the rich physics of spatially-confined exciton-polaritons and their condensates (nonlinear effects are out of our scope here) that have been the subject of substantial interest in microcavities with conventional crystalline-semiconductor quantum wells [35, 36]. It needs to be mentioned that our calculations did not involve unverified fundamental assumptions but rather essentially relied on the framework of macroscopic Maxwell equations and experimentally measured room-temperature optical susceptibilities of TMDs. It is remarkable that sizable binding exceeding 10 meV can be achieved just with micron-scale-sized patterns of monolayered materials but we note that some enhancement might also be possible with bi- or few-layered materials (compare the experimental results on χ from [27] in Fig. 2 for monolayer and trilayer of MoS2). In this regard, one may recall potential traps shallower than 1 meV resulting from the deposition of metal masks on the top of traditional microcavities [35, 37] or meV-scale potential modulations achieved with elaborated procedures to pattern the thickness of such microcavities [35, 36, 38, 39]. In fact, an interesting comparison can be made that the effect of the 2D layer polarizability on the cutoff frequency ωc in Eq. (1) is similar to the effect of the increased cavity thickness. The fabrication of micron-scale-sized TMD patterns may be well suited for the modern layer-by-layer deposition techniques. It is known that quality TMD materials can be grown via chemical vapor deposition [40–42] as well as via atomic layer deposition [41], the latter technique also being used for the growth of oxides [45, 46] applicable for cavity materials.
The spatial binding of cavity modes to strip-like patterns of 2D materials may be conceptually compared to the spatial confinement of waveguide modes to the ordinary dielectric waveguides [31]; it then becomes conceivable that various geometric structures and functionalities of the integrated (on-chip) photonics [31, 33, 34] can find their analogues in the material platform of cavity-embedded patterned 2D semiconductors. While our examples in this paper involved patterns positioned only in the central plane of the microcavity, even more structural variability is possible with 3D arrangements of patterned pieces within the cavity interior.
Of particular emphasis should be an opportunity to exercise optical control of the cavity modes bound to 2D material patterns, as schematically illustrated in Fig. 7 for the case of a coupler structure. It has been shown that the optical properties of the 2D semiconductors can be modulated [11–13] via absorption of the light above the optical gap (or, perhaps, indirectly via energy transfer [28, 47]). The illumination of a part of the 2D material structure by a modulated external light source would thus result in the effective change of that part’s optical susceptibility from the original χ value to a modified χmod (optically created “defects”). Such modification is expected to affect the propagation and spatial localization of the bound cavity modes. For the illustrative structure of Fig. 7, the modulation of the susceptibility could be used, e.g., to control the phase mismatch and optical power transfer [31]. While detailed practical aspects of such optical control remain to be studied, it is relevant to note that pump-probe experiments [12] on WS2 material (optical gap and lowest excitonic resonance around 2 eV) revealed that intense pump excitation at 2.4 eV can result in a dramatic change of the optical response of WS2 over a spectral range of hundreds of meV below the optical gap. This change was interpreted as consistent with bandgap renormalization [15] that is an order of magnitude larger than that found in conventional quantum-well systems.
Fig. 7 Schematically, not to scale: The configuration of two parallel strips of the 2D semiconductor inside the optical microcavity that is subjected to optical modulation. A part of one strip is illuminated by an external source; within the illuminated spot the optical susceptibility of the material is modified from the original value of χ to χmod.
In summary, our theoretical analysis suggests that embedding patterned 2D semiconductors within optical microcavities can be an attractive prospect for judicious engineering of photonic cavity modes and for realization of integrated photonics functionalities on a very different material platform.
Funding
This work was supported by the Department of Energy, Office of Basic Energy Science (DOE/OBES) grant DE-SC0010697.
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