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Abstract
The design of photonic structures plays a crucial role in the engineering of light-matter interactions. Planar microcavities have been widely used to establish strong light-matter coupling in semiconductor quantum wells, leading to intense research on exciton-polariton systems in the past few decades. However, planar cavities are limited in material compatibility, inflexible for mode engineering, and bulky for integration. Here we demonstrate dielectric slab photonic crystals as a flexible and compact platform for polaritons, where excitons are strongly coupled to photons confined in the leaky modes of the slab. We show our structure is well-suited for van der Waals materials, features unusual adjustable dispersions, and allows for multi-wavelength operation on a single chip.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When excitons in semiconductors are strongly coupled to photons, new quasi-particles called exciton-polaritons are formed [1, 2]. In vertical Fabry Perot (FP) microcavities, polaritons have a meta-stable ground state at the zero in-plane wavenumber with a very light effective mass and robust coherence. Such microcavity polaritons allow room-temperature Bose-Einstein condensation and other many-body phenomena at high temperatures, potentially enabling polaritronic devices such as low threshold lasers and ultrafast optical switches [3–5]. However, high quality vertical FP cavities are made of distributed Bragg reflectors, which require a large numbers of wavelength-scale epitaxial-layers that are closely lattice-matched to the quantum wells containing the excitonic medium. Consequently, such polariton cavities are difficult to make and are limited to a handful of material choices. High quality polariton cavities have been made with GaAs and CdTe systems, but they require low temperature operation. In high temperature materials, polaritons have been realized with GaN [6], ZnO [7], and organics [8], although typically with relative low cavity quality and large inhomogeneous broadening of excitons, which makes it challenging to observe many-body coherent phenomena. Furthermore, the multi-layer construction of FP-cavities makes them bulky and inflexible, and it is difficult to integrate multiple components on a single chip.
Recently, monolayer transition metal dichalcogenides crystals (TMDCs) have emerged as a promising candidate for both high-temperature polariton physics and increased flexibility in device integration. They are direct-bandgap semiconductors [9, 10] with exceptionally large exciton binding energies and oscillator strengths [11], therefore supporting excitons and polaritons at room temperature [12–17].
Uniquely, these 2D crystals and heterostructures can be placed on substrates without requiring lattice-matching and are robust against surface effects of the substrate [18]. These properties suggest new possibilities to control matter-light interactions in these materials with diverse photonic structures. It has been demonstrated that photonic crystal defect cavities coupled to 2D materials can be used to enhance both absorption [19] and emission [20], as well as second harmonic generation [21–23]. Recently TMDC-polaritons have been created in slab photonic crystals (PhCs) [13]. In this paper, we provide the design principles of such PhC-polariton systems and show that they enable dispersion engineering, and multi-color integration on a single chip, therefore providing a flexible and compact platform for 2D material polaritons.
2. Guided mode resonances
We will restrict the discussion to the simplest example of one-dimensional (1D) periodic PhCs, so as to clarify the basic properties and design principles of the slab-PhC polariton system. Extension to more complicated 1D patterns and 2D PhCs is straightforward conceptually as well as fabrication-wise, but opens up a wide design space for mode-engineering. For the 1D PhC, we will consider a semi-infinite SiO2 substrate (n = 1.46) with a Si3N4 (n=2.02) PhC grown on top. As shown in Fig. 1(a), the PhC is characterized by a period Λ, thickness t, modulation fill factor ℱΛ = wg/Λ, and thickness fill factor ℱt = (1 − tg/t). All calculations are done using the RCWA method.
Fig. 1 (a) A schematic of the proposed PhC structure. (b) A reflectance spectrum of the guided mode resonance showing the characteristic asymmetric Fano lineshape (Λ = 420 nm, t = 100 nm, ℱΛ = .9, ℱt = 0) (c) The distribution of the amplitude of the electric field for the guided mode resonance in (b).
We utilize guided mode resonances in the PhC, which occur when light is incident on a weakly modulated PhC that essentially acts as a periodic waveguide [24]. The incident light can be coupled into the waveguide mode and propagates along while energy is coupled out of the waveguide due to the periodic modulation [25], resulting in the formation of sharp, polarization-dependent resonances. Assuming the index modulation can be treated as a weak perturbation, these resonances are formed when the external radiation and the waveguide mode satisfy the following phase matching condition [26]:
where ω is the angular frequency, k‖ is the in-plane wavevector, kx,y are the wavenumbers of the external radiation in the x̂/ŷ directions, m is an integer, and βj is the effective propagation constant of the jth slab waveguide mode. The resonance wavelength can be adjusted by modifying Λ, which changes the frequency that satisfies Eq. (1), or by adjusting the grating parameters such as t, ℱΛ, or ℱt, which alters the waveguide dispersion β(ω).The fundamental mode manifests as a narrow-band peak in the reflectance spectrum, with the peak reflectance approaching unity, as shown in Fig. 1(b). This is due to destructive interference of the transmitted light [25, 27]. The resonance features a characteristic Fano lineshape due to interference between energy reflected through two distinct pathways: energy is reflected directly from the slab as well as indirectly reflected after coupling to the guided resonances of the slab [28,29]. The linewidth of the resonances can become very narrow, in particular when the magnitude of the index modulation (controlled here by ℱΛ and ℱt) is very small [30].
This type of PhC structure has been used in a variety of applications, such as general-purpose narrow-band optical filters [27], biosensors [31], and distributed feedback lasers [32]. In this letter, we focus on using the fundamental TE0 mode for strong-coupling.
3. Photonic design for strong coupling
The thin slab results in strong vertical confinement of the waveguide mode and large electric field enhancement (Fig. 1(c)), which makes the 1D PhC a good platform for polaritons. The electric field of the resonant mode is confined to the high-index slab and decays exponentially into the surrounding regions. The 2D materials can be incorporated simply by exfoliation onto the top of the PhC, as shown in Fig. 1(a). Note that propagating waveguide-polaritons can also be formed in unmodulated waveguides [33], which do not have a meta-stable ground state and do not couple out of the system except at the ends of the waveguide. Here we focus on polaritons with a meta-stable ground state within the light cone, by using PhCs.
The evanescent coupling between the PhC mode and the 2D material excitons can be tuned by adjusting the thickness of the slab. This can be understood as being analogous to the evanescent field overlap in an asymmetric slab waveguide. When the waveguide core is much thicker than the operating wavelength, the field is mostly confined within the guiding layer with low evanescent leakage as well as low normalized peak amplitude. Reducing the thickness, the evanescent leakage increases while the normalized peak amplitude also increases due to tighter confinement, leading to higher field amplitude at the surface of the waveguide. This can be quantified by appropriately normalizing the electric fields according to [34]
The electric field is confined in the vertical direction and unbound in the in-plane directions, and the integral is evaluated over a finite volume. As the waveguide thickness is decreased to much smaller than the wavelength, there will be a high amount of field leakage, and the peak amplitude of the field will decrease as well since the waveguide confinement is no longer sufficient to localize the field. Thus, it follows there is an optimal thickness for the field-amplitude enhancement. This is seen in Fig. 2(a), where we plot the E field, normalized according to Eq. (2) and averaged in the plane of the monolayer.Fig. 2 (a) Variation of the average normalized field amplitude at the monolayer (black) with tg, compared with the polariton splitting vs. tg obtained from (c) (red dots); (b–c) The absorption spectrum versus the PhC thickness at high (b) and low (c) temperatures. The period is tuned to maintain zero-detuning with the exciton. All calculations are at normal incidence.
With a large field enhancement and the large oscillator strength of TMDCs, strong coupling is readily achieved once we incorporate TMDCs on the PhC. We consider the example of a monolayer WS2, coupled to a PhC with ℱt = 0 and ℱΛ = .9, with the period Λ adjusted such that we obtain a guided mode resonance at zero in-plane wave-number that matches the exciton resonance. We model the exciton resonance of a single .8 nm thick monolayer of WS2 as a Lorentz oscillator:
with background static permittivity εB = 20, exciton resonance EX, oscillator strength f =1.59 eV2, and linewidth Γ=30 meV, consistent with experimentally measured results [16]. The resonances of this WS2-PhC coupled system are shown in Fig. 2(b). Normal mode splitting is evident. The splitting varies with the thickness of the grating, proportional to the field strength enhancement, as shown in Fig. 2(a). A maximum splitting of about 30 meV is obtained at maximum field enhancement, comparable to monolithic DBR structures near room temperature [14], and 13 meV smaller than the splitting in Bloch-Surface Wave structures [17], as expected by the improved field confinement in the latter [35]. The corresponding optimal thickness is at t = 100 nm, in agreement with Eq. (2). Although we are primarily interested in room temperature polaritons in this letter, for comparison we show the same plot at low temperatures in Fig. 2(c) by reducing Γ to 5 meV, which shows a similar result with a slightly higher splitting of about 35 meV. The coupling can be increased by moving the TMDCs to the interior of the PhC, where the field amplitude is higher, and by using more than one monolayer. By embedding three WS2 monolayers in the center of the Si3N4 layer of the PhC, each separated by 2 nm sheets of hBN, the polariton mode splitting is increased to over 70 meV. While we have used Si3N4 here due to its prevalence as a common optical material, higher index materials can also be used to increase the coupling [35].Due to the coexistence of the direct and indirect transmission/reflection pathways discussed above, the structure exhibits both weak coupling of excitons to the free-space continuum of optical modes, clearly visible in Fig. 2(b), and the split upper and lower polaritons modes formed due to strong coupling between the exciton and the confined resonant mode.
4. Dispersion properties
Unlike a DBR cavity, which has a fixed isotropic dispersion, a PhC allows for more complex dispersions [36]. As shown in Fig. 3, the dispersion of the polariton modes consists of highly anisotropic bands. Along one direction it is very steep, and along the other it is shallow. The anisotropy of the bands can be understood from the phase matching condition in Eq. (1). For dispersions along just the x̂ and ŷ directions, we have the following expressions:
The first equation simply results in a folded waveguide dispersion, and the second results in an approximately parabolic dispersion with a much lower group velocity. The dispersion anisotropy means that depending on the plane in k-space being considered, the polaritons have drastically different group velocities.Fig. 3 (a–b) Polariton dispersions along the two principle k directions; note the different scales on the k-axis. (c) A double-well dispersion, showing two separate local minima located at the first Brillouin zone edges kmin = ±π/Λ, well away from k = 0.
The device may also be designed to have unusual dispersions. For example, the polariton ground state can be shifted to k ≠ 0. As shown in the example in Fig. 3(c), the ground states are located at the edge of the PhC Brillouin zone at kx = π/Λ. For this structure, we use a center-coupled grating with Λ = 606 nm, t = 100 nm, ℱt=0, and ℱΛ=.96. By choosing Λ appropriately, the edge of the Brillouin zone can be tuned to a desired kx.
The 1D PhC can be generalized to two dimensions. The dispersion may be further tuned by breaking the symmetry in the ŷ direction to create a 2D PhC, or by introducing variations within a single period [37] to allow additional degrees of freedom in designing the dispersion relation for the system, such as massless Dirac disperions [38].
5. Multi-color, hybrid polariton systems on a single chip
Another unique benefit of the PhC platform is that different PhC cavities can be created side-by-side on the same slab of a uniform total thickness t, which allows for simple fabrication of multiple devices operating at different wavelengths on the same chip [39]. With 2D materials, the different PhC cavities can also couple to different materials. As an example, in Fig. 4 we tune the PhC resonance in the strong-coupling regime over 200 nm by varying only the period. The PhC has a fixed t = 100 nm, ℱΛ = .9, and ℱt = 0. For a period of Λ = 420 nm, this structure is equivalent to the structure in Sec. 4 for strong coupling with monolayer WS2. As we vary Λ from 400 nm to 540 nm, the resonance shifts from 600 nm to 800 nm, covering the exciton resonances of four common monolayer TMDCs. At the same time, a high finesse is maintained. Adjusting the exciton resonance to track the PhC resonance and assuming the same oscillator strength and linewidth of WS2, we compute the coupled modes in this structure. As shown in Fig. 4(b), normal mode splitting is maintained throughout the tuning range of over 200 nm. This shows that one can straightforwardly integrate photonic and polaritronic devices of WS2, WSe2, MoS2, MoSe2 and their heterostructures at various temperatures, simply by choosing an appropriate Λ at various locations on a single chip.
Fig. 4 (a) Tuning of a PhC resonance with the period Λ while all other grating parameters are fixed. The circles mark the resonance wavelength of the exciton resonances of the four commonly used monolayer TMDCs. Inset: A schematic showing the proposed multi-wavelength chip with multiple TMDC heterostructures emitting at different wavelengths. (b) The corresponding absorption spectrum of the TMDC-PhC system as Λ is tuned. Normal mode splitting is evident and maintained throughout the tuning range.
6. Conclusions
In conclusion, we have demonstrated slab PhCs as a simple and flexible structure for creating polaritons with monolayer TMDCs. The PhC resonators allow comparable exciton-photon coupling as in DBR cavities, while allowing on-chip integration, compact design, and a much greater degree of flexibility in engineering the properties of the polaritons, including generating complex dispersions and wavelength tuning. These features will facilitate the realization of novel phenomena and development of practical photonic devices with 2D material polaritons.
Funding
Army Research Office (ARO) MURI (W911NF-17-1-0312).
Acknowledgments
We thank Pavel Kwiecien and Victor Liu [40] for their open-source RCWA codes which were both used for calculations in this paper.
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