Abstract

The plasmon-phonon hybridization behavior between anisotropic phonon polaritons (APhP) of orthorhombic phase Molybdenum Trioxide (α − MoO3) and the plasmon-polaritons of Graphene layer – forming a van der Waals (vdW) heterostructure is investigated theoretically in this paper. It is found that in-plane APhP shows strong interaction with graphene plasmons lying in their close vicinity, leading to large Rabi splitting. Anisotropic behavior of biaxial MoO3 shows the polarization-dependent response with strong anti-crossing behavior at 0.55 eV and 0.3 eV of graphene’s Fermi potential for [100] and [001] crystalline directions, respectively. Numerical results reveal unusual electric field confinement for the two arms of enhanced hybrid modes: the first being confined in the graphene layer representing plasmonic-like behavior. The second shows volume confined zigzag pattern in hyperbolic MoO3. It is also found that the various plasmon-phonon hybridized modes could be wavelength tuned, simply by varying the Fermi potential of the graphene layer. The coupling response of the hybrid structure is studied analytically using the coupled oscillator model. Furthermore, we also infer upon the coupling strength and frequency splitting between the two layers with respect to their structural parameters and interlayer spacing. Our work will provide an insight into the active tunable property of hybrid van der Waals (vdW) structure for their potential application in sensors, detectors, directional spontaneous emission, as well as for the tunable control of the propagating polaritons in fields of flat dispersion where strong localization of photons can be achieved, popularly known as the flatband optics.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical polaritons, are quasiparticles formed by the coupling of electromagnetic waves with oscillating charges [1] that can be either free electrons in metal forming surface plasmon polaritons (SPPs) [2] or crystal vibrations in polar dielectrics giving rise to phonon polaritons (PhPs) [3]. Sub-wavelength nature of polaritons facilitates the study of interaction at nanoscale and can be altered with material or structural parameters. Metal based SPPs suffer from significant ohmic losses, limited range of operation, and lack tunability [4]. One optimum solution to overcome the limitations of SPPs is provided by light coupled ionic oscillations – Phonon Polaritons – which are unsusceptible to ohmic losses. Dispersion relation of such polaritonic materials can range from elliptical to hyperbolic [5], with the latter providing extreme confinement of photonic density of states [6], negative refraction [7], sub-diffractional imaging [8], and enhanced spontaneous emission [9]. Hyperbolic response in artificially engineered materials often known as metamaterials (MM) is realized from alternate metal-dielectric layers [10] or metal-wire grids [11]. However, stringent control of fabrication parameters and intrinsic ohmic loss of metals further adds a fundamental limit to their performance. The discovery of two-dimensional (2-D) materials [12] in the past decade eventually led to the development of hybrid van der Waals (vdW) [13] materials – layered atomically thick Two-Dimensional (2-D) materials bonded in out-of-plane direction via weak van der Waal forces. These vdW materials host a wide variety of polaritonic modes having properties of its each constituent layer [14,15]. Graphene [16], the eldest plasmonic material of this 2-D class, holds an additional advantage of strong in-plane conductivity that can be tuned by chemical doping and electrostatic gating, which finds application in actively tunable IR [17] and THz devices [18]. Another class of 2-D materials includes recently discovered naturally occurring hyperbolic materials (NHMs) [19] such as hexagonal Boron Nitride (hBN) [20] and molybdenut Trioxide ($\alpha -MoO_3$) [21], which support PhPs having low loss [22], extraordinary propagating high-k modes and longer phononic lifetimes [23]. In stark contrast to the uniaxial hyperbolic response of hBN, $\alpha -MoO_3$ provides a biaxial response which further allows an additional degree of freedom for control of electromagnetic waves for flatband optics [24].

Orthorhombic-phase Molybdenum Trioxide ($\alpha -MoO_3$) is a naturally occurring biaxial hyperbolic crystal with its phononic response extending from mid-IR to far terahertz (THz) regime [25] (9-38 $\mathrm {\mu }$m wavelength). Commonly, in hyperbolic materials, a frequency region known as Restrahalen Band (RB) is defined as bounded by transverse ($\omega _{TO}$) and longitudinal optical phonon frequency ($\omega _{LO}$) [26]. In this range, the permittivity response of polar dielectrics is negative, at least in one crystalline direction, and thus imparts metal-like high reflectivity and supports localized as well as propagating polaritons similar to the SPPs. The nonequivalent bond length of Mo-O along three crystal axes imparts a 7$\%$ in-plane lattice mismatch giving rise to its strong in-plane optical anisotropy [27]. Utilizing these additional advantages of broadband hyperbolic response and giant in-plane anisotropy of MoO$_3$, its applications in lithography-free polarization devices such as polarizers and waveplates [28], perfect absorbers [29], and molecular sensors [30,31] have been reported. However, their phononic response, which depends on the crystal structure and atomic composition, can only be tuned with the number of layers [32] and other structural changes [33]. Switching in propagation direction of phononic modes alongwith enhanced confinement have been achieved by substrate mediated hybrid polaritons [34]. Since these are passive techniques and once designed cannot be altered, therefore it limits their use for development into actively tunable devices. This limits their use for their development into actively tunable devices. Active tunability via external control can allow better utilization of hyperbolic phonon polaritons (HPhPs) for practical applications. Efforts have been laid for specific control of HPhPs through ion intercalation and carrier injection, but all these offer narrow range tunable response along with high optical losses [3537]. Another method explores interaction of evanescent fields generated from volume confined HPhPs at the interface, which is highly sensitive to surrounding dielectric materials [38]. Tuning the dispersion and propagation properties of HPhPs by graphene [33,39] or phase change material like vanadium dioxide (VO$_2$) [40,41] have been demonstrated earlier. Formation of hybrid plasmonic-phononic modes by the hybridization effect provided tunability with propagation lengths improved from that of non-hybridized states [39]. Hybridization between plasmons of anisotropic black phosphorus and graphene has also been studied to provide direction-sensitive hybridized modes whose tunable response has been achieved via graphene [42]. In another report [43], 2D/3D vdWs heterostructure made up of hBN and germanium have been proposed for its potential application as gate-tunable superlens. Also, patterning of such hybrid structures can further allow subwavelength localization of hybrid modes with a quality factor as high as $\sim$ 85 [44]. Inspired by these methods of active control and improved performance, here, we report a hybridization study of anisotropic phonon polaritons of $\alpha -MoO_3$ with tunable graphene plasmons.

In this work, interaction of anisotropic phonons of MoO$_3$ with graphene plasmons is studied from a hybridization perspective. The substantial in-plane anisotropic property provides an additional degree of freedom for controlling interactions at nanoscale. Patterning of such crystals will allow free-space electromagnetic waves to couple and produce polaritons that can further interact with surrounding materials to create hybridized modes. Strong coherent coupling between the naturally occurring biaxial hyperbolic crystal of MoO$_3$ and surface plasmons of graphene is numerically studied and analytically verified. We find that splitting of resonance response is not just a result of absorption in RB but is rather produced by strong coupling between plasmons and phonons. Furthermore, the effect of change in Fermi potential of graphene on hybridized modes and that of structural parameters on the coupling strength is studied. The broadband hyperbolic response of $\alpha -MoO_3$ and tunable characteristics of graphene, creating a tunable hybrid nanostructure, can be extended for low-loss, tunable mid-infrared to far-THz applications including THz Time-Domain Spectroscopy to investigate molecular signatures of aromatic hydrocarbons [45,46], replacing THz domain lossy metallic antennas, thermal emitters [47] and lithographically fabricated hypercrystals [48,49].

2. Design and numerical modeling

To study the hybridization between the phonons of $\alpha -MoO_3$ and plasmons of graphene sheet, we have developed a theoretical model and studied it using finite element methods (FEM). High-momenta plasmonic and phononic modes cannot be excited directly from incident electromagnetic wave due to momentum and energy mismatch [50]. Here, we propose a model consisting of MoO$_3$ nanoribbons separated from a continuous graphene layer by a dielectric spacer of refractive index 1.7 and thicknesses of 50 nm. The nanoribbons so formed will provide momentum compensation allowing incident light to couple into the structure and will produce non-propagating modes with ultra-high confinement [6]. The structure is made periodic in the x-direction with periodicity P=300 nm and is extended to infinity along the y-direction. The width of MoO$_3$ nanoribbon is fixed at 210 nm. The structural parameters are judicially chosen so that graphene’s plasmonic response and the phononic response of MoO$_3$ lie close to each other. An electromagnetic wave of power 1W polarized perpendicular to nanoribbon is normally incident upon the hybrid structure, as shown in Fig. 1(a). The optical response of $\alpha -MoO_3$ is governed by its lattice vibrations in mid-infrared (MIR) as well as in the far THz region showing hyperbolic response in over six frequency regions [25,51]. The dielectric function of such anisotropic phononic crystals is a tensor and is given by the Lorentz model [52]

$$\varepsilon_i = \varepsilon_{\infty,i} \prod_{j=1}^{N_j} \frac{1 + (\omega_{ij}^{LO})^2 + (\omega_{ij}^{TO})^2}{(\omega_{ij}^{TO})^2 - \omega^2 - i \omega \Gamma_{ij}}$$
where, i=x,y and z represents three crystallographic axes [100], [001] and [010] respectively. $\varepsilon _{\infty ,i}$ is the high-frequency permittivity tensor, and N$_j$ gives the number of lattice oscillations in i$^{th}$ direction. $\Gamma _{ij}$ is the linewidth, $\omega _{ij}^{TO}$ and $\omega _{ij}^{LO}$ are the vibrational frequency of optical phonons along the transverse and longitudinal direction of light propagation, respectively. Parameters for the Lorentz oscillator model are given in Table 1, showing RB response from 262-1010 cm$^{-1}$. Since the dielectric response of $\alpha -MoO_3$ is superior in 818-963 cm$^{-1}$,544-850 cm$^{-1}$ and 956-1006 cm$^{-1}$ phonon range for x,y, and z direction respectively, we have considered these ranges for the present work. Its dielectric response is shown in Fig. 1(b).

 figure: Fig. 1.

Fig. 1. Optical properties of molybdenum trioxide and structure under study. (a) Proposed hybrid structure periodic in x- direction and extended to infinity along y- direction, (b) Permittivity of $\alpha -MoO_3$

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Tables Icon

Table 1. Parameters of $\alpha$-MoO$_3$ for Lorentz oscillator model [25,51]

Graphene being a 2-D material, is modeled as a single layer conductive sheet for the complete range of operation. At low frequencies (away from near-infrared (NIR)), its intraband transitions contribute towards the overall performance, and thus its conductivity in reduced form is given by the Drude-like response [53]

$$\sigma(\omega) = \frac{e^2 E_F}{\pi \hbar^2} \frac{i}{\omega + i\tau^{{-}1}}$$
where, e is the electronic charge, E$_F$ is the Fermi level of graphene, $\hbar$ is the reduced Planck’s constant, $\tau$ is the relaxation time, and $\omega$ is the incident optical angular frequency. Relaxation time $\tau$, shows dependence on E$_F$ and is given as $\tau = \mu _c E_F/e\nu _f^2$, where $\nu _f$ is the Fermi velocity and is usually 1$\times$10$^6$ m/s and $\mu _c$ is the carrier density which typically lies in the range [54] of 1-20 $\times$ 10$^3$ cm$^2$/V.s, therefore in this study, we have taken 7$\times$10$^3$ cm$^2/V.s$ for continuous graphene layer.

Firstly, the anisotropic phononic response of MoO$_3$ nanoribbons formed along [100] and [001] crystalline direction without graphene is studied for both x- and y- polarization incidence, and the absorption spectra are obtained as shown in Fig. 2(a). When the structure is periodic along [100] ([001]) crystalline direction, two absorption peaks – one for x-polarized light at 25.886 THz (19.434 THz) and the other for y-polarized at 16.33 THz (24.52 THz) appear. Increment in normalized absorption response of grating $\varepsilon _{[100]}$ for x-polarized light is due to the combined resonance of grating and phononic response. However, the response for y-polarization is clearly governed by intrinsic absorption along [001] direction, which can be confirmed from the imaginary part of dielectric constant in the respective direction. The same holds true when the structure is made periodic along the [001] direction, i.e., response of the structure for the incident light polarized along the length of grating is dictated by the intrinsic property of material, while that for polarization orthogonal to the grating is the resultant response of nanoribbons and phononic oscillations. For [001] direction, the nanoribbon response is not optimized; therefore, the absorption spectra do not show much enhancement in our proposed structure. Other resonances for both crystal directions are additional phononic modes that are excited due to higher-order diffractive modes of the grating. Electromagnetic modes confined in the volume of MoO$_3$ nanoribbon at resonance frequency are shown in Fig. 2(b) for grating $\varepsilon _{[001]}$ and grating $\varepsilon _{[100]}$. The zigzag confinements are phononic modes corresponding to the hyperbolic response of material itself. Another feature that can be noted is the narrow band absorption with a full width half maximum (FWHM) of 129 nm (grating $\varepsilon _{[100]})$ and 320 nm (grating $\varepsilon _{[001]})$, corresponding to quality (Q) factors of 89.77 and 57.36 respectively, which is enabled by the longer lifetime of phonon polaritons. Electric field plots for y-polarization incidence have not been included because no confinement occurs for this case. The polarization-dependent varying response of MoO$_3$ nanoribbons indicated by different resonance wavelengths thus confirms strong in-plane anisotropy of $\alpha -MoO_3$.

 figure: Fig. 2.

Fig. 2. Response of MoO$_3$ without graphene layer (a) Normalized absorption for grating in [100] (red curve) and [001] (blue curve) crystalline direction for x- (solid line) and y- (dashed line) polarization in the absence of graphene, and (b) represents electric field plot for grating with x-polarization at the resonance frequency.

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3. Results and discussion

3.1 Hybridization of phonon-plasmon and their tunability

The hybridization behavior of phonon-plasmons and its dynamically tunable response by varying the Fermi potential (E$_F$) of the graphene layer is studied in this section. Nanostructures made of hyperbolic crystals support extremely confined high-momentum modes [55]. The reason behind this enhancement can be considered as confinement of guided modes of MoO$_3$ slab by the structural boundary of the designed grating. Surface plasmons in a continuous graphene layer can be excited once the wave vector of diffractive order modes of MoO$_3$ nanoribbon matches that of the in-plane wave vector of plasmonic modes [56].

For an initial understanding of the hybridization behavior of anisotropic phonon and plasmons, E$_F$=0.4 eV is chosen for the graphene layer placed beneath MoO$_{3,[100]}$ grating. Two cases for x- and y-polarization are studied. For incident light polarized across the grating (x-polarized), two absorption peaks appear, one at 21.79 THz with absorption $\sim$ 18$\%$ and the other at 26.14 THz with $\sim$ 38$\%$ absorption (Fig. 3(a)). The normalized electric field plot of hybridized modes formed by coupling between the plasmonic response of graphene and the phononic response of grating coupled MoO$_3$ is shown in Figs. 3(b) and 3(c). It is clearly evident that response at lower frequency exhibits more plasmonic behavior, indicating confinement in graphene layer and at higher frequency volume-confined high-wavevector (high-$\kappa$) modes in MoO$_3$ nanoribbons, similar to that shown in Fig. 2(b) are supported. But when polarization is along the grating length, i.e., y-polarization, only a single absorption peak unaffected by the presence of graphene layer is observed (Fig. 3(a)). This is due to the absence of confined photonic states along the grating that plasmons in the graphene layer could not be excited, as shown in Fig. 3(d). Having known that plasmons can be excited by grating only, we restrict our further study to just x-polarized light. Next, we vary the Fermi potential of graphene from 0.1-1 eV to study the tunable property of the proposed structure. The absorption spectra for grating in both crystalline directions are shown in Figs. 4(a) and 4(b), respectively. Two absorption bands are observed for both cases. Using the terminology reported earlier, these hybrid modes can be classified as surface plasmon-phonon polaritons [33] (SP$^3$ found outside RB) and hyperbolic plasmon-phonon polaritons [57] (HP$^3$ found inside RB). The RB confined by $\omega _{TO}$ on lower frequency edge and by $\omega _{LO}$ on higher frequency edge has been marked for both the cases with dotted lines to show bounded region. The formation of these hybrid modes is the result of coupling between graphene-plasmons and MoO$_3$-phonons. It is seen that with an increase in Fermi potential, hybrid mode experiences an increment in energy and thus moves towards a higher frequency range. The rise in Fermi potential imparts more energy to graphene plasmons, and thus stronger coupling can allow more efficient energy exchange. Electric field plots for both crystalline directions are shown in Figs. 4(c), 4(d) and 4(e), 4(f), along with an energy diagram to depict a strong coupling response. Colors indicated in the energy diagram (Fig. 4(g,h) represents two different hybrid modes – black color resembles HP$^3$ and red color shows SP$^3$ where the length of line indicates which hybrid mode is prominent.

 figure: Fig. 3.

Fig. 3. Hybridized plasmon-phonon response along with hybrid modes. (a) Normalized absorption plot for hybrid graphene-MoO$_{3,[100]}$ grating for x- and y- polarization incidence, respectively. Normalized electric field plot for x- polarization incidence at (b) 21.79 THz showing graphene-like hybrid response, and (c) 26.144 THz showing MoO$_3$ like hybrid response, and (d) normalized electric field plot for y- polarization. A Fermi potential of 0.4 eV is assumed in simulations. The color bar represents the strength of normalized electric field.

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 figure: Fig. 4.

Fig. 4. Tunable response with varying Fermi potential of graphene. Normalized absorption and electric field plots for MoO$_{3,[100]}$ (a), (c) and (d) and MoO$_{3,[001]}$ grating (b), (e) and (f) with varying Fermi potential along with energy hybridization representation in (g) and (h) respectively.

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An analytical model based on coupled oscillator model (COM) has been used to verify the above-described hybridization response [58]. Considering phonons of MoO$_3$ and plasmons of graphene as the source of two oscillations coupled to each other, their hybridization can be explained fully by the COM. A new set of hybridized frequencies $\omega _\pm$ for plasmon-phonon coupling can be obtained from individual oscillator frequencies as,

$$\omega_\pm^2 = {\frac{(\omega_{plamson}^2 + \omega_{phonon}^2) \pm \sqrt{\omega_{plasmon}^2 - \omega_{phonon}^2)^2 + 4\eta_i^2}}{2}}$$
where $\eta _{i}$ is the coupling frequency obtained from splitting energy of modes. Plasmonic frequency for graphene [59] is given as
$$\omega_{gr}=q \sqrt{ \frac{\nu_f (n \pi) N^{\frac{1}{2}}}{\hbar \varepsilon_0 (\varepsilon_{spa} + \varepsilon_{sub})P}}$$
and phononic frequency ($\omega _{phonon}$) is equivalent to the $\omega _{TO}$ frequency of MoO$_3$ for a specified RB band coupling. Here n=1 is taken for fundamental plasmonic mode. For phononic frequency, grating coupled phonon frequency in the absence of graphene layer is obtained from simulation and set as $\omega _{phonon,[100]}$ = 25.886 THz and $\omega _{phonon,[100]}$ = 19.434 THz. By comparing the analytically (Figs. 5(c) and 5(f)) and numerically obtained absorption spectra for hybridized modes plots (Figs. 5(a) and 5(d)), it is evident that strong coherent coupling [60] occurs between MoO$_3$ nanoribbons and graphene layer. An essential feature of hybridization is the anti-crossing behavior of energy-separated absorption bands. It represents a typical Rabi splitting phenomenon resulting from the hybridization of two oscillating systems. This anti-crossing behavior for grating along [100]- and [001]- crystalline directions is observed fermi potential at 0.55 eV and 0.3 eV, respectively. The difference in the Fermi potential of the graphene layer for efficient coupling in the two cases is due to strong in-plane anisotropic response of MoO$_3$. Also, enhanced oscillation strength for [001]- direction produces coupled response at lower Fermi potential in addition with wider mode splitting of 12.42 meV as compared to 10.35 meV for [100] grating. Another essential point to be noted is that the hybrid phononic response is more pronounced in the limited frequency region bounded by $\omega _{TO}$ at lower frequency and $\omega _{LO}$ at higher frequency end. This is clear evidence that phonons are supported inside RB only. Outside RB-2 (818-963 cm$^{-1}$), the response is mainly contributed by plasmons and is not much enhanced in the simulations. Similarly, the energy splitting at 0.3 eV for grating in [001] direction with hybridized phononic frequency is shown in Fig. 5(d). The discrepancies in the numerical (Figs. 5(a) and 5(d)) and analytical (Figs. 5(c) and 5(f)) results are mainly due to change in a simulation environment which is not included in the analytical study. We did not include higher-order plasmonic and phononic modes in the analytical study to reduce complexity.

 figure: Fig. 5.

Fig. 5. Normalized absorption spectra representing strong coherent coupling. (a) and (b) Normalized absorption plot of hybrid structure made up of MoO$_{3,[100]}$ and graphene obtained numerically for x- and y- polarization incidence, (c) Analytically mapped dispersion relation between MoO$_{3,[100]}$ , graphene plasmons and hybrid response obtained from Eq. (3) with splitting energy of 10.35 meV, (d), (e), and (f) shows numerically and analytically obtained plots obtained MoO$_{3,[001]}$ with splitting energy of 12.42 meV. False color map of normalized absorption.

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Thus, we can conclude that when the incident light is polarized across the grating width, it produces enhancement of photonic density of states, which in turn excites plasmons in the graphene layer and produce hybridized modes. The difference in hybridized frequencies of these modes for [100] and [001] crystalline direction is the contribution of in-plane anisotropic response of MoO$_3$ crystal. Additionally, hybrid phononic modes are not supported beyond RBs. These hybrid modes can be tuned by varying the Fermi potential of graphene.

The tuning of hybrid modes finds its potential application on phonon polariton based sensing. As plasmon based sensing suffers from ohmic losses due to the presence of electrons in metals [6164], phonon polaritons provided a path for lossless sensing using surface enhanced infrared absorption (SEIRA) spectroscopy. Phonon polaritons are advantageous over plasmons – as they are immune to losses and possess high momentum. SEIRA has been used for phonon polariton based sensing to detect low polarizable analytes [65,66]. By adding graphene to the system and changing its Fermi potential by adding external energy, it is possible to tune the overall phonon-polariton resonance response, thereby opening the possibility of detection of multiple analytes as described in [67]. Thus, a theoretical study of hybridization between the phonon polaritons of MoO$_3$ and tunable plasmon polaritons of graphene could further allow the field of molecular sensing via SEIRA to grow.

3.2 Effect of structural parameters

Plasmonic and phononic frequencies show dependency on structural parameters, and thus hybrid structures consisting of such oscillators follow the same route. To investigate the dependence of hybridization response on structural parameters, we perform simulations for varying periodicity, the width of grating, and separation between MoO$_3$ grating and graphene. Studies have been conducted for MoO$_3$ grating along [100] crystalline direction, but the same can be extended for other directions, i.e. [001]. Firstly, the periodicity of structure is varied by keeping other parameters constant. From Eq. (4), it is clear that graphene plasmonic frequency depends on two parameters – E$_F$,P. In Fig. 6(a), periodicity-dependent plasmonic frequency is mapped for varying Fermi potential. Horizontal line parallel to x-axis represents $\omega _{phonon}$, and their intersection point defines the Fermi potential to achieve strong coherent coupling.

 figure: Fig. 6.

Fig. 6. Dependency of hybrid response on structural parameters (a) Analytically computed plasmonic frequency of graphene from Eq. (4), (b) Numerically obtained absorption plot for MoO$_3$ grating – graphene hybrid structure for varying P (each plot is frequency v/s E$_F$), (c) Absorption of hybrid structure for varying width of MoO$_3$ grating at E$_F$=0.55 eV. Plots are shifted along vertical axis for clarity, (d) Mode splitting with respect to separation between graphene and MoO$_3$ grating with inset showing surface plot of normalized absorption (upper) for d=100 nm and (lower) for d=20 nm. All the plots are obtained for MoO$_{3,[100]}$ grating for x-polarization.

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It can be observed that, as periodicity increases, the plasmonic frequency of graphene decreases (due to inverse relation with periodicity), and thus more Fermi potential is required to produce effective coupling. Figure 6(b) shows absorption spectra for different periodicity. It can be verified that amount of mode splitting and energy required to achieve strong coherent coupling obeys Fermi potential scaling exactly from what has been obtained analytically. It is clearly observable from absorption plots that as periodicity increases, Fermi level also increases along with improved mode splitting.

Next, dependency of phononic frequency on the width of grating is analyzed. Here, the width of grating is varied by keeping all other structural parameters constant. Periodicity is taken to be 300 nm, spacing between graphene layer and MoO$_3$ grating is 50 nm. All the simulations are performed with a Fermi potential of 0.55 eV as the maximum coupling is achieved at this energy. Usual behavior commonly reported for many resonating structures, i.e., wavelength scales linearly with the size is observed. As shown in Fig. 6(c), for an increase in w, the entire hybrid modes show red shift, which is the direct influence of the grating resonant frequency being reduced. After studying the dependency of individual oscillators on structural parameters, the effect of separation (between the graphene-MoO$_3$) on coupling strength is studied. We varied the separation distance from 10-100 nm in steps of 10 nm and plotted the response of coupling strength for varying ’d’. When separation is less than or equal to 50 nm, efficient coupling occurs. However, as the separation is increased, significantly above 100 nm, their fields could no longer influence each other, and thus no coupling can be seen (Fig. 6(d)). Therefore, it is clear that coupling between the two oscillators can only occur when the field decays of both lies in close vicinity to each other.

4. Conclusion

In summary, a theoretical study of hybridization response of long-IR phonons of MoO$_3$ with graphene plasmons is done with the help of finite element method. The small volume of MoO$_3$ nanoribbons indicates that the high oscillator strength of miniaturized hyperbolic structures is enough to excite plasmonic response of graphene. Strong coherent coupling can be achieved between the two as long as both lies in the vicinity of others evanescent field. Here we have studied the change in response of hybrid structure by varying parameters. It is found that variation in the Fermi potential of graphene imparts tunablility to hybrid modes. Also, dependency of plasmonic and phononic response on structural parameters is studied. It is observed that graphene’s plasmonic frequency decreases with an increase in periodicity, and coherent coupling occurs at higher Fermi potential. The phononic response is dominated by the grating width and exhibits a red shift with an increase in width. Effect of separation distance on coupling strength reveals that increment in the depth of graphene layer reduces the coupling phenomenon and allows the two oscillators to produce the uncoupled response. Furthermore, a large-area continuous graphene sheet allows effective tuning of hybrid modes via electrostatic gating. A wide range of tunable polarization-dependent responses can be obtained if the structure were optimized consecutively for both crystalline directions. Our study provides insight into the hybrid behavior of anisotropic hyperbolic material, which can be tuned with graphene for application in tunable MIR and THz devices, opening new avenues for planar nanophotonic applications.

Funding

Department of Science and Technology, Ministry of Science and Technology, India (DST/NM/TUE/QM-1/2019); Sponsored Research and Industrial Consultancy (IIT/SRIC/EC/DSZ).

Acknowledgments

Basudev Lahiri wants to acknowledge the Department of Science and Technology (DST), Government of India for funding support through the Nano mission research grant of DST/NM/TUE/QM-1/2019 along with the funding support provided by IIT Kharagpur through the Institute Scheme for Innovative Research and Development (ISIRD).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publically available at this time but may be obtained from the authors on reasonable request.

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2021 (9)

S. Dixit, N. R. Sahoo, A. Mall, and A. Kumar, “Mid infrared polarization engineering via sub-wavelength biaxial hyperbolic van der waals crystals,” Sci. Rep. 11(1), 6612–6619 (2021).
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C. Wei and T. Cao, “A tunable ultrasensitive plasmonic biosensor based on α − MoO3/graphene hybrid architecture,” J. Phys. D: Appl. Phys. 54(23), 234005 (2021).
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Q. Zhang, Q. Ou, G. Hu, J. Liu, Z. Dai, M. S. Fuhrer, Q. Bao, and C.-W. Qiu, “Hybridized hyperbolic surface phonon polaritons at α-moo3 and polar dielectric interfaces,” Nano Lett. 21(7), 3112–3119 (2021).
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L. Wang, J. Liu, B. Ren, J. Song, and Y. Jiang, “Tuning of mid-infrared absorption through phonon-plasmon-polariton hybridization in a graphene/hbn/graphene nanodisk array,” Opt. Express 29(2), 2288–2298 (2021).
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G. Lu, C. R. Gubbin, J. R. Nolen, T. Folland, M. J. Tadjer, S. De Liberato, and J. D. Caldwell, “Engineering the spectral and spatial dispersion of thermal emission via polariton–phonon strong coupling,” Nano Lett. 21(4), 1831–1838 (2021).
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T. V. de Oliveira, T. Nörenberg, G. Álvarez-Pérez, L. Wehmeier, J. Taboada-Gutiérrez, M. Obst, F. Hempel, E. J. Lee, J. M. Klopf, I. Errea, A. Y. Nikkitin, S. C. Kehr, P. Alonso-González, and L. M. Eng, “Nanoscale-confined terahertz polaritons in a van der waals crystal,” Adv. Mater. 33(2), 2005777 (2021).
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D. Yoo, F. de León-Pérez, M. Pelton, I.-H. Lee, D. A. Mohr, M. B. Raschke, J. D. Caldwell, L. Martín-Moreno, and S.-H. Oh, “Ultrastrong plasmon–phonon coupling via epsilon-near-zero nanocavities,” Nat. Photonics 15(2), 125–130 (2021).
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R. Kumari, A. Yadav, S. Sharma, T. D. Gupta, S. K. Varshney, and B. Lahiri, “Tunable van der waal’s optical metasurfaces (voms) for biosensing of multiple analytes,” Opt. Express 29(16), 25800–25811 (2021).
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2020 (9)

Z. Yuan, R. Chen, P. Li, A. Y. Nikitin, R. Hillenbrand, and X. Zhang, “Extremely confined acoustic phonon polaritons in monolayer-hbn/metal heterostructures for strong light–matter interactions,” ACS Photonics 7(9), 2610–2617 (2020).
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S. Sharma, R. Kumari, S. K. Varshney, and B. Lahiri, “Optical biosensing with electromagnetic nanostructures,” Rev. Phys. 5, 100044 (2020).
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Y. Li, K. Tantiwanichapan, A. K. Swan, and R. Paiella, “Graphene plasmonic devices for terahertz optoelectronics,” Nanophotonics 9(7), 1901–1920 (2020).
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Y. Wu, Q. Ou, Y. Yin, Y. Li, W. Ma, W. Yu, G. Liu, X. Cui, X. Bao, J. Duan, G. Álvarez-Pérez, Z. Dai, B. Shabbir, N. Medhekar, X. Li, C.-M. Li, P. Alonso-González, and Q. Bao, “Chemical switching of low-loss phonon polaritons in α − MoO3 by hydrogen intercalation,” Nat. Commun. 11(1), 2646 (2020).
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M. Chen, X. Lin, T. H. Dinh, Z. Zheng, J. Shen, Q. Ma, H. Chen, P. Jarillo-Herrero, and S. Dai, “Configurable phonon polaritons in twisted α − MoO3,” Nat. Mater. 19(12), 1307–1311 (2020).
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G. Álvarez-Pérez, T. G. Folland, I. Errea, J. Taboada-Gutiérrez, J. Duan, J. Martín-Sánchez, A. I. Tresguerres-Mata, J. R. Matson, A. Bylinkin, M. He, W. Ma, Q. Bao, J. I. Martín, J. D. Caldwell, A. Y. Nikitin, and P. Alonso-González, “Infrared permittivity of the biaxial van der waals semiconductor α − MoO3 from near-and far-field correlative studies,” Adv. Mater. 32(29), 1908176 (2020).
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G. Hu, J. Shen, C.-W. Qiu, A. Alú, and S. Dai, “Phonon polaritons and hyperbolic response in van der waals materials,” Adv. Opt. Mater. 8(5), 1901393 (2020).
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H. Zhang, Z. Jiao, and E. Mcleod, “Tunable terahertz hyperbolic metamaterial slabs and super-resolving hyperlenses,” Appl. Opt. 59(22), G64–G70 (2020).
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K.-D. Xu, J. Li, A. Zhang, and Q. Chen, “Tunable multi-band terahertz absorber using a single-layer square graphene ring structure with t-shaped graphene strips,” Opt. Express 28(8), 11482–11492 (2020).
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2019 (4)

Z. Zheng, N. Xu, S. L. Oscurato, M. Tamagnone, F. Sun, Y. Jiang, Y. Ke, J. Chen, W. Huang, W. L. Wilson, A. Ambrosio, S. Deng, and H. Chen, “A mid-infrared biaxial hyperbolic van der waals crystal,” Sci. Adv. 5(5), eaav8690 (2019).
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A. Fali, S. T. White, T. G. Folland, M. He, N. A. Aghamiri, S. Liu, J. H. Edgar, J. D. Caldwell, R. F. Haglund, and Y. Abate, “Refractive index-based control of hyperbolic phonon-polariton propagation,” Nano Lett. 19(11), 7725–7734 (2019).
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Q. Zhang, Z. Zhen, C. Liu, D. Jariwala, and X. Cui, “Gate-tunable polariton superlens in 2d/3d heterostructures,” Opt. Express 27(13), 18628–18641 (2019).
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Q. Zhang, Z. Zhen, Y. Yang, G. Gan, D. Jariwala, and X. Cui, “Hybrid phonon-polaritons at atomically-thin van der waals heterointerfaces for infrared optical modulation,” Opt. Express 27(13), 18585–18600 (2019).
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2018 (8)

M. Tamagnone, A. Ambrosio, K. Chaudhary, L. A. Jauregui, P. Kim, W. L. Wilson, and F. Capasso, “Ultra-confined mid-infrared resonant phonon polaritons in van der waals nanostructures,” Sci. Adv. 4(6), eaat7189 (2018).
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M. Autore, P. Li, I. Dolado, F. J. Alfaro-Mozaz, R. Esteban, A. Atxabal, F. Casanova, L. E. Hueso, P. Alonso-González, J. Aizpurua, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Boron nitride nanoresonators for phonon-enhanced molecular vibrational spectroscopy at the strong coupling limit,” Light: Sci. Appl. 7(4), 17172 (2018).
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J. Nong, W. Wei, W. Wang, G. Lan, Z. Shang, J. Yi, and L. Tang, “Strong coherent coupling between graphene surface plasmons and anisotropic black phosphorus localized surface plasmons,” Opt. Express 26(2), 1633–1644 (2018).
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S. Bandaru, G. Saranya, N. J. English, C. Yam, and M. Chen, “Tweaking the electronic and optical properties of α − MoO3 by sulphur and selenium doping–a density functional theory study,” Sci. Rep. 8(1), 10144 (2018).
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Z. Zheng, J. Chen, Y. Wang, X. Wang, X. Chen, P. Liu, J. Xu, W. Xie, H. Chen, S. Deng, and N. Xu, “Highly confined and tunable hyperbolic phonon polaritons in van der waals semiconducting transition metal oxides,” Adv. Mater. 30(13), 1705318 (2018).
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W. Ma, P. Alonso-González, S. Li, A. Y. Nikitin, J. Yuan, J. Martín-Sánchez, J. Taboada-Gutiérrez, I. Amenabar, P. Li, S. Vélez, C. Tollan, Z. Dai, Y. Zhang, S. Sriram, K. Kalantar-Zadeh, S.-T. Lee, R. Hillenbrand, and Q. Bao, “In-plane anisotropic and ultra-low-loss polaritons in a natural van der waals crystal,” Nature 562(7728), 557–562 (2018).
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A. J. Giles, S. Dai, I. Vurgaftman, T. Hoffman, S. Liu, L. Lindsay, C. T. Ellis, N. Assefa, I. Chatzakis, T. L. Reinecke, J. G. Tischler, M. M. Fogler, J. H. Edgar, D. N. Basov, and J. D. Caldwell, “Ultralow-loss polaritons in isotopically pure boron nitride,” Nat. Mater. 17(2), 134–139 (2018).
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M. D. Goldflam, I. Ruiz, S. W. Howell, J. R. Wendt, M. B. Sinclair, D. W. Peters, and T. E. Beechem, “Tunable dual-band graphene-based infrared reflectance filter,” Opt. Express 26(7), 8532–8541 (2018).
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2017 (5)

M. N. Gjerding, R. Petersen, T. G. Pedersen, N. A. Mortensen, and K. S. Thygesen, “Layered van der waals crystals with hyperbolic light dispersion,” Nat. Commun. 8(1), 320–328 (2017).
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T. Low, A. Chaves, J. D. Caldwell, A. Kumar, N. X. Fang, P. Avouris, T. F. Heinz, F. Guinea, L. Martin-Moreno, and F. Koppens, “Polaritons in layered two-dimensional materials,” Nat. Mater. 16(2), 182–194 (2017).
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J. Duan, R. Chen, J. Li, K. Jin, Z. Sun, and J. Chen, “Launching phonon polaritons by natural boron nitride wrinkles with modifiable dispersion by dielectric environments,” Adv. Mater. 29(38), 1702494 (2017).
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T. Galfsky, J. Gu, E. E. Narimanov, and V. M. Menon, “Photonic hypercrystals for control of light–matter interactions,” Proc. Natl. Acad. Sci. 114(20), 5125–5129 (2017).
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H. Hajian, A. Ghobadi, S. A. Dereshgi, B. Butun, and E. Ozbay, “Hybrid plasmon–phonon polariton bands in graphene–hexagonal boron nitride metamaterials [invited],” J. Opt. Soc. Am. B 34(7), D29–D35 (2017).
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2016 (3)

T. Galfsky, Z. Sun, C. R. Considine, C.-T. Chou, W.-C. Ko, Y.-H. Lee, E. E. Narimanov, and V. M. Menon, “Broadband enhancement of spontaneous emission in two-dimensional semiconductors using photonic hypercrystals,” Nano Lett. 16(8), 4940–4945 (2016).
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J. Chae, B. Lahiri, and A. Centrone, “Engineering near-field seira enhancements in plasmonic resonators,” ACS Photonics 3(1), 87–95 (2016).
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J. D. Caldwell, I. Vurgaftman, J. G. Tischler, O. J. Glembocki, J. C. Owrutsky, and T. L. Reinecke, “Atomic-scale photonic hybrids for mid-infrared and terahertz nanophotonics,” Nat. Nanotechnol. 11(1), 9–15 (2016).
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2015 (5)

K. Korzeb, M. Gajc, and D. A. Pawlak, “Compendium of natural hyperbolic materials,” Opt. Express 23(20), 25406–25424 (2015).
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J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons,” Nanophotonics 4(1), 44–68 (2015).
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P. Li, M. Lewin, A. V. Kretinin, J. D. Caldwell, K. S. Novoselov, T. Taniguchi, K. Watanabe, F. Gaussmann, and T. Taubner, “Hyperbolic phonon-polaritons in boron nitride for near-field optical imaging and focusing,” Nat. Commun. 6(1), 7507 (2015).
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E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner, M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. Koppens, and R. Hillenbrand, “Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity,” Nat. Photonics 9(10), 674–678 (2015).
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2014 (4)

S. Dai, Z. Fei, Q. Ma, A. Rodin, M. Wagner, A. McLeod, M. Liu, W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H.. Castro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. Fogler, and D. N. Basov, “Tunable phonon polaritons in atomically thin van der waals crystals of boron nitride,” Science 343(6175), 1125–1129 (2014).
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D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
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P. Miró, M. Audiffred, and T. Heine, “An atlas of two-dimensional materials,” Chem. Soc. Rev. 43(18), 6537–6554 (2014).
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2013 (5)

A. K. Geim and I. V. Grigorieva, “Van der waals heterostructures,” Nature 499(7459), 419–425 (2013).
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J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
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F. Cataldo, D. García-Hernández, and A. Manchado, “Far-and mid-infrared spectroscopy of complex organic matter of astrochemical interest: coal, heavy petroleum fractions and asphaltenes,” Mon. Not. R. Astron. Soc. 429(4), 3025–3039 (2013).
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W. Gao, G. Shi, Z. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13(8), 3698–3702 (2013).
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B. Lahiri, S. G. McMeekin, M. Richard, and N. P. Johnson, “Enhanced fano resonance of organic material films deposited on arrays of asymmetric split-ring resonators (a-srrs),” Opt. Express 21(8), 9343–9352 (2013).
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2012 (1)

K. Michaelian, Q. Wen, B. Billinghurst, J. Shaw, and V. Lastovka, “Far-and mid-infrared photoacoustic spectra of tetracene, pentacene, perylene and pyrene,” Vib. Spectrosc. 58, 50–56 (2012).
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2011 (1)

R. Kotyński, T. Stefaniuk, and A. Pastuszczak, “Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers,” Appl. Phys. A 103(3), 905–909 (2011).
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2009 (2)

M. Lapine, D. Powell, M. Gorkunov, I. Shadrivov, R. Marqués, and Y. Kivshar, “Structural tunability in metamaterials,” Appl. Phys. Lett. 95(8), 084105 (2009).
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B. Lahiri, A. Z. Khokhar, M. Richard, S. G. McMeekin, and N. P. Johnson, “Asymmetric split ring resonators for optical sensing of organic materials,” Opt. Express 17(2), 1107–1115 (2009).
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2008 (2)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104(8), 084314 (2008).
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Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16(20), 15439–15448 (2008).
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2007 (1)

Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. Hwang, S. D. Sarma, H. Stormer, and P. Kim, “Measurement of scattering rate and minimum conductivity in graphene,” Phys. Rev. Lett. 99(24), 246803 (2007).
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2004 (1)

J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot–semiconductor microcavity system,” Nature 432(7014), 197–200 (2004).
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2003 (2)

K. E. Oughstun and N. A. Cartwright, “On the lorentz-lorenz formula and the lorentz model of dielectric dispersion,” Opt. Express 11(13), 1541–1546 (2003).
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D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
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Abate, Y.

A. Fali, S. T. White, T. G. Folland, M. He, N. A. Aghamiri, S. Liu, J. H. Edgar, J. D. Caldwell, R. F. Haglund, and Y. Abate, “Refractive index-based control of hyperbolic phonon-polariton propagation,” Nano Lett. 19(11), 7725–7734 (2019).
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Adam, S.

Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. Hwang, S. D. Sarma, H. Stormer, and P. Kim, “Measurement of scattering rate and minimum conductivity in graphene,” Phys. Rev. Lett. 99(24), 246803 (2007).
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Data availability

Data underlying the results presented in this paper are not publically available at this time but may be obtained from the authors on reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Optical properties of molybdenum trioxide and structure under study. (a) Proposed hybrid structure periodic in x- direction and extended to infinity along y- direction, (b) Permittivity of $\alpha -MoO_3$
Fig. 2.
Fig. 2. Response of MoO$_3$ without graphene layer (a) Normalized absorption for grating in [100] (red curve) and [001] (blue curve) crystalline direction for x- (solid line) and y- (dashed line) polarization in the absence of graphene, and (b) represents electric field plot for grating with x-polarization at the resonance frequency.
Fig. 3.
Fig. 3. Hybridized plasmon-phonon response along with hybrid modes. (a) Normalized absorption plot for hybrid graphene-MoO$_{3,[100]}$ grating for x- and y- polarization incidence, respectively. Normalized electric field plot for x- polarization incidence at (b) 21.79 THz showing graphene-like hybrid response, and (c) 26.144 THz showing MoO$_3$ like hybrid response, and (d) normalized electric field plot for y- polarization. A Fermi potential of 0.4 eV is assumed in simulations. The color bar represents the strength of normalized electric field.
Fig. 4.
Fig. 4. Tunable response with varying Fermi potential of graphene. Normalized absorption and electric field plots for MoO$_{3,[100]}$ (a), (c) and (d) and MoO$_{3,[001]}$ grating (b), (e) and (f) with varying Fermi potential along with energy hybridization representation in (g) and (h) respectively.
Fig. 5.
Fig. 5. Normalized absorption spectra representing strong coherent coupling. (a) and (b) Normalized absorption plot of hybrid structure made up of MoO$_{3,[100]}$ and graphene obtained numerically for x- and y- polarization incidence, (c) Analytically mapped dispersion relation between MoO$_{3,[100]}$ , graphene plasmons and hybrid response obtained from Eq. (3) with splitting energy of 10.35 meV, (d), (e), and (f) shows numerically and analytically obtained plots obtained MoO$_{3,[001]}$ with splitting energy of 12.42 meV. False color map of normalized absorption.
Fig. 6.
Fig. 6. Dependency of hybrid response on structural parameters (a) Analytically computed plasmonic frequency of graphene from Eq. (4), (b) Numerically obtained absorption plot for MoO$_3$ grating – graphene hybrid structure for varying P (each plot is frequency v/s E$_F$), (c) Absorption of hybrid structure for varying width of MoO$_3$ grating at E$_F$=0.55 eV. Plots are shifted along vertical axis for clarity, (d) Mode splitting with respect to separation between graphene and MoO$_3$ grating with inset showing surface plot of normalized absorption (upper) for d=100 nm and (lower) for d=20 nm. All the plots are obtained for MoO$_{3,[100]}$ grating for x-polarization.

Tables (1)

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Table 1. Parameters of α -MoO 3 for Lorentz oscillator model [25,51]

Equations (4)

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ε i = ε , i j = 1 N j 1 + ( ω i j L O ) 2 + ( ω i j T O ) 2 ( ω i j T O ) 2 ω 2 i ω Γ i j
σ ( ω ) = e 2 E F π 2 i ω + i τ 1
ω ± 2 = ( ω p l a m s o n 2 + ω p h o n o n 2 ) ± ω p l a s m o n 2 ω p h o n o n 2 ) 2 + 4 η i 2 2
ω g r = q ν f ( n π ) N 1 2 ε 0 ( ε s p a + ε s u b ) P

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