Epsilon-Near-Zero Strong Coupling in Metamaterial-Semiconductor

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Epsilon-Near-Zero Strong Coupling in Metamaterial-Semiconductor Hybrid Structures Young Chul Jun, John Reno, Troy Ribaudo, Eric A Shaner, Jean-Jacques Greffet, Simon Vassant, François Marquier, Michael B. Sinclair, and Igal Brener Nano Lett., Just Accepted Manuscript • DOI: 10.1021/nl402939t • Publication Date (Web): 14 Oct 2013 Downloaded from http://pubs.acs.org on October 17, 2013

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Epsilon-Near-Zero Strong Coupling in Metamaterial-Semiconductor Hybrid Structures

Young Chul Jun1, 2, 3,*, John Reno1, Troy Ribaudo2, Eric Shaner2, Jean-Jacques Greffet4, Simon Vassant4, Francois Marquier4, Mike Sinclair2, and Igal Brener1, 2,† 1

Center for Integrated Nanotechnologies, Sandia National Laboratories, NM 87185, USA 2

Sandia National Laboratories, NM 87185, USA

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Department of Physics, Inha University, Incheon 402-751, Republic of Korea

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Laboratoire Charles Fabry, Institut d’Optique, Univ. Paris-Sud, CNRS, 2 av Fresnel, 91127 Palaiseau, France * [email protected], † [email protected]

Abstract We present a new type of electrically tunable strong coupling between planar metamaterials and epsilonnear-zero modes that exist in a doped semiconductor nanolayer. The use of doped semiconductors makes this strong coupling tunable over a wide range of wavelengths through the use of different doping densities. We also modulate this coupling by depleting the doped semiconductor layer electrically. Our hybrid approach incorporates strong optical interactions into a highly tunable, integrated device platform.

Keywords: Nano-optics, Metamaterials, Semiconductors, Strong coupling, Optoelectronics, Infrared

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Epsilon-near-zero (ENZ) materials exhibit highly unusual and intriguing optical properties. It has been shown that perfect coupling can be achieved between two waveguides through a very narrow ENZ channel of arbitrary shape and length1-4. The associated large field enhancement of highly squeezed fields in the thin ENZ channel was also considered for enhanced nonlinear optical interactions, such as optical switching and bistability5. ENZ materials were also employed for antenna directivity and radiation pattern control6, 7, thermal emission control8, and coherent perfect absorption9. Recently it was shown that thin layers of an ENZ material can support a new type of guided modes near the epsilon zero frequency, called ENZ modes10, 11. These modes do not couple to free space but can be excited locally with an electric field normal to the layers (TM) and cause enhanced optical fields inside the layer. For a symmetric structure, the ENZ modes are similar to long range (LR) surface plasmon polariton (SPP) modes in thin metal films. Both have the same field symmetry (i.e. symmetric normal electric field); however, the electric field magnitude is maximum inside the film for ENZ modes, and minimum for LR SPP modes 11. It has also been pointed out that ENZ modes can exist in non-symmetric configurations10. Strong field enhancement in ENZ modes can be useful for enhancing light-matter interactions at the nanoscale. In this paper, we demonstrate a novel type of optical strong coupling in a hybrid structure consisting of planar metamaterials (MMs) and a doped semiconductor ENZ nanolayer. To the best of our knowledge, this type of strong coupling has never been reported before. This coupling is enabled by the generation of TM field components in the near field of the MM resonators. The strong coupling between the MM resonance and ENZ modes is evidenced by a very large spectral splitting as the resonance frequency of the MM resonators is varied using geometric scaling. A clear anti-crossing in the measured spectra is observed at room temperature. Finally, to demonstrate the usefulness of this phenomenon for potential device applications, we dynamically modulate this coupling by depleting the doped ENZ layer electrically. Because the zero crossing of the permitiviy (Re[ε(ω)] = 0) in a doped semiconductor originates from the plasmon contribution of the electrons, the ENZ frequency can be tuned in a wide range of infrared (IR) frequencies by controlling the doping density. Therefore, our approach naturally incorporates strong optical interactions into a highly tunable, integrated device platform. These results provide a path to a new type of plasmonic devices12-14, which can be valuable for a variety of imaging and sensing applications15.

Figure 1a shows the schematic of our hybrid structure. A planar MM layer is patterned on a semiconductor substrate containing an ultra-thin ENZ layer (30 nm n+ GaAs, ND ~ 5.5 x 1018 cm-3). Incident light is directed normal to the sample surface – i.e. it does not have an electric field component normal to the ENZ interface. But, it is polarized orthogonal to the split ring resonator (SRR) gap in order to excite its fundamental resonance. The resonantly excited SRRs produce a strong normal electric field component to the interface, which is further intensified at the ENZ layer due to the boundary condition ε1E1┴ = ε2E2┴. Figure 1b shows the dielectric constant of our ENZ layer (see Fig. S1 in Supporting Information for the detailed wafer structure) as a function of frequency, measured using spectral ellipsometry (J.A. Woollam IR VASE). The real part of the dielectric constant becomes zero in the mid-IR around 780 cm-1.

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We first show that this hybrid structure should excite ENZ modes efficiently. We consider a simpler case that can provide more physical intuition. Light scattering from a deep subwavelength particle provides a wide range of wavevector components and can be modeled as dipole emission. In fact, emission from a dipole above a planar multilayer can be studied analytically. In this calculation, we can obtain the emitted dipole power as a function of in-plane wavevector, and thus we can directly compare it with the dispersion relation of the multilayer structure. Here we perform dipole emission calculations for a simple 3-layer structure (Fig. 2a, Air/ENZ/Substrate) and compare it with the dispersion relation of the ENZ mode (Fig. 2b). We find that the dipole emission spectrum lies on top of the ENZ mode dispersion exactly (Fig. 2c), suggesting that the dipole emission couples to the ENZ modes efficiently. The dispersion relation for a TM mode in an asymmetric 3-layer structure can be obtained by matching field components and applying boundary conditions at each interface: ( where

(

)

(

)

)(

)

(1),

. The solutions of this relation for several different ENZ thicknesses are

plotted in Fig. 2b (here, the ENZ layer is n-doped GaAs with a doping density ND ~ 5.5 x 1018 cm-3). For a very thin layer, these modes are termed ENZ modes10, 11. The ENZ mode lies around the epsilon-zero frequency (Re[ε(ω)] = 0) for a small wavevector region, but the mode dispersion red-shifts for larger wavevectors. It finally approaches the surface plasmon frequency (Re[ε(ω)] = -1) in the very large wavevector region. As shown in Fig. 2b, as the ENZ thickness increases, the dispersion curve red-shifts and approaches the surface plasmon frequency (~ 745 cm-1). Now we consider dipole emission above a 3-layer structure, as shown in Fig. 2a. The dipole is oscillating normal to the surface and placed above the ENZ layer. The emitted dipole power can be obtained as follows16, 17: ̂

∫ where the indices 1~3 correspond to each layer, plane and out-of-plane wavevectors, normalized distance is defined as ̂

√

and , and

(

(2), )

are the normalized in-

is the 3-layer reflection coefficient17. The

, where d is the dipole-surface distance. The dipole emission power is

normalized to that of a dipole in free-space (P0). We use experimentally measured dielectric constants for the ENZ (n+ GaAs) layer, and assume a fixed dielectric constant ε = 10.89 for the GaAs substrate. The integrand of Eq. 2 gives the wavevector spectrum of dipole emission. We calculated the dipole power spectrum for a range of dipole emission frequencies (640 cm-1 ~ 890 cm-1) and obtained 2-dimensional plots of the emission power density spectrum for several different ENZ thicknesses (t) and dipole-surface distances (d). As shown in Fig. 2c, the dipole emission has resonant distribution in both frequency and wavevector domains. When the ENZ thickness t increases, the resonance frequency slightly red-shifts. But as the dipole-surface distance d increases, it blue-shifts. In both cases, the dipole power distribution changes in the frequency and wavevector domains. We can superimpose dispersion curves on the dipole emission spectra for comparison. The white lines in Fig. 2c are the ENZ dispersion curves for a given ENZ thickness t. We can see that the dipole emission spectrum lies on top of the ENZ dispersion exactly, suggesting that the dipole emission couples to the ENZ modes. A

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subwavelength structure like an SRR provides near-field components with a wide range of wavevectors, and therefore the SRR should excite the ENZ mode too. We have strong field enhancement in the ENZ layer due to the boundary condition ε1E1┴ = ε2E2┴. From the measured dielectric constants (Fig. 1b), we find that we can have a very large field intensity enhancement for the normal electric field component (about 70 times) at the ENZ frequency (Fig. S2, Supporting Information). We verified this field enhancement using numerical simulations too (Fig. S3, Supporting Information). We use this remarkable field enhancement in ENZ modes to demonstrate a novel type of optical strong coupling that can have far reaching consequences for optoelectronic devices. Arrays of gold SRR MMs were patterned using a JEOL JBX-6300FS electron-beam lithography system, followed by electron beam evaporation of 5 nm Ti and 55 nm Au and lift-off process (Fig. 3a). The substrate includes a 30 nm n+ doped GaAs layer grown by molecular beam epitaxy (MBE) (Wafer A in Fig. S1). We use a modified SRR as our MM, because it has strong field enhancement at two gaps, though other MM structures could be used too. We performed Fourier transform infrared (FTIR) measurements at room temperature to characterize our sample. Transmission spectra through the MM samples were measured with a Bruker IFS 66v/S Fouriertransform infrared spectrometer (FTIR) using a DTGS detector. FTIR spectra were referenced to a bare substrate region which did not have SRR structures. A series of SRR MMs with different SRR scale factors were fabricated, so that the MM resonance frequency gradually shifted across the ENZ frequency. When the MM resonance matched the ENZ frequency (780 cm-1) of the doped semiconductor layer, we observed a clear spectral splitting (Scale 1.6 in Fig. 3b). Clear anti-crossing in the transmission spectra was observed as well. The obtained spectral splitting was as large as 195 cm-1. This kind of avoided crossing has been observed in various physical systems and is indicative of strong coupling between two coupled resonant modes 18, 19. We can estimate the coupling strength in our hybrid MM/ENZ structure. From the two-coupled damped oscillator model, we obtain the following relation for spectral splitting (Δω) and coupling strength (V) (e.g. see Chapter 4 in Ref. [20]): (

√

)

(3).

-1

The ENZ damping rate was determined to be

cm by fitting a Lorentzian line shape to numerically

simulated absorption spectrum of the ENZ mode: ( )

(

)

(4),

where A is a parameter describing the absorption strength. In the numerical simulation, we used experimentally measured dielectric constants for the ENZ layer. Here, the SRR resonance was far away from the ENZ frequency (i.e. we excited the ENZ mode by an off-resonant SRR), so that the resonant interaction is negligible. The MM damping rate was obtained by similar Lorentzian fitting. We used numerically simulated SRR absorption spectrum without the ENZ layer (i.e. we obtained intrinsic loss of the SRR resonance at the epsilon zero frequency) and obtained

cm-1. From the splitting Δω = 190 cm-1, we obtain the coupling

strength of V = 95 cm-1. This coupling strength can be tuned further by adjusting the barrier or the ENZ layer thickness. MM designs can be also improved to increase the interaction strength further.

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We also performed numerical simulations (3-dimensional finite difference time domain, or 3D FDTD) and verified this strong coupling behavior (Fig. 3c). We used experimentally measured dielectric constants for the ENZ (n+ GaAs) and SRR (gold) layers. A broadband pulse was incident from the top and polarized orthogonal to the gap. Transmission was measured on the substrate side. We repeated simulations for different SRR scale factors. The SRR resonance peak gradually red-shifted as the scale factor increased, and clear anti-crossing behavior was obtained (Fig. 3c). Numerical simulations agree remarkably well with experimental data. The spectral splitting from simulations (~ 190 cm-1) was very close to the experimental one. In both cases, the anticrossing was centered at the ENZ frequency. We also calculated the absorption (A) spectra by obtaining transmission (T) and reflection (R) from the same structure (A = 1 – T –R), and observed a similar spectral splitting (Fig. S4, Supporting Information). This anti-crossing in the absorption spectra is a necessary and sufficient condition for strong coupling20. We prepared another MBE grown semiconductor substrate with a lower doping level (wafer B in Fig. S1, n+ GaAs, ND ~ 2.2 x 1018 cm-3) and fabricated a similar MM/semiconductor hybrid sample. Due to the lower doping density, now the ENZ frequency shifted to a lower frequency (~ 530 cm -1). We measured the transmission through this sample and obtained a similar spectral splitting and anti-crossing (Fig. 3d). This was again verified by numerical simulations (Fig. 3e). We could obtain the same strong coupling in a different material system too (n+ InAs, ND ~ 1.1 x 1019 cm-3, Fig. S5, Supporting Information). These experimental and simulation results clearly show that this ENZ strong coupling is a general phenomenon and the frequency where anti-crossing occurs can be tuned over a wide range of wavelengths by altering semiconductor materials or doping levels.

Finally, to demonstrate the usefulness of this novel phenomenon for potential device applications, we dynamically modulate this coupling by depleting the doped ENZ layer electrically. To test this concept, we fabricated interconnected SRR arrays on Wafer B (Fig. 4). Using IV (current-voltage) and CV (capacitancevoltage) measurements, we first verified that the depletion width gradually increases with a negative bias (Fig. S6, Supporting Information). Upon the application of a negative bias, we are able to deplete the doped layer and effectively reduce the ENZ layer thickness. This weakens the coupling between the MM resonators and the ENZ mode. Figure 5a shows the SEM and optical microscope images of the fabricated electrical device. Here, the interconnected MM layer is also used as a metal gate to deplete the doped semiconductor layer electrically21, 22. Both metal gate and ohmic contacts were defined by optical lithography, metal deposition, and lift-off processes. The outside ohmic contact had electron-beam deposition of 8 nm Ni / 27 nm Ge / 54 nm Au / 14 nm Ni / 150 nm Au in sequence, followed by rapid thermal annealing at 380 °C for 30 seconds with a forming gas. The inner metal gate contact had electron-beam deposition of 5 nm Ti and 150 nm Au. The SRR arrays were connected to this metal gate via electrical bus lines and these are surrounded by the outer ohmic contact. Finally, both ohmic and metal gate contacts were wire-bonded to a chip carrier for electric biasing. We performed room temperature FTIR transmission measurements with voltage biases (Fig. 5b). For electric biasing, the Keithley 2400 source meter was used during FTIR transmission measurements. Because of the scaling chosen, the optical coupling at zero bias was only evidenced as a weak shoulder (i.e. we are at an off-

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resonant position somewhat away from the exact anti-crossing point). We obtained clear electrical tuning of the transmission spectrum (SRR Scale factor 2.4). At zero bias (black curve), the spectrum was broadened in the high-frequency region of 500 ~ 700 cm-1 due to optical coupling (indicated by a blue arrow in Fig. 5b). The MM resonance was also shifted to a lower-frequency region, compared to an original/uncoupled MM resonance position, due to this coupling (indicated by a red arrow in Fig. 5b). These two arrows correspond to the two branches of anti-crossed modes. With negative bias voltages, we depleted carriers and reduced this coupling. Thus, the spectrum became more symmetric (i.e. returning back to the original position of the isolated/uncoupled MM resonance) at -3.5 V (red curve). This is a gradual change with biasing. At -2 V, the spectrum was in between those measured at 0 V and -3.5 V (dotted green curve). Leakage current was very small during this measurement (e.g. ~ 2 μA at -2 V), thus eliminating the possibility of thermal effects. Moreover, thermal heating rather induces a red-shift of the spectrum23. In our previous work21, 22, we studied electrical tuning of MM resonances due to a permittivity change of the substrate. In that work, SRRs were made on a thick (~ 700 nm) doped semiconductor substrate, and the device was operating far away from the ENZ frequency of the doped semiconductor. We could observe small “red-shift” (< 10 cm-1) with negative biases and this red-shift can be easily understood from the fact that the SRR is a LCresonator (

√ ) and the capacitance increases with a reverse bias due to the increased substrate

permittivity in the depletion region (C ~ εd/A). In our current work, we have SRRs on top of an ultra-thin (~ 30 nm) doped semiconductor layer and novel optical coupling between them is induced. And, with negative biases, we have large “blue-shift” instead of small “red-shift”. This opposite trend and bigger spectral shift clearly shows that electrical tuning is not due to a simple permittivity change in the substrate, but indeed resulting from voltage-tunable optical coupling. Recently, ENZ modes at the longitudinal optical phonon frequency of gallium arsenide (GaAs) were used for electrical control of reflectivity

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. However, this phonon frequency is an intrinsic property of the material

and cannot be altered unless another material is used. Also, this scheme involved a delicate quantum well intersubband transition to trigger electrical tuning, which is much more complicated than our simple tuning mechanism based on carrier depletion in a doped semiconductor. Our work also offers alternative ways for tuning and controlling the resonant behavior of MMs. Previously, most work on semiconductor-based tunable MMs (primarily at THz frequencies24) relied on changing the permittivity near the metal traces through carrier depletion. This gets increasingly difficult at higher frequencies since high doping layers are needed and the depletion width is reduced accordingly. ENZ modes actually require very thin layers of a plasmonic material and we could observe our strong coupling for thicknesses of tens of nanometers, which can be completely depleted at the carrier densities used here. Therefore, our approach and the phenomena discussed here are fundamentally different from previous works24: our tuning mechanism involves electrically tunable optical coupling to a new type of electromagnetic wave (ENZ mode). This mechanism requires the use of an extremely thin doped layer. The IR spectral range is technologically important for a number of applications15, including chemical/biological sensing, thermal imaging, and free-space optical communication. Therefore, we expect that

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this novel optical strong coupling and its electrical tuning can find exciting, new applications for chip-scale active IR devices.

References and Notes (1) Silveirinha, M.; Engheta, N. Phys. Rev. Lett. 2006, 97 (15), 157403. (2) Edwards, B.; Alù, A.; Young, M.; Silveirinha, M.; Engheta, N. Phys. Rev. Lett. 2008, 100 (3), 033903. (3) Liu, R.; Cheng, Q.; Hand, T.; Mock, J. J.; Cui, T. J.; Cummer, S. A.; Smith, D. R. Phys. Rev. Lett. 2008, 100 (2), 023903. (4) Adams, D. C.; Inampudi, S.; Ribaudo, T.; Slocum, D.; Vangala, S.; Kuhta, N. A.; Goodhue, W. D.; Podolskiy, V. A.; Wasserman, D. Phys. Rev. Lett. 2011, 107 (13), 133901. (5) Argyropoulos, C.; Chen, P.-Y.; D’Aguanno, G.; Engheta, N.; Alù, A. Phys. Rev. B 2012, 85 (4), 045129. (6) Enoch, S.; Tayeb, G.; Sabouroux, P.; Guerin, N.; Vincent, P. A. Phys. Rev. Lett. 2002, 89 (21), 213902. (7) Alù, A.; Silveirinha, M. G.; Salandrino, A.; Engheta, Phys. Rev. B 2007, 75 (15), 155410. (8) Molesky, S.; Dewalt C. J.; Jacob Z. Opt. Express 2013, 21 (S1), A96-A110. (9) Feng, S.; Halterman, K. Phys. Rev. B 2012, 86 (16), 165103. (10) Vassant, S.; Hugonin, J.-P.; Marquier, F.; Greffet, J.-J. Opt. Express 2012, 20 (21), 23971-23977. (11) Vassant, S.; Archambault, A.; Marquier, F.; Pardo, F.; Gennser, U.; Cavanna, A.; Pelouard, J. L.; Greffet, J.-J. Phys. Rev. Lett. 2012, 109 (23), 237401. (12) Schuller, J. A.; Barnard, E. S.; Cai, W.; Jun, Y. C.; White, J. S.; Brongersma, M. L. Nat. Mater. 2010, 9 (3), 193-204. (13) Zheludev, N. I.; Kivshar, Y. S. Nat. Mater. 2012, 11 (11), 917-924. (14) Kildishev, A. V.; Boltasseva, A.; Shalaev, V. M. Science 2013, 339 (6125), 1232009. (15) Law, S.; Podolskiy, V.; Wasserman, D. Nanophotonics 2012, DOI:10.1515/nanoph-2012-0027. (16) Ford, G. W.; Weber, W. H. Phys. Rep. 1984, 113 (4), 195-287. (17) Jun, Y. C.; Briggs, R. M.; Atwater, H. A.; Brongersma, M. L. Opt. Express 2009, 17 (9), 7479-7490. (18)

me , D. . Vernon, . C.

ulvaney, P.; Davis, T. J. Nano Lett. 2010, 10 (1), 274–278.

(19) Baudrion, A.-L.; Perron, A.; Veltri, A.; Bouhelier, A.; Adam, P.-M.; Bachelot, R. Nano Lett. 2013, 13 (1), 282–286. (20) Kavokin, A.; Baumberg, J.; Malpuech, G.; Laussy, F. Microcavities 2007, Oxford University: New York. (21) Jun, Y. C.; Gonzales, E.; Reno, J.; Shaner, E.; Gabbay, A.; Brener, I. Opt. Express 2012, 20 (2), 1903-1911. (22) Jun, Y. C.; Brener, I. J. Opt. 2012, 14 (11), 114013. (23) Shaner, E. A.; Cederberg, J. G.; Wasserman, D. Appl. Phys. Lett. 2007, 91 (18), 181110. (24) Chen, H.-T.; Padilla, W. J.; Zide, J. N. O.; Gossard, A. C.; Taylor, A. J.; Averitt, R. D. Nature 2006, 444 (7119), 597-600.

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Supporting Information Available Wafer structure, field enhancement in an ENZ nanolayer, anti-crossing in the absorption spectra, strong coupling in an n+ InAs system, capacitance measurement and calculation, Figures S1–S7. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgements This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. YCJ acknowledges the support from the MSIP (Ministry of Science, ICT&Future Planning), Korea, under the ITRC (Information Technology Research Center) support program (NIPA-2013-H0301-13-1010) supervised by the NIPA (National IT Industry Promotion Agency).

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Figure 1.

Figure 1 (a) Schematic of the metamaterial-semiconductor hybrid structure. The semiconductor substrate includes a doped semiconductor ENZ nanolayer. (b) Dielectric constant of n+ GaAs (ND ~ 5.5 x 1018 cm-3) as a function of frequency, measured by ellipsometry. We have field intensity enhancement due to the boundary condition ε1E1┴ = ε2E2┴ at the n+ GaAs / GaAs interface for the normal electric field component. If ε2 goes to zero, E2┴ should diverge. But the electric field magnitude still remains finite due to the non-zero imaginary part of the dielectric constant. Nevertheless, we can still obtain very large field enhancements near the ENZ frequency.

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Figure 2.

Figure 2 (a) Geometry of the 3-layer structure that we study for the dipole emission and dispersion relation. The ENZ layer is n-doped GaAs with a doping density ND ~ 5.5 x 1018 cm-3. (b) Dispersion relations of ENZ modes for several different t’s. (c) Dipole emission power spectra for different ENZ thicknesses (t) and the emittersurface distances (d). The color scale is dipole emission power density normalized by free space emission. White lines are the ENZ mode dispersion curves for each t.

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Figure 3.

Figure 3 (a) Scanning electron microscope image of gold SRR MM (Scale factor 1, Red scale bar: 2 μm). L = 720 nm, W = 130 nm,

= 110 nm, and the period is 1.4 μm for Scale factor 1. (b) FTIR transmission spectra for

a series of SRR scale factors (Wafer A, ND ~ 5.5 x 1018 cm-3). Clear anti-crossing was observed at room temperature. (c) Numerically simulated transmission spectra. (d) FTIR transmission spectra (Wafer B, ND ~ 2.2 x 1018 cm-3). Due to a lower doping level, the ENZ frequency shifted to a lower frequency (~ 530 cm-1). Anticrossing was observed again. (e) Numerically simulated transmission spectra.

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Figure 4.

Figure 4 Schematic of the electrically tunable MM/ENZ hybrid device. Optical coupling between MM and ENZ layers can be electrically tuned by depleting the doped semiconductor ENZ layer.

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Figure 5.

Figure 5 (a) Scanning electron microscope and optical microscope images of the device. The MM layer (1 mm x 1 mm in size) is connected to the metal gate by electrical bus lines (Red scale bar: 5 μm). (b) FTIR transmission measurement with biases. With negative biases, we deplete carriers in the ENZ layer, and the spectrum becomes more symmetric (i.e. returning back to the original position of the isolated/uncoupled MM resonance). Two arrows correspond to the two branches of anti-crossed modes. In our MM device, we have a patterned (split-ring resonator) metal contact. So, the depletion region in the actual device should grow in all directions around the patterned metal gate (generating a cylindrical or spherical region of carrier depletion). This complication may have created a smaller effect on electrical tuning than expected from optical spectra (Fig. 3).

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