Abstract

Polaritons in polar-dielectrics and van der Waals (vdW) materials provide a channel for strong photon confinement. Precise control of their propagation could lead to deep sub-wavelength photonic devices. Here, we report negative refraction of hybrid surface phonon-hyperbolic polaritons (SPh-HP) at the interface of two-dimensional (2D) van der Waals layers such as hexagonal boron nitride (h-BN) and 3D semiconductors such as germanium and silicon carbide. These hybrid polariton modes have naturally negative group velocity arising from the intrinsic Type-I hyperbolicity of h-BN resulting in negative refraction at interfaces with positive group velocity. Using this phenomenon, we demonstrate an in-plane superlensing effect in an ultrathin (~10 nm) slab with spatial confinement of long Infrared wavelengths to below 200 nm focal spots. We further demonstrate electrical tunability of the superlens by controlling the Fermi level of graphene, thereby offering potential for miniaturized infrared to THz modulators, photodetectors as well as logic switches.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Light can be confined and manipulated at the nanoscale using hybrid light-matter excitations known as polaritons. Particularly, surface phonon polaritons (SPhPs) supported on polar crystals [1–4], such as silicon carbide (SiC), hexagonal boron nitride (h-BN) and quartz, have inherent long lifetimes (~1ps) and low optical loss [5], in contrast to surface plasmon polaritons (SPPs) in metals, which are lossy with a lifetime of ~10 femtoseconds [6]. The recent emergence of polaritons in atomically-thin van der Waals materials (vdWs) enables new avenues for light-matter interactions across visible to terahertz spectral ranges [7–11], with unique optical control beyond what can be achieved using traditional plasmonic counterparts [12,13]. This is mainly due to the “all-surface”, passivated nature of the van der Waals materials allowing them to be stacked arbitrarily and the atomically-thin nature making them amenable to be easily tuned by electric fields and changes in the dielectric environment. For instance, the tunable and highly confined SPPs which were predicted and later observed in graphene and black phosphorus (BP) [14–16], have led to applications such as molecular sensing, tunable THz generation and electro-optical light modulation [17–20]. More recently, rapid progress in fabrication techniques [21,22] of vdW heterostructures has also led to diverse reports of 2D materials based nanophotonic and optoelectronics [23], such as tunable saturable absorbers [24] formed by heterostructuring of Bi2Te3-graphene, as well as broadband photodetectors based on BP [25]. As opposed to the lossy plasmon-polaritons, phonon polaritons are less lossy due to the charge neutral and bosonic nature of the phonons and therefore have longer propagation lengths and lifetimes. h-BN which is isostructural to graphene but an electronic insulator is a natural hyperbolic materials due to its layered structure and supports hyperbolic phonon (HP) polaritons which have lower losses as compared to SPPs yet remain highly confined in thin flakes of h-BN [26,27]. These volume confined HP polaritons allow for ray-like propagation, exhibiting high quality factors and hyperlensing effects [28,29]. Confinement factors (β = λ0p), (defined as the ratio of polariton wavelength to free space photon propagation wavelength), of over 50 have been achieved for both graphene plasmons [30] and h-BN phonon polaritons [31,32], which means that infrared light (5-20 ums) can be modulated with wavelength down to a few hundred nanometers. This ability to localize light to achieve deeply subwavelength control is of great technological significance as it allows the integration of the merits of both electronics and photonics at high device density into a single technology [33]. However, direct coupling of polaritons requires artificially patterning deep-subwavelength resonators with precise edges and uniformity to achieve high quality factors and minimize scattering using advanced lithography techniques. Owing to the limits of lithography and patterning, this approach renders optical control, such as negative refraction, superlens and wave-guiding difficult to realize in vdW materials systems. Thus far there have been limited number of studies reporting control over propagation of graphene plasmons [14,34] and h-BN phonon-polaritons [35].

Negative refraction and superlensing have been the most appealing physical phenomena pertaining to the field of nanophotonics and metamaterials over the past decade [36–40], which have now been predicted and demonstrated in a variety of metamaterial systems, across the electromagnetic spectrum [41–43]. In-plane negative refraction of SPPs was first predicted [44] and experimentally demonstrated [43] in metal-dielectric-metal (MIM) metamaterial structures and in photonic crystals [39,41]. However, the damping by optical losses in plasmonic modes and the complexity of fabrication for the dielectric photonic crystals has saturated progress in those systems [6]. More recently, the prediction of in-plane negative refraction in vdW materials by Lin et al. [45] has renewed interest in this area, particularly since the low-loss, highly confined and tunable nature of the vdW polaritons presents new opportunities for applications such as near-field nano-imaging, and tunable, integrated optics. In their proposed lateral heterojunction of h-BN slab and graphene monolayer, the h-BN slab supports HP polariton with negative group velocity. When travelling to the other side of the interface i.e graphene, the refracted graphene SPP stays on the same side of the normal breaking Snell’s law, resulting in negative refraction in-plane. However, the HP modes are volume confined in h-BN slab, whereas the SPP modes are surface confined in graphene monolayer, two types of mode-profile are different and the confiments are mismatch result the reflection and scattering at h-BN-graphene interface. To overcome this mismatch and loss owing to scattering, a more efficient design for polariton volume confinement is desired. Further, in order to make practical device applications from negative refraction of vdW polaritons, a field or carrier density dependent modulating device platform must be conceptualized and engineered.

In this letter, we theoretically propose and computationally demonstrate a platform based on a 2D/3D vdW heterojunction of graphene and h-BN on polar, bulk SiC substrate, where all polariton modes are volume confined within same thickness, which reduces the reflection at the negative and positive medium interfaces. First, due to the strong coupling of h-BN’s HP and SiC’s SPhP, h-BN slab on SiC supports hybrid SPh-HP modes, and features natural negative group velocity because of the intrinsic Type-I hyperbolicity in h-BN. To demonstrate the proof of concept, the hybrid SPh-HP when transmitted to a high-index dielectric such as Ge with same thickness, the Ge slab on SiC supports dielectric tailored SPhP mode with positive velocity, the sign flipping of group velocity between two modes leads to the negative refraction at the h-BN/Ge interface. Second, we using graphene spaced from SiC by an insulator instead of Ge to construct an graphene/gap/SiC heterostructure, due to the strong coupling between graphene’s plasmon (g-) and SiC’s surface phonon, a hybrid mode called as g-SPhP confined in the gap spacer. The gap spacer structure is realistic and has been experimentally demonstrated and grpahene and h-BN based van der Waals heterostrutures. The strong coupling with SiC’s phonon also reduces the loss of pure graphene plasmons [45]. Further, since both graphene and SiC feature negative permittivities at these infrared frequencies, g-SPhP is analogous to a gap waveguide mode in a typical metal-insulator-metal (MIM) structure, thus the mode profiles of g-SPhP can provide superior overlap with hybrid SPh-HP mode in h-BN/SiC. Finally, by exploiting the electrical tunability of graphene, we demonstrate an in-plane gate-tunable superlens whose focal lengths and imaging resolutions can be actively tuned via electrostatic gating of graphene.

2. In-plane negative refraction in h-BN-Ge/SiC heterostructure

Figure 1(a) shows the schematic of in-plane negative refraction between two hybrid phonon polaritons. The heterostructure is composed of two slabs (h-BN and Ge) on the SiC substrate. The dielectric function of SiC and h-BN are shown in Fig. 1(c), both are extracted by a Lorentz (TL) oscillator model for polar dielectric crystals [5]:

 figure: Fig. 1

Fig. 1 (a) Schematic of in-plane negative refraction between hybrid surface phonon-hyperbolic phonon polariton (SPh-HP) and tailored surface phonon polariton (SPhP). (b) Left part: the hybrid SPh-HP has negative group velocity, and the Ez distributions (mushroom pattern) inside and outside h-BN have opposite sign; Right part: tailored SPhP has positive group velocity, the Ez distributions inside and outside Ge have same sign. (c) Real permittivity for h-BN and SiC, the overlapping frequencies between h-BN’s first reststrahlen band (εx/y>0, εz<0) and SiC’s reststrahlen band (ε<0) from 797 to 835 cm−1. (d) Type-I hyperbolic responses (solutions for Eq. (1) at ω = 800 cm−1 (blue solid curve) and 820 cm−1 (blue dashed curve).

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ε(ω)=ε(1+ωLO2ωTO2ωTO2ω2iωγ)

The polariton modes can be supported over a spectral range referred to as the ‘Reststrahlen’ band, between the longitudinal (LO) and transverse optic (TO) phonon frequencies, where the permittivity is negative (ε<0). For SiC, ε = 6.56, ωLO = 973 cm−1, ωTO = 797 cm−1, and γ = 4.76 cm−1. Specifically, h-BN is a natural hyperbolic material, meaning that the components of its permittivity are the same in the basal plane (εt≡εx = εy) but have opposite signs (εt εz<0) in the direction normal to the basal plane (εz). For the in-plane permittivity, ωTOx = 1370 cm−1, ωLOx = 1610 cm−1, and γx = 4.87 cm−1; for the out-plane permittivity, ωTOz = 780 cm−1, ωLOz = 835 cm−1, and γz = 2.95 cm−1. Owing to this anisotropic feature, the group velocity of HP polaritons is negative within h-BN’s first Reststrahlen band (εx/y>0, εz<0), named as Type-I polaritons [31,46].

Here, we focus on the overlapping frequencies between h-BN’s first Reststrahlen band and SiC’s Reststrahlen band from 797 to 835 cm−1. Thus, due to the strong coupling of HP mode in h-BN and the SPhP mode on SiC, the left h-BN/SiC region in Fig. 1(a) supports hybrid SPh-HP mode. The mode analysis (performed using 2D solver Lumerical Mode Solutions) shows that such hybrid SPh-HP neutralizes the characters of two precursor modes with the resulting z-components of the electric field distributions (Ez) (visualized by the left mushroom pattern in Fig. 1(b)) having opposite sign inside and outside h-BN, at the air/h-BN interface. The Ez distribution inside h-BN is more based on the HP mode, while the Ez distribution outside in air is more like SiC based SPhP mode. However, the part inside the medium plays a more important role, because most y-components of the magnetic field distributions (Hy) are confined inside h-BN (Fig. 3(c)). Therefore, the group velocity of SPh-HP remains negative, attributed to the Type-I hyperbolicity [26] of h-BN’s HP polaritons:

kz2/εx+(kx2+ky2)/εz=(ω/c)2

For h-BN’s first reststrahlen band εx = εy, Re(εx,y)<0, Re(εz)>0. The solutions of Eq. (2) in this case are plotted in Fig. 1(d) at 800 cm−1 (solid curve) and 820 cm−1 (dashed curve), respectively. This hyperbolic response would lead to exotic behavior in the alignment of the wavevector k (green arrow in Fig. 1(d)) and Poynting vectors P (red arrow in Fig. 1(d)). The Poynting vector P is orthogonal to the wavevector k, and has a fixed angle relative to the z axis. As a result, the phase velocity (vp//k) and the group velocity (vg//P) have opposite directions with respect to the x axis (Fig. 1(b)), and hence lead to negative group velocity for SPh-HP mode in left h-BN region. Furthermore, with increasing wavenumber ω, the decrease of k (blue arrow in Fig. 1(d)) results in a larger periodicity (smaller effective indices) of the HP rays (dashed red arrows in Fig. 1(b)), also indicating that the group velocity vg = ∂ω/∂kx of SPh-HP modes have negative sign (decreasing slop). As seen in Fig. 1(b), the right polariton is exactly a dielectric tailored SPhP mode excited on SiC surface, but remains mainly confined into the Ge slab due to the higher index, both the Ez distributions inside and outside Ge keep the same sign, and the group velocity is positive since both P and k are at the same direction. Next we optimize our structure’s geometric parameters to configure SPh-HP and tailored SPhP modes to satisfy both the momentum (keff) and the energy (ω) matching at the same thickness, to minimize reflection and allow negative refraction.

The momentum and the energy match between two polaritons are important for negative refraction, which are associated with their dispersion responses (ω vs kx in the direction of propagation). We analytically derive a transfer matrix formalism (see Appendix A.) to trace the dispersions of polaritons in above h-BN-Ge/SiC heterostructure. To better demonstrate the coupling between SPh-HP, we first compare the dispersions of the two precursor polaritons (SPhP and HP), respectively. As shown in Fig. 2(a), the multi-order HP modes have negative dispersion (decreasing slop), indicating that the group velocity vg = ∂ω/∂kx and the phase velocity vp = ω/kx have opposite sign. Figure 2(b) shows the dispersion of SPhP mode on bulk SiC, the mode has positive dispersion (increasing slop), indicating positive vg. Upon comparing the magnitude of kx in Figs. 2(a) and 2(b), we observe that there is difference of roughly an order of magnitude which means that the negative refraction cannot be achieved between these two modes due to the large momentum mismatch. As shown in Fig. 2(c), only the first-order HP mode is strongly coupled with the SPhP mode for the complete h-BN/SiC heterostructure, and high-order HP modes disappear because of the symmetries mismatch of the modal fields with SPhP [47]. The hybrid SPh-HP mode neutralizes the characters of both precursor polaritons resulting in the dispersion first climbing up in frequency and then declining as a function of kx. The left part with increasing slop has positive vg, in which the SPhP mode plays a dominant role; while the right part with decreasing slop has negative vg, in which HP mode is dominant. Likewise, the SPhP mode needs to the tailored and confined to increase the group velocity by placing a high index dielectric above it. Figure 2(d) shows the dispersion of tailored SPhP modes in Ge/SiC heterostructure, the high-index Ge slab increases the wave-vector kx (or confine factor neff) of the SPhP mode in Fig. 2(b), so that the tailored SPhP mode can realize momentum match with the negative HP-SPhP mode in Fig. 2(c).

 figure: Fig. 2

Fig. 2 Theoretically calculated dispersions of the polariton modes, (a) hyperbolic phonon polaritons (HP) in the h-BN (400 nm)/SiO2 heterostructure, (b) surface phonon polariton (SPhP) on SiC substrate, (c) hybrid SPh-HP mode in the h-BN (500 nm)/ SiC heterostructure, and (d) tailored SPhP in the Ge (500 nm)/SiC heterostructure. Inset is the Ez field mode profile.

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The effective index (same as confinement factor β), defined as neff = kx/k0, represents the reduced polariton wavelength λp relative to the free-space value λ0. As expected, the neff of the hybrid SPh-HP modes are dependent on the thickness of the h-BN slab, i.e., dh-BN = 300, 400, 500, and 600 nm, as plotted by the blue triangle curves in Fig. 3(b), respectively. We observe that, at a specific wavenumber, the neff decreases with the increase of dh-BN. In contrast, for Ge a larger thickness dGe results in a larger neff, as shown by the purple circle curves in Fig. 3(b). We chose an iso-thickness intersection point (ω = 801cm−1, and neff = 1.98) for dh-BN = dGe = 600 nm to demonstrate the negative refraction phenomenon. Regarding the reflection for in-plane polariton waves at a discontinuous interface, it is widely accepted and well known that the overlap of mode profiles can be used to estimate the reflectance qualitatively [44,45]. We use this approach to estimate and minimize reflectance in our structure. To suppress the reflection, one way is to maximize the mode overlap between the incident and transmitted mode profiles at the interface. As shown in Figs. 3(c) and 3(d), the magnetic (Hy) distributions of two polaritons overlap very well thereby minimizing reflection, both modes can be considered as volume confined modes.

 figure: Fig. 3

Fig. 3 (a) 2D-view (y = 0 plane) of the complete heterostructure in Fig. 1(a). (b) Dependence of the real effective indices neff of the hybrid SPh-HP (blue curves) and the tailored SPhP (purple curves) modes on the wavenumber with different slab thicknesses, purple stars are intersection points. Mode profile of the magnetic Hy field for (c) hybrid SPh-HP mode within h-BN region and (d) tailored SPhP mode within Ge region at the neff matched intersection point (ω = 801cm−1, and neff = 1.98), the thicknesses of both h-BN and Ge slabs are 600 nm.

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Three-dimensional finite difference time domain simulation (Lumerical FDTD 3D solver) is implemented to validate above predictions. The schematic of the negative refraction is shown in Fig. 4(a), a mode source is used to launch the SPhP-HP mode from the left side with an angle θ = 30° to the normal plane at ω = 801 cm−1. The perfectly matched layer boundary conditions are employed in all x, y and z directions, the mesh size is set as 5nm (much smaller than the incident wavelength 1248nm) in all directions to guarantee accurate results. Figure 4(b) shows the cross-sectional view of electric fields (Ez) above the SiC substrate at xz plane. The electric field patterns inside and outside h-BN have opposite signs (π phase shift) in the left h-BN region. This is due to the fact that the electric field distribution is more characteristic of the h-BN based HP mode with negative vg, in which the energy flows (P, red arrow in Fig. 4(c)) and the wave-vector (k, green arrow in Fig. 4(b)) are in the opposite direction. In contrast the field distribution outside is more like the SiC based SPhP mode with positive vg, i.e. the wavevector and the power flow are in the same direction. However, a majority of the energy flow occurs confined inside h-BN with minimal leakage outside as verified by the Hz distribution in Fig. 3(c). This renders the overall sign of vg for hybrid SPhP-HP mode negative.

 figure: Fig. 4

Fig. 4 (a) Schematic of the in-plan negative refraction in lateral h-BN/Ge heterostructure above SiC substrate, the hybrid SPh-HP is launched by a mode source from h-BN side with incident angle 30°. Cross sectional view of (b) Ez field distribution and (c) Poynting vectors at xz plane. Top view of in-plan Ez field distributions (d and f) and Poynting vectors (e and g) outside (300 nm above top surface) in air and inside (300 nm below top surface) in h-BN/Ge, respectively. The power flow (wavevector) directions are marked by the red (green) arrows.

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In the right region, the electric field distribution (Ez) patterns both inside and outside Ge have the same sign, the power flow (P, red arrow in Fig. 4(c)) and the wave-vector (k, green arrow in Fig. 4(b)) are in the opposite direction, which indicates the positive group velocity of such of tailored SPhP mode. Figures 4(d) and 4(f) are the top view of in-plan Ez distributions outside (300 nm above top surface) in air and inside (300 nm below top surface) h-BN/Ge, respectively. The mirror symmetric Ez distribution along the middle interface (between h-BN and Ge) is a clear indication of in-plane negative refraction between two polaritons. At the interface, the outside Ez has π phase shift between two polaritons, but the inside Ez has zero phase shift. Figures 4(e) and 4(g) show the corresponding Poynting vectors for negative refraction outside and inside two slabs. These results concur and confirm with our analysis: first, negative refraction arises from the sign flipping of vg at the interface between two materials (h-BN and Ge); second, more power is confined and flows inside two slabs as compared to the power leaking out in air, guaranteeing the negative refraction even for the leaked out modes, despite the energy flows (Fig. 4(c)) in opposite directions across the middle (h-BN/Ge) interface; finally, the strong overlap between the two mode profiles minimizes the reflection at the interface.

3. Gate-tunable superlens in h-BN-graphene/SiC heterostructure

The above example shows the possibility of negative refraction in-plane using hybrid SPh-HP modes and tailored SPhP modes. However, as evident from the figures, the film thicknesses of h-BN and Ge are large ~500-600 nm and hence the confinement factors are small. The key application of negative refraction is the superlens and the key advantage of van der Waals materials is ultrahigh quality at small thicknesses resulting in superior confinement. The key idea is to take full advantage of these attributes of the material and physical phenomena for deeply subwavelength lensing of THz radiation with nanometer resolution. The first and simplest approach is reducing the thickness of h-BN. By decreasing the h-BN thickness (dh-BN) down to 10 nm, the confinement factors neff of hybrid SPh-HP modes can go up to 100. However, it is non-trivial to increase confinement factors of the tailored SPhP modes since as the thickness of the high index dielectric (Ge) is decreased the mode leaks out and confinement factors reduce.To overcome such a large index mismatching, we use monolayer graphene instead of Ge to construct the right side of the lateral-junction, the schematic of a superlens configuration is shown in Fig. 5(a).

 figure: Fig. 5

Fig. 5 (a) Schematic of the superlens hybrid structure. (b) The dispersions of two type polaritons, g-SPhPs are electric tunable by controlling graphene’s Fermi level, both the thickness of h-BN and Gap is 10nm. Mode profiles of the magnetic Hy field for (c) hybrid SPh-HP mode within h-BN region and (d) g-SPhP mode within graphene region at the frequency of ω = 817 cm−1, the graphene Fermi level is μc = 0.16 ev, corresponds to a momentum matched neff = 80. (e) The figure of merit (γ−1) for graphene polaritons on different substrates, μc = 0.3 ev.

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Graphene plasmons are especially attractive for the proposed application, because of their comparable light confinement with h-BN polaritons. Here the graphene is free-standing above the top gate, while 10 nm suspension gap may be a practically challenging design, we propose that a dielectric spacer would be effective as well, similar as the recent work by suspending graphene with few-nanometer-thick Al2O3 on metal substrate [48]. Due to the strong coupling between graphene’s plasmon and SiC’s surface phonon, a hybrid mode confined in the gap spacer, called as g-SPhP. The dispersion of the hybrid SPh-HP and g-SPhP are shown in Fig. 5(b), high confinement factors (>100) are observed for both polaritons at the thicknesses of h-BN and gap height are set as 10nm. The dispersion of g-SPhP can be electrically controlled by tuning the Fermi-level of graphene to satisfy the energy and momentum match with hybrid SPh-HPs at different energies/frequencies or wavenumbers. For instance, at the frequency of ω = 817 cm−1, we obtain an effective index neff = 80 for both SPhP-HP and g-SPhP, at graphene Fermi level μc = 0.16 eV. The mode profile of g-SPhP (Fig. 5(d)) is analogous to a gap waveguide mode in a typical metal-insulator-metal (MIM) structure since both graphene and SiC feature negative permittivities at these THz frequencies. In comparison to a prior study of negative refraction between volume confined h-BN HP polaritons and surface confined graphene plasmon polaritons [45], where reflection existed at the interface arising from the mode confinement mismatch (from volume to surface) between the two polaritons, here this gap volume confined g-SPhP can provide superior mode-profile overlap with hybrid SPh-HP mode in h-BN/SiC (Fig. 5(c)). In addition, graphene plasmons hybridized with SiC’s surface phonons may largely reduce the loss. The figure of merit (FOM) for polaritons propagation damping, represented by the magnitude of the ratio γ-1 = Re(kp)/Im(kp), also indicates the propagation losses. As seen in Fig. 5(e), γ−1 for graphene plasmon on SiO2 (black line) is limited to 15, because of SiO2 suffering large optical loss at infrared frequencies. Due to the same honeycomb lattice, h-BN has been demonstrated as the best encapsulation layer of graphene, γ−1 of graphene plasmon on h-BN (blue line) can achieve 50. Here, in our heterostructure of graphene on SiC, γ−1 of such a hybid g-SPhP modes (purple line) extend to 100, which is closed to the low-loss graphene plasmon at cryogenic temperatures [30]. The calculatedγ−1 for hybrid SPh-HP is 25, near to the pure low-loss h-BN HP on SiO2 (γ-1~20) that have been experimentally reported at Mid-infrared frequencies [1,27].

To illustrate the superlens effect, a dipole source is placed 400 nm away from the h-BN-Graphene interface at the left h-BN side to launch the SPh-HP mode. Experimentally, such a mode can be launched using a metallic tip such as the one used in near-field scanning optical microscopy (NSOM). The top-view of the in-plane |E|2 and Hy fields are shown in Fig. 6(a) and 6b. As can be clearly seen, all angle negative refraction occurs at the h-BN-Graphene interface, resulting in the g-SPhP mode in the right side graphene region showing a mirror image of the dipole (Hy of Fig. 6(b)). The normalized Hy field (Fig. 6(c)) at the image plane (vertical cut by the black dashed line in Fig. 6(b)) shows the resolution of the superlens. With a FWHM of the peak equaling 125 nm for a free space wavelength of ~12240 nm (817 cm−1) the superlens is highly effective and suggests deep-subwavelength focusing. An interesting aspect of using graphene and polaritons mode in graphene for deep sub-wavelength focusing is the ability to electrically tune graphene. We demonstrate this in our in-plane superlens by changing the Fermi-level of graphene to tune the lens properties. For instance, if keep the frequency fixed at ω = 817 cm−1, but change the graphene Fermi level (from 0.16 ev to 0.41 ev), we can electrically tune the focal length of the superlens, as can be compared in Figs. 6(a) and 6(d). Changing (increasing) the Fermi level changes (decreases) the effective index of graphene and because of this the transmission from high-index (nh-BN = 80) to low-index region (ngra = 42), the refract angle θt increasing leads to the decrease of focal length f. This is exactly opposite to the case of conventional lenses in which the larger index material results in smaller focal length of the lens. This phenomenon demonstrates the opposite working rules of negative refraction. The totally focal intensity (Fig. 6(d)) is lower than that in Fig. 6(a), due to the momentum (index) mismatch at the interface which results in loss of some light via reflection. Further, the lower effective index of graphene (ngra = 42) also increases the refracted wave period (Hy in Fig. 6(e)), and correspondingly decreases the resolution of the superlens (FWHM = 178 nm).

 figure: Fig. 6

Fig. 6 (a, b) The distributions of in-plan |E|2 and Hy field (5nm above SiC substrate) to illustrate the superlens effects, at the frequency of ω = 817 cm−1, the graphene Fermi level is μc = 0.16 ev, corresponds to a momentum matched neff = 80 for both polaritons. (c) The normalized Hy field at the image plane (black dashed line in (d)), the full width at half maximum (FWHM) is 125 nm. (d, e) The distributions of in-plan |E|2 and Hy field, and f) the normalized Hy field at the image plane, ω = 817 cm−1and μc = 0.41 ev.

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In addition, having independent control over both illuminated frequency and the Fermi level, for the intersection points in Fig. 5(b), the momentum and energy matching can be achieved for multiple values which allows tuning of the superlens resolution over a wide range, as shown in Fig. 7, the decrement of μc results a larger index of g-SPhP, thereby increases the resolution at the imaging plane. Overall, due to the combined advantages of tunability, low loss, and superior spatial confinement provided by these polaritons at a 2D/3D heterointerface, our proposed superlens platform is promising for highly miniaturized device design for active control of infrared-THz light in the deep sub-wavelength limit. Thus, its primary potential for applications lies in integrated flat optics, where novel wave guiding and optical switching is greatly desired for on-chip photonic circuits.

 figure: Fig. 7

Fig. 7 The distributions of in-plan |E|2 and Hy field to illustrate the electrical tunability of the superlens. (a) ω = 824cm−1, μc = 0.41ev, corresponds to a momentum matched neff = 45 for both polaritons, the full width at half maximum (FWHM) is 200 nm. (b) ω = 817 cm−1, μc = 0.16 ev, neff = 80 and FWHM = 125 nm. (c) ω = 811 cm−1, μc = 0.11ev, neff = 105 and FWHM = 102 nm.

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4. Conclusion

In summary, we have theoretically demonstrated the strong-coupling between HP and SPhP polaritons in h-BN/SiC heterostructure. We have used the hybrid SPh-HP modes featuring negative group velocity, in combination with a dielectric tailored SPhP to achieve in-plane negative refraction in a lateral heterojunction of h-BN/Ge supported on SiC. We extend this concept into a practical superlens design of h-BN-graphene/SiC heterojunction that is ultrathin in nature (~10 nm) and has beyond-diffraction-limited resolution (more than 80 times smaller than the incident wavelength, FWHM<200nm) for THz frequencies. This is primarily achieved due to the superior spatial confinement of hybrid polaritons in h-BN and graphene. Finally, we have shown that the focal length and the resolution of the superlens can be electric tuned via the graphene Fermi-level. Our proposed superlens platform has potential applications ranging from infrared nano-imaging to highly miniaturized photonic circuits.

Appendix A Dispersions of the hybrid phonon-plasmon polaritons

We follow an approach similar to the one used by Huber [15] to trace the dispersions of interface polaritons in black phosphorus heterostructures by the imaginary part of the Fresnel reflection coefficient rp. Because the modes of polaritons are the singularity poles in the coefficient of rp, we can visualize the dispersion of these hybridized polaritons via a false-color plot of Im(rp) as a function of kx and ω [45,49]. The coefficient rp for a multilayered system can be calculated using the transfer matrix formalism. In our case, as shown in Fig. 8, three layers are included, layer 1 (z>d, air), layer 2 (0 <z<d, h-BN, Ge or air Gap), and layer 3 (-H<z<0, SiC substrate), where d is the thickness of layer-2 slabs. A matrix M can be used to describe the interaction of electromagnetic waves in the heterostructure:

 figure: Fig. 8

Fig. 8 The schematic of three-layer heterostructure, layer 1 (z>d, air), layer 2 (0 <z<d, h-BN, Ge or air Gap), and layer 3 (z<0, SiC substrate), where d is the thickness of layer-2 slabs. The following derivations give the dispersions of hybrid SPh-HP, tailored-SPhP and g-SPhP, respectively.

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M=[MaaMabMbaMbb]=R1,2T2R2,3

Where matrices Ri,j describe the reflection at each interface between layers i and j, and the matrix T2 describes the propagation of the electromagnetics wave through layer 2. The matrices R1,2, R2,3 and T2 are given by

R1,2=1t1,2[1r1,2r1,21],R2,3=1t2,3[1r2,3r2,31],T2=[eikz2d00eikz2d]

Where, ri,j (ti,j) are the Fresnel reflection (transmission) coefficients for a single interface between two infinite half spaces; kzi is the out-of-plane k-vector of the electromagnetic wave in layer i, such as:

kz1=ω2c2ε1kx2,kz2=ω2c2ε2xkx2ε2xε2z,kz3=ω2c2ε3kx2

Where ε1 and ε3 are the relative permittivity of layer 1 and 3, respectively. Since h-BN is a hyperbolic material, its relative permittivity is characterized by an anisotropic diagonal tensor [ε2x, ε2y, ε2z], and the reflection and transmission coefficients are expressed as,

ri,j=εxjkziεxikzjεxjkzi+εxikzj,ti,j=2εxjkziεxjkzi+εxikzj

The reflection coefficient rp for the whole heterostructure is then given as a ratio of two matrix components of M in Eq. (3),

rp=MbaMaa

Finally, the dispersions of hybrid SPh-HP mode in h-BN/SiC region are visualized by using a false-color map of rp, as shown in Fig. 2(c) of the main text. Similarly, if layer 2 is set as Ge material, we can solve the dispersion of tailored SPhP mode in the Ge/SiC region, as shown in Fig. 2(d). For g-SPhP in graphene/gap/SiC materials, the surface conductivity (σ) of graphene would largely change the reflection and transmission coefficients at the interface between layer-1 and layer-2, thus the corresponding derivations for r1,2 and t1,2 should be replaced by,

r1,2=εx2kz1εx1kz2+σkz1kz2/ωε0εx2kz1+εx1kz2+σkz1kz2/ωε0,t1,2=2εx2kz1εx2kz1+εx1kz2+σkz1kz2/ωε0

The dispersions of g-SPhP are electric tunable by controlling the graphene’s Fermi level, as shown in Fig. 5(b) of the main text.

Appendix B Optical conductivity of graphene

In this work, we use random phase approximation (RPA) to model the optical conductivity of graphene [50,51]. The conductivity is given by:

σ(ω)=e2EFπ2iω+iT1+e242(θ(ω2EF)+iπlog(|ω2EFω+2EF|))

where e is the unit electric charge, ℏ is the reduced Planck constant, θ denotes a step function, EF is the graphene Fermi level, and T=300 K. The electron relaxation time τ is determined by τ=μEF ∕evF2, wherein the carrier mobility is assumed as μ=104cm2V−1s−1 and the vF ≈ 106m/s is the Fermi velocity.

Funding

Fund of China Academy of Engineering Physics, National Natural Science Foundation of China (11374318 and 11674312), Penn Engineering, U. S. Army Research Office (W911NF1910109).

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References

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2018 (10)

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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A. M. Dubrovkin, B. Qiang, H. N. S. Krishnamoorthy, N. I. Zheludev, and Q. J. Wang, “Ultra-confined surface phonon polaritons in molecular layers of van der Waals dielectrics,” Nat. Commun. 9(1), 1762 (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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V. E. Babicheva, S. Gamage, L. Zhen, S. B. Cronin, V. S. Yakovlev, and Y. Abate, “Near-field Surface Waves in Few-Layer MoS2,” Acs Photon. 5(6), 2106–2112 (2018).
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B. Yao, Y. Liu, S. W. Huang, C. Choi, Z. Xie, J. F. Flores, Y. Wu, M. Yu, D. L. Kwong, and Y. Huang, “Broadband gate-tunable terahertz plasmons in graphene heterostructures,” Nat. Photonics 12(1), 22–28 (2018).
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Y. Kurman, N. Rivera, T. Christensen, S. Tsesses, M. Orenstein, M. Soljačić, J. D. Joannopoulos, and I. Kaminer, “Control of semiconductor emitter frequency by increasing polariton momenta,” Nat. Photonics 12(8), 495 (2018).
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G. X. Ni, A. S. McLeod, Z. Sun, L. Wang, L. Xiong, K. W. Post, S. S. Sunku, B. Y. Jiang, J. Hone, C. R. Dean, M. M. Fogler, and D. N. Basov, “Fundamental limits to graphene plasmonics,” Nature 557(7706), 530–533 (2018).
[Crossref] [PubMed]

P. Li, I. Dolado, F. J. Alfaro-Mozaz, F. Casanova, L. E. Hueso, S. Liu, J. H. Edgar, A. Y. Nikitin, S. Vélez, and R. Hillenbrand, “Infrared hyperbolic metasurface based on nanostructured van der Waals materials,” Science 359(6378), 892–896 (2018).
[Crossref] [PubMed]

D. Alcaraz Iranzo, S. Nanot, E. J. C. Dias, I. Epstein, C. Peng, D. K. Efetov, M. B. Lundeberg, R. Parret, J. Osmond, J.-Y. Hong, J. Kong, D. R. Englund, N. M. R. Peres, and F. H. L. Koppens, “Probing the ultimate plasmon confinement limits with a van der Waals heterostructure,” Science 360(6386), 291–295 (2018).
[Crossref] [PubMed]

Y. Jiang, X. Lin, T. Low, B. Zhang, and H. Chen, “Group-velocity-controlled and gate-tunable directional excitation of polaritons in graphene-boron nitride heterostructures,” Laser Photonics Rev. 12(5), 1800049 (2018).
[Crossref]

2017 (7)

X. Lin, Y. Yang, N. Rivera, J. J. López, Y. Shen, I. Kaminer, H. Chen, B. Zhang, J. D. Joannopoulos, and M. Soljačić, “All-angle negative refraction of highly squeezed plasmon and phonon polaritons in graphene-boron nitride heterostructures,” Proc. Natl. Acad. Sci. U.S.A. 114(26), 6717–6721 (2017).
[Crossref] [PubMed]

A. Woessner, Y. Gao, I. Torre, M. B. Lundeberg, C. Tan, K. Watanabe, T. Taniguchi, R. Hillenbrand, J. Hone, M. Polini, and F. H. L. Koppens, “Electrical 2π phase control of infrared light in a 350-nm footprint using graphene plasmons,” Nat. Photonics 11(7), 421–424 (2017).
[Crossref]

S. C. Dhanabalan, J. S. Ponraj, Z. Guo, S. Li, Q. Bao, and H. Zhang, “Emerging trends in phosphorene fabrication towards next generation devices,” Adv. Sci. (Weinh.) 4(6), 1600305 (2017).
[Crossref] [PubMed]

D. Jariwala, T. J. Marks, and M. C. Hersam, “Mixed-dimensional van der Waals heterostructures,” Nat. Mater. 16(2), 170–181 (2017).
[Crossref] [PubMed]

M. A. Huber, F. Mooshammer, M. Plankl, L. Viti, F. Sandner, L. Z. Kastner, T. Frank, J. Fabian, M. S. Vitiello, T. L. Cocker, and R. Huber, “Femtosecond photo-switching of interface polaritons in black phosphorus heterostructures,” Nat. Nanotechnol. 12(3), 207–211 (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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F. Hu, Y. Luan, M. Scott, J. Yan, D. Mandrus, X. Xu, and Z. Fei, “Imaging exciton–polariton transport in MoSe 2 waveguides,” Nat. Photonics 11(6), 356–360 (2017).
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2016 (7)

D. N. Basov, M. M. Fogler, and F. J. García de Abajo, “Polaritons in van der Waals materials,” Science 354(6309), aag1992 (2016).
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G. X. Ni, L. Wang, M. D. Goldflam, M. Wagner, Z. Fei, A. S. Mcleod, M. K. Liu, F. Keilmann, B. Özyilmaz, and A. H. C. Neto, “Ultrafast optical switching of infrared plasmon polaritons in high-mobility graphene,” Nat. Photonics 10(4), 244–247 (2016).
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J. S. Ponraj, Z.-Q. Xu, S. C. Dhanabalan, H. Mu, Y. Wang, J. Yuan, P. Li, S. Thakur, M. Ashrafi, K. Mccoubrey, Y. Zhang, S. Li, H. Zhang, and Q. Bao, “Photonics and optoelectronics of two-dimensional materials beyond graphene,” Nanotechnology 27(46), 462001 (2016).
[Crossref] [PubMed]

S. C. Dhanabalan, J. S. Ponraj, H. Zhang, and Q. Bao, “Present perspectives of broadband photodetectors based on nanobelts, nanoribbons, nanosheets and the emerging 2D materials,” Nanoscale 8(12), 6410–6434 (2016).
[Crossref] [PubMed]

Z. Sun, A. Martinez, and F. Wang, “Optical modulators with 2D layered materials,” Nat. Photonics 10(4), 227–238 (2016).
[Crossref]

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).
[Crossref] [PubMed]

B. Zhu, G. Ren, B. Wu, Y. Gao, H. Li, and S. Jian, “Nanofocusing of hybrid plasmons-phonons-polaritons in a graphene-hexagonal boron nitride heterostructure,” Opt. Lett. 41(19), 4578–4581 (2016).
[Crossref] [PubMed]

2015 (10)

S. Dai, Q. Ma, M. K. Liu, T. Andersen, Z. Fei, M. D. Goldflam, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, G. C. A. M. Janssen, S. E. Zhu, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Graphene on hexagonal boron nitride as a tunable hyperbolic metamaterial,” Nat. Nanotechnol. 10(8), 682–686 (2015).
[Crossref] [PubMed]

E. Yoxall, M. Schnell, A. Y. Nikitin, O. Txoperena, A. Woessner, M. B. Lundeberg, F. Casanova, L. E. Hueso, F. H. L. Koppens, and R. Hillenbrand, “Direct observation of ultraslow hyperbolic polariton propagation with negative phase velocity,” Nat. Photonics 9(10), 674–678 (2015).
[Crossref]

A. Woessner, M. B. Lundeberg, Y. Gao, A. Principi, P. Alonso-González, M. Carrega, K. Watanabe, T. Taniguchi, G. Vignale, M. Polini, J. Hone, R. Hillenbrand, and F. H. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14(4), 421–425 (2015).
[Crossref] [PubMed]

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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S. Dai, Q. Ma, T. Andersen, A. S. Mcleod, Z. Fei, M. K. Liu, M. Wagner, K. Watanabe, T. Taniguchi, M. Thiemens, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov, “Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolic material,” Nat. Commun. 6(1), 6963 (2015).
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H. Mu, Z. Wang, J. Yuan, S. Xiao, C. Chen, Y. Chen, Y. Chen, J. Song, Y. Wang, Y. Xue, H. Zhang, and Q. Bao, “Graphene–Bi2Te3 heterostructure as saturable absorber for short pulse generation,” Acs Photon. 2(7), 832–841 (2015).
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A. C. Ferrari, F. Bonaccorso, V. Fal’ko, K. S. Novoselov, S. Roche, P. Bøggild, S. Borini, F. H. Koppens, V. Palermo, N. Pugno, J. A. Garrido, R. Sordan, A. Bianco, L. Ballerini, M. Prato, E. Lidorikis, J. Kivioja, C. Marinelli, T. Ryhänen, A. Morpurgo, J. N. Coleman, V. Nicolosi, L. Colombo, A. Fert, M. Garcia-Hernandez, A. Bachtold, G. F. Schneider, F. Guinea, C. Dekker, M. Barbone, Z. Sun, C. Galiotis, A. N. Grigorenko, G. Konstantatos, A. Kis, M. Katsnelson, L. Vandersypen, A. Loiseau, V. Morandi, D. Neumaier, E. Treossi, V. Pellegrini, M. Polini, A. Tredicucci, G. M. Williams, B. H. Hong, J. H. Ahn, J. M. Kim, H. Zirath, B. J. van Wees, H. van der Zant, L. Occhipinti, A. Di Matteo, I. A. Kinloch, T. Seyller, E. Quesnel, X. Feng, K. Teo, N. Rupesinghe, P. Hakonen, S. R. Neil, Q. Tannock, T. Löfwander, and J. Kinaret, “Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems,” Nanoscale 7(11), 4598–4810 (2015).
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D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “APPLIED PHYSICS. Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (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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J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotechnol. 10(1), 2–6 (2015).
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2014 (3)

S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. 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. 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).
[Crossref] [PubMed]

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5(1), 5221 (2014).
[Crossref] [PubMed]

F. J. Garcia de Abajo, “Graphene plasmonics: challenges and opportunities,” Acs Photon. 1(3), 135–152 (2014).
[Crossref]

2012 (2)

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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2011 (1)

F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

2007 (1)

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007).
[Crossref] [PubMed]

2006 (2)

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006).
[Crossref] [PubMed]

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
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2005 (2)

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

2004 (1)

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92(12), 127401 (2004).
[Crossref] [PubMed]

2003 (1)

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Figures (8)

Fig. 1
Fig. 1 (a) Schematic of in-plane negative refraction between hybrid surface phonon-hyperbolic phonon polariton (SPh-HP) and tailored surface phonon polariton (SPhP). (b) Left part: the hybrid SPh-HP has negative group velocity, and the Ez distributions (mushroom pattern) inside and outside h-BN have opposite sign; Right part: tailored SPhP has positive group velocity, the Ez distributions inside and outside Ge have same sign. (c) Real permittivity for h-BN and SiC, the overlapping frequencies between h-BN’s first reststrahlen band (εx/y>0, εz<0) and SiC’s reststrahlen band (ε<0) from 797 to 835 cm−1. (d) Type-I hyperbolic responses (solutions for Eq. (1) at ω = 800 cm−1 (blue solid curve) and 820 cm−1 (blue dashed curve).
Fig. 2
Fig. 2 Theoretically calculated dispersions of the polariton modes, (a) hyperbolic phonon polaritons (HP) in the h-BN (400 nm)/SiO2 heterostructure, (b) surface phonon polariton (SPhP) on SiC substrate, (c) hybrid SPh-HP mode in the h-BN (500 nm)/ SiC heterostructure, and (d) tailored SPhP in the Ge (500 nm)/SiC heterostructure. Inset is the Ez field mode profile.
Fig. 3
Fig. 3 (a) 2D-view (y = 0 plane) of the complete heterostructure in Fig. 1(a). (b) Dependence of the real effective indices neff of the hybrid SPh-HP (blue curves) and the tailored SPhP (purple curves) modes on the wavenumber with different slab thicknesses, purple stars are intersection points. Mode profile of the magnetic Hy field for (c) hybrid SPh-HP mode within h-BN region and (d) tailored SPhP mode within Ge region at the neff matched intersection point (ω = 801cm−1, and neff = 1.98), the thicknesses of both h-BN and Ge slabs are 600 nm.
Fig. 4
Fig. 4 (a) Schematic of the in-plan negative refraction in lateral h-BN/Ge heterostructure above SiC substrate, the hybrid SPh-HP is launched by a mode source from h-BN side with incident angle 30°. Cross sectional view of (b) Ez field distribution and (c) Poynting vectors at xz plane. Top view of in-plan Ez field distributions (d and f) and Poynting vectors (e and g) outside (300 nm above top surface) in air and inside (300 nm below top surface) in h-BN/Ge, respectively. The power flow (wavevector) directions are marked by the red (green) arrows.
Fig. 5
Fig. 5 (a) Schematic of the superlens hybrid structure. (b) The dispersions of two type polaritons, g-SPhPs are electric tunable by controlling graphene’s Fermi level, both the thickness of h-BN and Gap is 10nm. Mode profiles of the magnetic Hy field for (c) hybrid SPh-HP mode within h-BN region and (d) g-SPhP mode within graphene region at the frequency of ω = 817 cm−1, the graphene Fermi level is μc = 0.16 ev, corresponds to a momentum matched neff = 80. (e) The figure of merit (γ−1) for graphene polaritons on different substrates, μc = 0.3 ev.
Fig. 6
Fig. 6 (a, b) The distributions of in-plan |E|2 and Hy field (5nm above SiC substrate) to illustrate the superlens effects, at the frequency of ω = 817 cm−1, the graphene Fermi level is μc = 0.16 ev, corresponds to a momentum matched neff = 80 for both polaritons. (c) The normalized Hy field at the image plane (black dashed line in (d)), the full width at half maximum (FWHM) is 125 nm. (d, e) The distributions of in-plan |E|2 and Hy field, and f) the normalized Hy field at the image plane, ω = 817 cm−1and μc = 0.41 ev.
Fig. 7
Fig. 7 The distributions of in-plan |E|2 and Hy field to illustrate the electrical tunability of the superlens. (a) ω = 824cm−1, μc = 0.41ev, corresponds to a momentum matched neff = 45 for both polaritons, the full width at half maximum (FWHM) is 200 nm. (b) ω = 817 cm−1, μc = 0.16 ev, neff = 80 and FWHM = 125 nm. (c) ω = 811 cm−1, μc = 0.11ev, neff = 105 and FWHM = 102 nm.
Fig. 8
Fig. 8 The schematic of three-layer heterostructure, layer 1 (z>d, air), layer 2 (0 <z<d, h-BN, Ge or air Gap), and layer 3 (z<0, SiC substrate), where d is the thickness of layer-2 slabs. The following derivations give the dispersions of hybrid SPh-HP, tailored-SPhP and g-SPhP, respectively.

Equations (9)

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ε(ω)= ε (1+ ω LO 2 ω TO 2 ω TO 2 ω 2 iωγ )
k z 2 / ε x +( k x 2 + k y 2 )/ ε z = (ω/c) 2
M=[ M aa M ab M ba M bb ]= R 1,2 T 2 R 2,3
R 1,2 = 1 t 1,2 [ 1 r 1,2 r 1,2 1 ], R 2,3 = 1 t 2,3 [ 1 r 2,3 r 2,3 1 ], T 2 =[ e i k z2 d 0 0 e i k z2 d ]
k z1 = ω 2 c 2 ε 1 k x 2 , k z2 = ω 2 c 2 ε 2x k x 2 ε 2x ε 2z , k z3 = ω 2 c 2 ε 3 k x 2
r i,j = ε xj k zi ε xi k zj ε xj k zi + ε xi k zj , t i,j = 2 ε xj k zi ε xj k zi + ε xi k zj
r p = M ba M aa
r 1,2 = ε x2 k z1 ε x1 k z2 +σ k z1 k z2 /ω ε 0 ε x2 k z1 + ε x1 k z2 +σ k z1 k z2 /ω ε 0 , t 1,2 = 2 ε x2 k z1 ε x2 k z1 + ε x1 k z2 +σ k z1 k z2 /ω ε 0
σ(ω)= e 2 E F π 2 i ω+i T 1 + e 2 4 2 ( θ(ω2 E F )+ i π log( | ω2 E F ω+2 E F | ) )

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