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Gate-tunable negative refraction of mid-infrared polaritons

Negative refraction provides an attractive platform to manipulate mid-infrared and terahertz radiation for molecular sensing and thermal radiation applications. However, its implementation based on available metamaterials and plasmonic media presents challenges associated with optical losses, limited spatial confinement, and lack of active tunability in this spectral range. Here, we demonstrate gate-tunable negative refraction at mid-infrared frequencies using hybrid topological polaritons in van der Waals heterostructures with high spatial confinement. We experimentally visualize wide-angle negatively-refracted surface polaritons on {\alpha}-MoO3 films partially decorated with graphene, undergoing planar nanoscale focusing down to 1.6% of the free-space wavelength. Our atomically thick heterostructures outperform conventional bulk materials by avoiding scattering losses at the refracting interface while enabling active tunability through electrical gating. We propose polaritonic negative refraction as a promising platform for infrared applications such as electrically tunable super-resolution imaging, nanoscale thermal manipulation, and molecular sensing.


Introduction
Negative refraction has been extensively investigated in optics(1-3), nanoelectronics(4), acoustics (5), and magnetism(6) as a counter-intuitive physical phenomenon-bending of electromagnetic fields in the 'wrong' direction-that holds strong potential for applications such as subwavelength imaging and cloaking (7). The last two decades have witnessed substantial progress in the study of negative refraction, typically implemented using metallic metamaterials (8,9), dielectric photonic crystals (10,11), and hyperbolic metamaterials (12,13) that are composed of periodic arrays of subwavelength unit cells and possess extraordinary optical properties beyond those existing in conventional materials. The metamaterial concept invoked in these structures (i.e., their behavior as homogeneous media for small unit cells) limits their ability to strongly confine light. As an alternative, metal plasmons have also been demonstrated to undergo negative refraction in the ultraviolet (14), visible (15), and near-infrared (16) spectral regions. This approach is limited by ohmic losses at visible and higher frequencies as well as poor spatial confinement of plasmons in the infrared range. Deepsubwavelength negative refraction at mid-infrared and terahertz frequencies has therefore remained as a challenge, despite the interest arising from its potential in the manipulation of signals associated with molecular vibrations and thermal emission in such spectral domains.
The emergence of van der Waals (vdW) two-dimensional materials has introduced a new degree of freedom in controlling light at the nanoscale over a wide spectral range by leveraging the strong optical confinement associated with their polaritonic modes (17)(18)(19)(20)(21).
These materials exhibit a combination of tunability, low losses, and ultrahigh confinement that makes them appealing for the design of nanophotonic devices. Understandably, recent theoretical studies have proposed the use of vdW polaritons to achieve deep subwavelength mid-infrared negative refraction, for example, in periodic arrays of graphene (22), or using subtly planar heterostructures formed by graphene and hexagonal boron nitride (h-BN) (23). Nevertheless, the extreme spatial confinement of polaritons in these structures makes it difficult to tailor their dispersion relations and achieve negative refraction. In addition, reflection and scattering losses that are inherent to such structures also complicate the realization of this theoretical concept.
Here, we demonstrate deep-subwavelength mid-infrared negative refraction by constructing a vdW heterostructure consisting of an α-MoO3 film partially covered by monolayer graphene. This heterostructure provides hybrid topological polaritons with tunable contours by varying the graphene doping level, such that negative refraction takes place when combining hyperbolic and elliptic dispersion profiles on either side of the inplane interface. In addition, optical losses are reduced during refraction due to the unique single-atomic-layer nature of graphene and the strong modal-profile matching of the involved polaritons. By adjusting the position of a launching metal antenna used as a source of polaritons, we demonstrate reversible negative refraction over a wide range of incidence angles, allowing us to focus incident polaritons with either concave or convex wavefronts, without suffering undesired diffraction effects. Features associated with negative refraction are revealed by a scattering-type scanning near-field optical microscope (s-SNOM) equipped with a tunable quantum cascade laser (24,25). (B) Isofrequency contours (IFCs) of topological polaritons in hyperbolic (α-MoO3) and elliptic (graphene/α-MoO3) media. Negative refraction of a wave propagating in the former (solid orange arrow) takes place at the interface between the two media due to conservation of parallel wave vector (indicated by dashed arrows), which results in a transmitted wave propagating on the other side of the interface normal (solid green arrow). We define incidence and refraction angles θ1 and θ2 of the polaritons with wave vectors k1 and k2, as well as incidence and refraction angles φ1 and φ2 of the corresponding polariton Poynting vectors S1 and S2.
By conserving the polariton wave vector along the direction of the graphene edge, negative refraction is shown to take place when polaritons traverse the in-plane interface between the two regions ( Figure 1A). The Poynting vector S (directed along the energy flow, which is normal to the IFC) and wave vector k (normal to the wavefront) are not collinear for non-circular IFCs (Supplementary Figure 1 and Note 1), such as those of hyperbolic polaritons in α-MoO3 ( Figure 1B) (36). Since the boundary conditions at the interface (between α-MoO3 with and without covering graphene) only require conservation of the tangential wave vector component k∥ = k sinθ, where θ is the angle between the wave vector and the interface normal, the refracted wave can exhibit standard (positive) refraction for k, but negative refraction for S (with S∥ = S sinφ, where φ is the angle between the Poynting vector and the interface normal).
Negative refraction transforms the linear interface into a lens capable of focusing polaritons at a position determined from the relation between the incidence and refraction angles, which can in turn be obtained by inspecting the distribution of the z-z component  Near-field profiles along the x (left) and y (right) directions (see arrows in panels (A) and (B)). Black curves in the latter are Gaussian fits of the transverse focal profile. All experimental results are measured from an in-situ sample. The α-MoO3 thickness is d = 242 nm. The graphene is mechanically exfoliated and statically doped to EF = 0.5 eV by coverage with monolayer α-RuCl3. The illumination frequency is ω0 = 893 cm -1 (A) and 900 cm -1 (B), respectively. (F) Refraction (incidence) angle φ2 in graphene/α-MoO3 as a function of incidence (refraction) angle φ1 in α-MoO3 for various illumination frequencies from 880 to 940 cm -1 (gray shaded area surrounded by dashed curves). Color curves represent specific frequencies measured in our experiments. The reported in-plane negative refraction effect takes place over a wide range of incidence angles ( Figure 2F) as well as spectral range, where the opening angle β of PhPs in α-MoO3 increases with the illumination frequency and the focal length varies in opposite ways with frequency for the two refraction scenarios (Supplementary Figures 9,   10). We plot the relationship between incidence and refraction angles at different illumination frequencies ranging from 880 to 940 cm -1 . For incidence or refraction in bare α-MoO3, the incidence angle φ1 or the refraction angle φ2, respectively, are limited by the opening angle β because PhP propagation is prohibited beyond the hyperbolic region of the IFCs, whereas negative refraction can be generated for a wide range. Notably, in the Reststrahlen band III (higher light frequency from 972 to 1005 cm -1 ), a lateral heterostructure can also achieve nanoscale negative refraction arising from the negative group velocity associated with the reversed dispersion of PhPs in α-MoO3   Negative refraction achieved by engaging topological polaritons with different IFC shapes is distinct from that attained by using bulk materials with a negative index of refraction (8). We are essentially relying on a homogeneous film (α-MoO3) in which the addition of an atomically thin layer (graphene) radically changes the surface-mode characteristics to enable negative refraction. By changing the Fermi energy of graphene (and, correspondingly, the composition of the topological hybrid polaritons), we can actively and continuously transition from positive to negative refraction, whereas control of bulk materials remains as a challenge. However, to achieve a high degree of negative refraction, the required doping level is relatively high (~0.8 eV) and difficult to reach using a typical SiO2/Si back-gate. Therefore, we use bilayer graphene and adopt thinner α-MoO3 films in Figure 3 to decrease the demand for gate voltages and graphene doping level, although this results in a shorter propagation distance than in the thicker sample in

Conclusion
We have experimentally demonstrated mid-infrared negative refraction by designing a vdW heterostructure consisting of an extended α-MoO3 film that is half-covered by monolayer graphene. The observed nanoscale negative refraction and focusing, which relies on topological polaritons with tunable dispersion curves, is revealed through realspace mapping based on infrared nanoscopy. We leverage negative refraction to demonstrate nanoscale optical focusing, which, thanks to the high spatial confinement of polaritons and the atomic thickness of the employed vdW structures, results in deepsubwavelength focal spots with highly squeezed sizes of less than 60 times that of the corresponding illumination wavelength, as well as a 10-fold intensity enhancement and a ~90% transmission of negatively refracted energy (Supplementary Figure 8). Importantly, we show that negative refraction can be actively tuned by an electrostatic gate, resulting in the unique ability to control the wavefront of polaritons in situ and change focal spots and their associated nanoscale optical fields.
Considering the vast range of newly available two-dimensional polaritonic materials, we anticipate negative refraction of polaritons in other vdW heterostructures involving, for example, α-V2O5, black phosphorus, and nanostructured metasurfaces (e.g., based on isotope h-BN). The broad suite of existing materials could lead to polaritonic negative refraction covering the entire mid-infrared and terahertz region. The combined advantage of strong polariton-field confinement, flexible control over anisotropic polariton propagation and focusing, and tunability by material stacking as well as electric gating opens exciting avenues for negative refraction in optical and thermal applications.

Nanofabrication of the devices.
The α-RuCl3 /graphene/α-MoO3 heterostructures were exfoliated and stacked using a vdW assembly technique (Supplementary Figure 3). Initially, monolayer α-RuCl3, monolayer graphene, and α-MoO3 films (~242 nm) were taken from natural crystals and exfoliated on SiO2 (300nm)/Si and Au (60nm)/Si substrates, respectively. Then, α-RuCl3 and graphene were picked up from a SiO2/Si substrate by a deterministic dry-transfer process aided by a glass /polydimethylsiloxane (PDMS)/polycarbonate (PC) stamp at 100! ". After that, the stamp with α-RuCl3 /graphene was aligned with α-MoO3 and the temperature was raised to 180 " to release the heterostructure with PC and form α-RuCl3/graphene/α-MoO3 heterostructures. The samples were subsequently immersed in chloroform for 10 min and rinsed in IPA for 3 min to eliminate any residual PC. We note that monolayer α-RuCl3 has been reported to produce high and homogeneous doping in monolayer graphene with mobilities ~4900 cm 2 / (V s) at a large hole density of 3 × 10 13 cm −2 , while having minimal optical absorption in the mid-infrared range (38).
The Au antenna arrays (3 μm × 250 nm × 50 nm) were patterned on the sample using 100 kV electron-beam lithography (EBL) (Vistec 5000+ES, Germany) on PMMA950K resist (thickness of ~350 nm). For fabrication of Au antennas, a layer of 50 nm thickness of Au was deposited by electron-beam evaporation in a vacuum chamber under a pressure of 5×10 -6 Torr. We deliberately did not prepare an adhesion layer (usually 5 nm thickness of Ti or Cr), so that the gold antennas on the sample could be moved away by an AFM tip (Supplementary Figure 4). Electron-beam evaporation was also used to deposit a 60 nm-thick gold film onto a low-doped Si substrate. To remove any residual organic materials, the samples were immersed in a hot acetone bath at 80 °C for 25 min and subject to a gentle rinse of IPA for 3 min, followed by nitrogen gas drying and thermal baking.

Near-field optical microscopy measurements.
For near-field measurements, a scattering SNOM setup (Neaspec GmbH) equipped with a tunable quantum cascade laser (890 to 2000 cm -1 ) was utilized. A Pt-coated AFM tip with a radius of ~25 nm (NanoWorld) was employed, with a tapping frequency and amplitude of ~270 kHz and ~30-50 nm, respectively. A p-polarized mid-infrared beam with a lateral spot size of 25 μm was aimed at the AFM tip, illuminating the antennas and a large area of the graphene/α-MoO3 samples. To effectively reduce background noise, a third-order demodulated-harmonic analysis of the near-field amplitude images was applied.

Calculation of dispersion and IFCs of hybrid plasmon-phonon polaritons.
The dispersion relation of hybrid polaritons was rigorously obtained from waveguide theory applied to graphene/α-MoO3 heterostructures, with the structures modeled as a 2D waveguide consisting of four stacked layers (see more details in Supplementary   Figure 1 and Note 1). In this analysis, air and the gold substrate were modeled in terms of their isotropic dielectric tensors, while α-MoO3 was described by anisotropic tensors.

Electromagnetic simulations.
A Finite Element Method software (COMSOL Multiphysics 5.5) was used to simulate the electromagnetic field.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

References
Supplementary Figure 1. Illustration of the device structure used in our theoretical model. Layers 1 to 4 correspond to air, graphene, α-MoO3, and Au, respectively. Each layer is described by its dielectric tensor.

Note 1. Calculation of the dispersion and IFCs of hybrid plasmon-phonon polaritons.
In the theoretical model, we treat the graphene/α-MoO3 heterostructure as a laterallyinfinite stratified medium consisting of four layers: > 0 (air, layer 1), − 1 < < 0 (graphene, layer 2), − 2 < < − 1 (α-MoO3, layer 3), and < − 2 (Au, layer 4). Each layer is normal to the axis and represented by its corresponding dielectric tensor. The air and Au layers are described by isotropic tensors diag ( !,# *, while graphene is modeled as a uniaxial anisotropic material (1), whose permittivity tensor is given by where the tangential permittivity is expressed as while the out-of-plane permittivity is set to Here, , is the free-space permittivity, $ represents the thickness of monolayer graphene, which is set to 0.33 nm (the interlayer spacing in graphite), and represents the optical conductivity of graphene taken from the local limit of the random-phase approximation (RPA) (2).
As α-MoO3 is a biaxial anisotropic material, its permittivity tensor is modeled as where 01 , 02 , and 0+ are the permittivity components along the principal axes (3).
where j = 2 and j = 3 denote the graphene and α-MoO3 layers, respectively. For a van der Waals structure of finite thickness, the solution for 2 ( ) can be expressed as Considering that the tangential components of ?⃗ and ???⃗ are continuous at the interfaces, we find the equations By substituting the fields in Eqs. (S8)-(S9) into the boundary conditions, we obtain the secular matrix equation Denoting the square matrix in this equation by M, we impose the vanishing of the determinant det{M} = 0 to guarantee the existence of a nonzero solution for the amplitude coefficients A, B, C, and D. Such condition leads to the transcendental equation where The solution to Eq. (S12) yields the mode dispersion relation, which can be solved for each value of the angle (see Eq. (S5)), thus generating a function ( ) that can be understood as for isofrequency contour (IFC) of the polaritons supported by the graphene/α-MoO3 heterostructure. Following this procedure, we find the numerical results that are shown in Figure 1B. As the tangential wave vectors of the interface for the refraction process are continuous (this is essentially Snell's Law), the relationship between the incidence and refraction angles can be obtained numerically, as illustrated in Supplementary Figure 2A.

Supplementary
For the bare α-MoO3 film, the hyperbolic polaritons in the Reststrahlen band II (816~976 cm -1 ) are highly oriented. When approaching the two asymptotes of the hyperbolic IFC, the number of available wave vectors of PhPs and the intensity of the electric field are significantly increased, as inferred upon examination of the Dyadic Green tensor (5). More precisely, the component of the Dyadic Green tensor of the bare α-MoO3 film reduces to where d and f represent the distance from the source point and the focal spot to the graphene edge, respectively.  Figure  2. The scale bar indicates 3 μm. (D) Three-dimensional representation of the polariton IFCs in α-MoO3 (blue) and graphene/α-MoO3 heterostructure (green), with experimentally measured frequencies represented by black curves. The opening angle β of PhPs in α-MoO3 increases with the illumination frequency, which weakens the focusing effect produced by negative refraction from α-MoO3 to graphene/α-MoO3 because of the increased spatial incidence range of PhPs in α-MoO3. In contrast, focusing should be enhanced with an increase of frequency for negative refraction from graphene/α-MoO3 to α-MoO3 due to the increase in refraction angle, which is related to the opening angle.