Abstract
Naturally existing inplane hyperbolic polaritons and the associated optical topological transitions, which avoid the nanostructuring to achieve hyperbolicity, can outperform their counterparts in artificial metasurfaces. Such plasmon polaritons are rare, but experimentally revealed recently in WTe_{2} van der Waals thin films. Different from phonon polaritons, hyperbolic plasmon polaritons originate from the interplay of free carrier Drude response and interband transitions, which promise good intrinsic tunability. However, tunable inplane hyperbolic plasmon polariton and its optical topological transition of the isofrequency contours to the elliptic topology in a natural material have not been realized. Here we demonstrate the tuning of the optical topological transition through Mo doping and temperature. The optical topological transition energy is tuned over a wide range, with frequencies ranging from 429 cm^{−1} (23.3 microns) for pure WTe_{2} to 270 cm^{−1} (37.0 microns) at the 50% Modoping level at 10 K. Moreover, the temperatureinduced blueshift of the optical topological transition energy is also revealed, enabling active and reversible tuning. Surprisingly, the localized surface plasmon resonance in skew ribbons shows unusual polarization dependence, accurately manifesting its topology, which renders a reliable means to track the topology with farfield techniques. Our results open an avenue for reconfigurable photonic devices capable of plasmon polariton steering, such as canaling, focusing, and routing, and pave the way for lowsymmetry plasmonic nanophotonics based on anisotropic natural materials.
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Introduction
Hyperbolic polaritons are a unique type of polariton that exhibits hyperbolic isofrequency contours (IFCs). They are advantageous over traditional isotropic or elliptic polaritons. With extreme anisotropy, the propagation of hyperbolic polaritons is highly directional^{1,2,3,4,5}. Meanwhile, they exhibit intense confinement and strong field enhancement, which enable subwavelength control of light–matter interactions, making them ideal candidates for applications such as sensing and energy conversion^{6}. In addition, the open geometry of the IFC in the momentum space leads to the theoretically infinite wavevectors and unprecedentedly high photonic density of states, which is particularly appealing in quantum applications like the enhancement of spontaneous emission^{7}.
Hyperbolic polaritons are typically found in manmade metamaterials, which require complicated nanofabrication^{8,9}. Fortunately, some anisotropic materials in nature have been discovered to host inplane hyperbolic polaritons, such as phonon polaritons in MoO_{3}^{10,11} and V_{2}O_{5}^{12}, and plasmon polaritons in WTe_{2}^{13}, which open up a plethora of opportunities in reconfigurable onchip integrated photonics^{14,15}. These findings have fueled an emerging research field termed lowsymmetry nanophotonics^{16,17,18,19}. Crucially, the tunability of the wavelength range of the hyperbolic IFCs (the hyperbolic regime) and the optical topological transition (OTT) of the IFCs to the elliptic topology^{20} are highly desirable to fulfill the potential. However, phonon polaritons, which are based on polar lattice vibrations, intrinsically show rather fixed Reststrahlen bands and materialspecific polariton dispersions. This certainly limits their tunability, although much efforts have been devoted to extrinsic schemes in MoO_{3} phonon polaritons, such as interfacing with suitable substrate^{21,22,23,24}, stacking with highly tunable graphene^{25,26,27,28,29}, and twisting bilayer structures^{30,31,32,33}. Progress has also been achieved through more disruptive measures, such as intercalation^{12,34}, but the phonon bands only shift a small amount^{12} (around 30 cm^{−1}), and hyperbolic polariton was not observed after intercalation.
On the other hand, plasmon polaritons, particularly in the twodimensional form, are easier to be tamed intrinsically^{35,36}, which has been exemplified by graphene plasmon polaritons^{37}. However, naturally existing inplane hyperbolic plasmon polaritons are rare, but have recently been demonstrated in WTe_{2} thin films^{13} in the frequency range of 429–632 cm^{−1}, with the elliptic regime of the IFCs below 429 cm^{−1}. Given the limited options for such materials, it’s even more imperative to tune the OTT energy to suit various applications. In fact, the plasmon dispersion in a natural hyperbolic plasmonic surface is typically governed by the anisotropy of both free carrier response and bound interband transitions^{13,38}, with the former for inductive and the latter for capacitive optical responses. Any variation of electronic properties, such as carrier density, effective mass anisotropy, frequency and strength of interband transition resonance, will give rise to a modulation of the hyperbolic regime^{38,39}. Though as promising as it sounds, however, the experimental demonstration of tunable inplane OTT of IFCs of plasmon polaritons in a natural material has not been realized up to date.
In this study, we report the first intrinsic tuning of such OTT in a broad wavelength range through chemical doping and temperature in a natural material. We reveal such tunability in Modoped WTe_{2} (Mo_{x}W_{1x}Te_{2}) thin films, a recently discovered layered TypeII Weyl semimetal with compositiondependent band structure^{40,41} and electric transport^{42}. Meanwhile, an innovative technique to track the topology based on the farfield polarization dependence of the localized surface plasmon resonances (LSPRs) in skew ribbons has been developed. This technique allows for efficient and accurate characterization of the topology of plasmon dispersion (IFCs) at a particular frequency in a single sample. Our study not only extends the hyperbolic regime in natural materials by other degrees of freedom, but also reveals the peculiar and informative polarization property of LSPRs in microstructures made from an anisotropic material.
Results
Sample fabrication and polarized IR spectra
We grew Mo_{x}W_{1x}Te_{2} crystals (x ≤ 0.5) in the semimetallic orthorhombic T_{d}phase using a chemical vapor transport technique with iodine as the transport agent (“Materials and methods” and Supplementary Note 1). The zigzag WW chains are along aaxis, with W atoms partially substituted by Mo after Mo doping^{43}, as displayed in Fig. 1a. Figure 1b shows the schematic illustration of a skew ribbon array patterned from an exfoliated single crystal film of Mo_{x}W_{1x}Te_{2} with a skew angle of \(\theta=33^\circ\) with respect to aaxis (“Materials and methods”), and illuminated by the normal incident light with a polarization angle ϕ. LSPRs can be excited in such ribbon arrays with farfield incident light. Figure 1c, d displays two typical extinction spectra (characterized by \(1T/{T}_{0}\), where T and \({T}_{0}\) are the transmission of light through the sample and the bare substrate, respectively) of such skew ribbon arrays with composition ratio x = 0.278 but different ribbon widths L (corresponding to the effective plasmon wavevector of \({\rm{\pi }}/L\)). The light polarization (ϕ = −14.8°, 24.5°) was selected as close as possible to where the plasmon resonance is most intense, with resonance frequencies (219 cm^{−1}, 462 cm^{−1}) in the elliptic and the hyperbolic regimes respectively (to be discussed below). Besides the plasmon resonance, the spectrum in Fig. 1d exhibits evident Drude response, in sharp contrast to that in Fig. 1c, suggesting that the polarization for maximal plasmon intensity in the hyperbolic regime deviates significantly from the perpendicular direction of ribbons. Such deviation was first reported in selfassembled carbon nanotubes^{44}. However, the implication on the topology of plasmon dispersion has not been revealed. Here, we show that such optimal polarization is fully dictated by the ratio of the imaginary parts of the anisotropic conductivities, and in turn can be utilized to determine the topology of plasmon dispersion.
Polarization dependence of LSPRs in skew ribbon arrays
When a skew ribbon array is illuminated, the plasmon resonance is most intense when the polarization of the incident light \({{\boldsymbol{E}}}_{{\rm{ext}}}\) is parallel to the polarization current density \({{\boldsymbol{J}}}_{{\rm{polar}}}\) (\({{\boldsymbol{J}}}_{{\rm{polar}}}=\partial {\boldsymbol{P}}/\partial t\), with \({\boldsymbol{P}}\) as the polarization vector), which is associated with the depolarization field \({{\boldsymbol{E}}}_{{\rm{depol}}}\) induced by the polarization charge. Note that \({{\boldsymbol{E}}}_{{\rm{depol}}}\) is always perpendicular to the ribbon edge due to the translation invariance of the polarization charge distribution along the edge (Fig. 1e, f), and \({{\boldsymbol{J}}}_{{\rm{polar}}}\) is not the conduction current related to the real part of the conductivity, which is responsible for the energy dissipation in the material. In an isotropic twodimensional material, the incident light with polarization perpendicular to the ribbon leads to the maximal plasmon resonance, since \({\boldsymbol{P}}\) and hence \({{\boldsymbol{J}}}_{{\rm{polar}}}\) are parallel to \({{\boldsymbol{E}}}_{{\rm{depol}}}\) due to the isotropic polarizability (conductivity) tensor. In ribbons made from an anisotropic film, however, \({\boldsymbol{P}}\) and \({{\boldsymbol{J}}}_{{\rm{polar}}}\) are not necessarily parallel to \({{\boldsymbol{E}}}_{{\rm{depol}}}\). Thus, as shown in Fig. 1e, f, the optimal polarization \({\phi }_{{\rm{\max }}}\) for plasmon excitation deviates from the perpendicular direction of the ribbon, the value of which is determined by the ratio of the imaginary parts of conductivities (\({\sigma }_{{aa}}^{{\rm{\text{'}\text{'}}}}\) and \({\sigma }_{{bb}}^{{\rm{\text{'}\text{'}}}}\)) and the skew angle \(\theta\) (Supplementary Note 2):
where \({\phi }_{\max }\) can be restricted in the range of −90° to 90°. Therefore, \({\phi }_{\max }\) has the same sign as θ in the elliptic regime since \({\sigma }_{{aa}}^{\prime\prime}{\sigma }_{{bb}}^{\prime\prime}\, >\, 0\) (Fig. 1e), but the opposite sign in the hyperbolic regime, for which \({\sigma }_{{aa}}^{\prime\prime}{\sigma }_{{bb}}^{\prime\prime}\, <\, 0\) (Fig. 1f). Particularly, for the plasmon polaritons at the OTT energy (\({\sigma }_{{bb}}^{{\prime\prime}}=0\)), the optimal light polarization coincides with aaxis.
To benchmark this scheme, we firstly use Eq. (1) to reexamine the topology of plasmon dispersion in WTe_{2} films, for which an OTT has been reported at about 429 cm^{−1} (23.3 microns in wavelength)^{13}. Skew ribbon arrays as in Fig. 1b with the same skew angle of θ = −33° but different ribbon widths were fabricated from WTe_{2} films. It should be noted that all of the studied skew ribbons in this paper have the same −33° skew angle (Supplementary Note 3). The polarizationresolved extinction spectra for two representative samples at 10 K are shown in Fig. 2a, b (“Materials and methods”). Resonance peaks of LSPRs can be observed at frequencies of 308 cm^{−1} in the elliptic regime (Fig. 2a) and 510 cm^{−1} in the hyperbolic regime (Fig. 2b), thus expected to have maximal plasmon intensity at polarization angles of different signs, according to Eq. (1). Note that, in Fig. 2b, spectra with polarization at around −33° (perpendicular to the ribbon) are nearly flat, indicating almost no plasmon absorption, in striking contrast to ribbons patterned from isotropic films. To see the polarization dependence more clearly, the extinction spectra in Fig. 2a, b are plotted as pseudocolor maps in Fig. 2c, d. The maximal plasmon intensity in Fig. 2c can be found below the zero line (red dashed line), while the plasmon resonance in Fig. 2d exhibits strongest absorption at an angle well above zero. The feature below 200 cm^{−1} comes from the Drude response of free carriers, whose maximum is always along the ribbon (ϕ = 57°), a common scenario for both isotropic and anisotropic ribbons (Supplementary Note 4). To extract the plasmon weight, spectra were fitted with the Drude–Lorentz model (“Materials and methods” and Supplementary Note 5), and the fitted plasmon weight is plotted in Fig. 2e, f as a function of the polarization angle. As shown in Fig. 2e (Fig. 2f), the polar angle \({\phi }_{{\rm{\max }}}\) is −11.7° (35.1°), which has the same (opposite) sign as the skew angle (−33°), consistent with the elliptic (hyperbolic) topology of IFCs.
More systematically, a series of samples with the same skew angle (\(\theta=33^\circ\)) and incremental plasmon frequencies ranging from 185 cm^{−1} to 510 cm^{−1} were fabricated by varying the ribbon width. The polarizationresolved absorption spectra due to the plasmon resonance are plotted as pseudocolor maps for several plasmon frequencies in Fig. 2g (“Materials and methods”). The angle for maximal plasmon intensity gradually evolves from negative to positive, crossing the zero line (aaxis) when the plasmon frequency coincides with the elliptic/hyperbolic boundary (429 cm^{−1}, determined by the plasmon dispersion in previous work^{13}). The measured frequencydependent polarization angle for the maximal plasmon intensity (ϕ_{max}) is displayed in Fig. 3a as brown dots, with errors from both angle measurements and fittings (Supplementary Note 5). In addition, the simulations of the extinction spectra in WTe_{2} skew ribbon arrays (\(\theta=33^\circ\)) with different ribbon widths were performed (Supplementary Note 6). The fitted results of the optimal \({\phi }_{{\rm{\max }}}\) are displayed as blue squares in Fig. 3a. The OTT of IFCs from the elliptic to the hyperbolic can be directly manifested by the sign change of the angle based on experiments and simulations, which agrees well with the calculations by Eq. (1) (black solid line) using the conductivities extracted from the plasmon dispersion in previous work^{13} (Supplementary Note 7), validating the capability of tracking the OTT both qualitatively and quantitatively. Note that when the skew angle is fixed, \({\phi }_{{\rm{\max }}}\) is solely determined by \(\frac{{\sigma }_{{bb}}^{{\rm{\text{'}\text{'}}}}\left(\omega \right)}{{\sigma }_{{aa}}^{{\rm{\text{'}\text{'}}}}\left(\omega \right)}\) at the plasmon frequency and will not be affected by the ribbon width or film thickness (assuming an unchanged band structure with sample thickness), enabling it to be a superior method to determine the topology of plasmon dispersion at individual frequencies.
Modopingdependent OTT of IFCs of plasmon polaritons
With the convenient toolkit at our disposal, we can now proceed to investigate the tuning of such OTT. Ribbon arrays with the same configuration as shown in Fig. 1b were fabricated on Modoped WTe_{2} films with various ribbon widths, resulting in different plasmon frequencies (Supplementary Note 8). The polarization dependence of plasmon polaritons was measured and the fitted \({\phi }_{{\rm{\max }}}\) values are summarized in Fig. 3b–c for 27.8% and 50% doping levels, respectively. A sign change for \({\phi }_{{\rm{\max }}}\) can be observed in Fig. 3b–c as the plasmon frequency increases, indicating an OTT in the doped samples as well. A large redshift of the zerocrossing point or the OTT frequency (indicated by the vertical black dashed line) occurs. It changes from 429 cm^{−1} (23.3 microns) for WTe_{2} to about 270 cm^{−1} (37.0 microns) for Mo_{0.5}W_{0.5}Te_{2}. This corresponds to a 38% redshift in frequency or a 1.6fold increase in wavelength for the OTT energy.
To further manifest the doping effect on the OTT, as an example, we compare the polarization dependence of plasmon polaritons with similar resonance frequencies of 345 ± 15 cm^{−1} for different doping levels. As shown in Fig. 3d–f, the polarization angle for the maximal plasmon intensity increases from negative to positive upon doping. This can be seen even more clearly in Fig. 3g–i, which shows that the angle \({\phi }_{{\rm{\max }}}\) increases from –9.5° for WTe_{2} films to 23.9° at 50% Mo doping, suggesting the OTT of IFCs from the elliptic to the hyperbolic by Mo doping. Note that the plasmon linewidth of these samples increases with doping (72 cm^{−1} of WTe_{2}, 112 cm^{−1} at 27.8% doping and 250 cm^{−1} at 50% doping). The increasing linewidth is primarily due to the stronger coupling between plasmon polaritons and interband transitions, which have lower energy at larger doping levels and hence are closer to the plasmon mode, as shown in Fig. 4g. Such couplinginduced broadening is manifested in WTe_{2} as well^{13}. As shown in Fig. 2g, the plasmon linewidth increases as the frequency approaches that of the interband transition. The linewidth of intrinsic plasmon resonances at lower frequencies, hence largely free of interband coupling, is more similar for all Mo_{x}W_{1x}Te_{2} samples (with resonance frequencies of 295.0, 291.7, 214.3 cm^{−1} and quality factors of 3.4, 3.1, 2.1 along aaxis at different doping levels respectively, Supplementary Note 9).
Hyperbolic regime determined by the optical absorption in Mo_{x}W_{1x}Te_{2} thin films
To delve into the physical mechanism of the tunable OTT, polarizationresolved extinction spectra of pristine films of Mo_{x}W_{1x}Te_{2} (x = 0, 0.278, 0.5, with thicknesses of 110, 80, 42 nm, respectively) were examined. The measured spectra (far and midIR ranges) and their fitting curves (Supplementary Note 10) are plotted in Fig. 4a–c. The corresponding extracted sheet optical conductivity (imaginary part) is displayed in Fig. 4d–f, with the generally expanding hyperbolic regime marked by the shaded area based on their signs. The IFCs of plasmon dispersion in Mo_{x}W_{1x}Te_{2} are also plotted according to the loss function^{13} to visualize the OTT (Supplementary Note 11). The lower boundary of the hyperbolic regime redshifts upon doping, consistent with the polarization behavior of LSPRs in skew ribbon arrays. In general, the topology of plasmon dispersion is determined by the optical response of both the intraband (Drude response) and interband transitions of carriers. The transition energy and oscillator strength of the interband transition resonance are determined by the specific band structure and the Fermi level, and the free carrier Drude response depends on the carrier density and effective mass. In our experiment, the extinction spectra in Fig. 4a–c share similar qualitative profiles. The primary difference in the farIR spectra is quantitative in nature, such as the peak position of the first interband transition resonance, as shown in Fig. 4g. The transition energy decreases from 726 cm^{−1} for WTe_{2} to about 488 cm^{−1} at 50% doping (Supplementary Note 10), as indicated by blue arrows for the spectra of baxis polarization in Fig. 4a–c, which brings the whole interband transition feature to lower frequencies. This salient doping dependence governs the redshift of the hyperbolic lower boundary, as the dielectric (capacitive) part of the optical conductivity at long wavelength is primarily attributed to interband transitions. It is also worth noting that the fitted Drude weight (normalized with thickness) along baxis slightly decreases upon Mo doping (Supplementary Note 10), playing a minor role in the redshift of the OTT energy.
Furthermore, by substituting Im(σ) in Fig. 4d–f into Eq. (1), the calculated frequency dependence of \({\phi }_{{\rm{\max }}}\) is plotted as gray dashed lines in Fig. 3a–c, which are consistent with the directly measured angles (brown dots) in skew ribbon arrays. Hyperbolic boundary frequencies derived from the polarization dependence of skew ribbons (red dots), film extinction measurements (black rightpointing triangles), and the plasmon dispersion in ref. ^{13} (blue pentagrams) are summarized in Fig. 4h. All these procedures show consistent doping dependence of the lower boundary, substantiating the redshift of the OTT energy.
Temperaturedependent OTT of IFCs of plasmon polaritons
As a typical semimetal, WTe_{2} exhibits strong temperature dependence in its electronic properties^{13,45}, enabling active tuning of the OTT. To fulfill this potential, the polarizationresolved extinction spectra of a 35nmthick WTe_{2} bare film at different temperatures (78 K, 130 K, 155 K, 180 K, 230 K, 300 K) were measured. The corresponding spectra (far and midIR ranges) and fitting curves are plotted in Fig. 5a–d. The anisotropy of interband transitions (between 700 and 1100 cm^{−1}) along two axes decreases, whereas the Drude weights increase due to more thermal carriers at higher temperatures. As a result, the lower boundary of the hyperbolic regime blueshifts from 428 cm^{−1} at 78 K to 553 cm^{−1} at 180 K, which is manifested by the extracted sheet optical conductivity in Fig. 5e–h and the temperature dependence of IFCs (Supplementary Note 11). The hyperbolic regime in the farIR range vanishes above 230 K. A kink appears in the temperature dependence of the hyperbolic regime at about 130 K (inset in Fig. 5l), which is consistent with the temperature dependence for the plasmon frequency and Drude weight in WTe_{2}^{13}. This is likely attributed to the temperatureinduced Lifshitz transition at about 147–160 K, where the two hole pockets move down in energy with respect to the Fermi surface and eventually disappear, resulting in no hole carriers at higher temperatures^{46,47}. Further, the polarization dependence of plasmon polaritons of the same devices as those shown in Fig. 3a at corresponding temperatures was measured. The polarization angle \({\phi }_{{\rm{\max }}}\) was extracted in the same way, which agrees well with the calculation of Eq. (1) as shown in Fig. 5i–l, demonstrating the temperatureinduced shifts of the OTT energy. Additional details are summarized in Supplementary Note 12. Compared to nearly temperatureindependent phonon polaritons, the hyperbolic regime in WTe_{2} plasmon polaritons shifts over a wide range with temperature. By combining Mo doing and temperature tuning, the hyperbolic regime overall covers the farIR spectrum from 13.5 microns (739 cm^{−1}) to 37.0 microns (270 cm^{−1}), which is 3.1 times broader than the hyperbolic wavelength range observed in pristine WTe_{2} films at 10 K^{13}.
Discussion
Characterization of the OTT in the farIR range with the farfield method
As a matter of fact, determining the topology of plasmon dispersion in the farIR range is a daunting task without the aforementioned polarizationbased method. In principle, both near and farfield techniques can probe the OTT in WTe_{2}. However, the nearfield scheme is not mature in the farIR range, especially with samples in the cryogenic conditions, even though it is widely and successfully employed in the midIR range to image inplane hyperbolic phonon polaritons^{3,4,5,10,11,12,16,21,22,23,25,26,27,28,29,30,31,32,33,34}. As a consequence, up to now, there is no nearfield imaging of hyperbolic plasmon polaritons in WTe_{2}. As for the farfield technique, previously we determined the OTT of plasmon polaritons in WTe_{2} through mapping the plasmon dispersion in the whole twodimensional momentum space^{13}, which was laborious and required numerous samples (each momentum q needs a ribbon array). Fortunately, with our polarizationbased farfield method, we can now determine the topology of the IFC at a particular frequency in a single sample without invoking the whole plasmon dispersion, which is truly advantageous.
Implications of the tuning of the OTT in WTe_{2}
By leveraging Mo doping and temperature, the hyperbolic regime expands 3.1 times than that in pristine WTe_{2}. This significant broadening demonstrates that hyperbolic plasmon polaritons can be manipulated more readily, which is fundamentally different from previously reported hyperbolic phonon polaritons. For instance, the fabrication of MoO_{3}/MoO_{3} twisted bilayers leads to a contraction of the hyperbolic regime^{30,31,32,33,48}. As a result, the expanded hyperbolic regime now covers nearly the entire farIR range. This expansion complements the existing hyperbolic phonon polaritons that predominantly reside in the midIR range. Moreover, the farIR range contains a multitude of intramolecular or intermolecular vibration modes (e.g., in proteins or DNA), making Mo_{x}W_{1x}Te_{2} an excellent candidate for biosensing and bioimaging applications^{49}. Furthermore, both the lower and higher boundaries of the hyperbolic regime, which correspond to the sigmanearzero points along b and aaxis, respectively, experience substantial shifts. When a material exhibits nearzero effective permittivity (conductivity), novel physical effects arise, such as field enhancement, tunneling through anomalous waveguides and transmission with small phase variation, which are also known as epsilonnearzero photonics^{50}. Thus, Mo_{x}W_{1x}Te_{2} naturally serves as an inplane tunable anisotropic sigmanearzero material for functional photonic devices.
Particularly, at the lower hyperbolic boundary, the IFC comprises two nearly parallel lines along baxis in the twodimensional momentum space, analogous to the socalled canalization regime in hyperbolic phonon polaritons^{3,30,32}. Thus, the tunability of the lower boundary allows for the canalization of the energy flow of hyperbolic polaritons over a wide spectral range. The estimated propagation length of canalized plasmon polaritons in WTe_{2} reaches 0.5 micron^{13}, which is comparable to that of isotropic plasmon polaritons in graphene^{51} and that of the canalized phonon polaritons in hBN metasurfaces^{3}. In addition, the doping procedure does not degrade the sample quality much, as suggested by the similar figure of merits (Supplementary Note 9). The lifetime of plasmon polaritons in Mo_{x}W_{1x}Te_{2} is ~0.05 picosecond (ps), comparable to that of plasmon polaritons in the undoped WTe_{2} (0.1 ps)^{13} and in graphene on SiO_{2}/Si substrates (about 0.05–0.1 ps)^{52}, though it is smaller than the lifetime of phonon polaritons in MoO_{3} (about 8 ps)^{10}. Future efforts can be devoted to increasing the sample quality through more meticulous growth and judicious choice of substrates.
In conclusion, our work demonstrates the inherent tunability of hyperbolic plasmon polaritons and the OTT in vdW surfaces by chemical doping and temperature over a wide range. The tuning mechanism involves both bound states and free carriers, providing more dimensions for manipulating OTT. Our experiments leverage a unique feature in the polarizationresolved extinction spectra of skew ribbons to determine the topology of IFCs, which can be of great use to investigate other anisotropic twodimensional materials.
Materials and methods
Mo_{x}W_{1x}Te_{2} crystal growth
Mo_{x}W_{1x}Te_{2} single crystals were grown by a chemical vapor transport technique with iodine as the transport agent. Stoichiometric mixtures of Mo, W, and Te powders were loaded into a quartz tube along with a small amount of iodine, which was subsequently sealed in vacuum and placed in a twozone furnace. The hot zone was maintained at 850 °C for 2 weeks, while the cold zone was kept at 750 °C. The composition of the final crystal was characterized using energy dispersive spectroscopy (EDS) with a scanning electron microscope.
Sample preparation and fabrication
Single crystal WTe_{2} was bought from HQ Graphene. Bare films of Mo_{x}W_{1x}Te_{2} (x ≤ 0.5) with thickness about 40–120 nm through the standard exfoliation technique were transferred to polycrystalline diamond substrates. The substrate has no polar phonon absorption and the transmission is about 70% in the farIR range. The thickness of the sample was determined by a stylus profiler (Bruker DektakXT) in conjunction with the optical contrast. The preferred sample size is 200 by 200 μm^{2}, with the side length longer than the farIR wavelength (~100 μm). Skew ribbon arrays were patterned using electron beam lithography (Zeiss Sigma SEM with Raith Elphy Plus), with the uncertainty in the skew angle of less than 0.5°. An intermediate skew angle of −33° was chosen in our study, because a toosmall angle results in a small \({\phi }_{{\rm{\max }}}\) as shown in Eq. (1), and a too large one leads to a more dramatic reduction of the maximal plasmon frequency achievable in that wavevector direction. Reactive ion etching (RIE) with SF_{6} gas was used to define the ribbons. If necessary, the surroundings of the sample were further etched away using the director writer UPG501 and RIE to ensure IR response only from the targeted sample.
FarIR optical spectroscopy
For the polarized farIR extinction spectra, we used a Bruker FTIR spectrometer (Vertex 70 v) integrated with a Hyperion 2000 microscope and a cryogenfree silicon bolometer system as the detector. The incident light was focused on Mo_{x}W_{1x}Te_{2} samples with a 15× IR objective. A THz polarizer was used to control the light polarization. The samples were cooled to 10 K in a heliumflow cryostat (Janis Research ST300) with vacuum at about \(5\) × 10^{−5} \(\text{mbar}\). Throughout the entire measurements, compressed dry air with dew point below −70 °C was purged to an enclosed space housing the cryostat. This procedure minimized the absorption of IR light by moisture in air and effectively increased the signal/noise ratio. The polarization dependence of plasmon polaritons in skew ribbon arrays was studied by rotating the polarizer with a step size of 11.3° from −90° to 90°. Hence, a total of 16 spectra were collected to extract each \({\phi }_{{\rm{\max }}}\).
Fitting of IR extinction spectra and plotting pseudocolor maps for plasmon spectra
The extinction spectrum is determined by the sheet optical conductivity \(\sigma \left(\omega \right)\) as follows^{52}:
where \({Z}_{0}\) is the vacuum impedance, ω is the frequency of light, and \({n}_{{\rm{s}}}\) is the refractive index of the substrate. The conductivity of the sample is expressed by the Drude–Lorentz model, where the Drude model describes free carriers and the Lorentz model accounts for the bound states such as plasmon resonance or interband transitions in our system:
In Eq. (3), D and \({S}_{{\rm{k}}}\) represent the spectrum weights, \({\omega }_{{\rm{k}}}\) represents the resonance frequency of the plasmon resonance or interband transitions, Γ and \({\varGamma }_{{\rm{k}}}\) are the corresponding FWHMs. The polarization angle \({\phi }_{{\rm{\max }}}\) was extracted by fitting the whole set of 16 polarization spectra with one Drude component and two Lorentz components (plasmon resonance, interband transitions, respectively). The spectral weights were fitting parameters and the Drude scattering rate, plasmon frequency and interband transition resonance frequency were kept the same for all spectra. Pseudocolor maps in Fig. 2g and Fig. 3d–f were plotted by substituting the fitted conductivities of plasmon resonances into Eq. (2) in turn. The plasmon weights \({S}_{{\rm{p}}}\) extracted above were fitted as cos^{2} \(\phi\) to obtain \({\phi}_{{{\max}}}\). The fitting details of the extinction spectra of Mo_{x}W_{1x}Te_{2} bare films in Fig. 4a–c are discussed in Supplementary Note 10.
Data availability
All data needed to evaluate the conclusions in the paper are present in the main text and the supplementary information.
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Acknowledgements
H.Y. is grateful to the financial support from the National Key Research and Development Program of China (Grant Nos. 2022YFA1404700 and 2021YFA1400100), the National Natural Science Foundation of China (Grant No. 12074085), the Natural Science Foundation of Shanghai (Grant No. 23XD1400200). C.W. is grateful to the financial support from the National Natural Science Foundation of China (Grant Nos. 12274030, 11704075) and the National Key Research and Development Program of China (Grant No. 2022YFA1403400). F.S. acknowledges the financial support from the National Key Research and Development Program of China (Grant No. 2017YFA0303203), the National Natural Science Foundation of China (Grant Nos. 92161201, 12025404, 11904165, and 12274208), the Natural Science Foundation of Jiangsu Province (Grant No. BK20190286). S.H. is grateful to the financial support from the China Postdoctoral Science Foundation (Grant No. 2020TQ0078). Part of the experimental work was carried out in Fudan Nanofabrication Lab.
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H.Y. and C.W. initiated the project and conceived the experiments. Y.X. prepared the samples, and performed the measurements and data analysis with assistance from C.W., Y.D., Yuqi Li, Q.X., S.H., Yuchen Lei., L.M., and J.Z. F.F. and F.S. grew and characterized the bulk single crystals of Mo_{x}W_{1x}Te_{2}. H.Y., C.W., and Y.X. cowrote the manuscript. H.Y. supervised the whole project. All authors commented on the manuscript.
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Xie, Y., Wang, C., Fei, F. et al. Tunable optical topological transitions of plasmon polaritons in WTe_{2} van der Waals films. Light Sci Appl 12, 193 (2023). https://doi.org/10.1038/s4137702301244w
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DOI: https://doi.org/10.1038/s4137702301244w
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