Tunable optical topological transitions of plasmon polaritons in WTe2 van der Waals films

Naturally existing in-plane hyperbolic polaritons and the associated optical topological transitions, which avoid the nano-structuring to achieve hyperbolicity, can outperform their counterparts in artificial metasurfaces. Such plasmon polaritons are rare, but experimentally revealed recently in WTe2 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 in-plane 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 WTe2 to 270 cm−1 (37.0 microns) at the 50% Mo-doping level at 10 K. Moreover, the temperature-induced 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 far-field 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 low-symmetry plasmonic nanophotonics based on anisotropic natural materials.


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 sub-wavelength control of light-matter interactions, making them ideal candidates for applications such as sensing and energy conversion 6 . Additionally, 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 man-made metamaterials, which require complicated nano-fabrication 8,9 . Fortunately, some anisotropic materials in nature have been discovered to host in-plane hyperbolic polaritons, such as phonon polaritons in MoO3 10,11 and V2O5 12 , and plasmon polaritons in WTe2 13 , which open up a plethora of opportunities in reconfigurable on-chip integrated photonics 14,15 . These findings have fueled an emerging research field termed low-symmetry 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 material-specific polariton dispersions.
On the other hand, plasmon polaritons, particularly in the two-dimensional form, are easier to be tamed intrinsically 35,36 , which has been exemplified by graphene plasmon polaritons 37 . However, naturally existing in-plane hyperbolic plasmon polaritons are rare, but have recently been demonstrated in WTe2 thin films 13 in the frequency range of 429 cm -1 to 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 in-plane 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 Mo-doped WTe2 (MoxW1-xTe2) thin films, a recently discovered layered Type-II Weyl semimetal with composition-dependent band structure 40,41 and electric transport 42 .
Meanwhile, an innovative technique to track the topology based on the far-field 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.

Sample fabrication and polarized IR spectra
We grew MoxW1-xTe2 crystals (x ≤ 0.5) in the semi-metallic orthorhombic Td-phase using a chemical vapor transport technique with iodine as the transport agent (Materials and methods, Supplementary Note 1). The zigzag W-W chains are along a-axis, with W atoms partially substituted by Mo after Mo-doping 43 , as displayed in Fig. 1a. Fig. 1b shows the schematic illustration of a skew ribbon array patterned from an exfoliated single crystal film of MoxW1-xTe2 with a skew angle of = −33° with respect to a-axis (Materials and methods), and illuminated by the normal incident light with a polarization angle . LSPRs can be excited in such ribbon arrays with far-field incident light. 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 self-assembled 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 ext is parallel to the polarization current density polar ( polar = ⁄ , with as the polarization vector), which is associated with the depolarization field depol induced by the polarization charge. Note that 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 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 two-dimensional material, the incident light with polarization perpendicular to the ribbon leads to the maximal plasmon resonance, since and hence polar are parallel to depol due to the isotropic polarizability (conductivity) tensor. In ribbons made from an anisotropic film, however, and polar are not necessarily parallel to depol . Thus, as shown in Fig. 1e-f, the optimal polarization 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 ( ′′ and ′′ ) and the skew angle (Supplementary Note   2): where max can be restricted in the range of -90° to 90°. Therefore, max has the same sign as θ in the elliptic regime since ′′ ′′ > 0 (Fig. 1e), but opposite sign in the hyperbolic regime, for which ′′ ′′ < 0 (Fig. 1f). Particularly, for the plasmon polaritons at the OTT energy ( ′′ = 0), the optimal light polarization coincides with a-axis.
To benchmark this scheme, we firstly use Eq. 1 to reexamine the topology of plasmon dispersion in WTe2 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

Mo-doping-dependent 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

Characterization of the OTT in the far-IR range with the far-field method
As a matter of fact, determining the topology of plasmon dispersion in the far-IR range is a daunting task without the aforementioned polarization-based method. In principle, both near-and far-field techniques can probe the OTT in WTe2. However, the near-field scheme is not mature in the far-IR range, especially with samples in the cryogenic conditions, even though it is widely and successfully employed in the mid-IR range to image in-plane hyperbolic phonon polaritons 3-5,10-12,16,21-23,25-34 . As a consequence, up to now, there is no near-field imaging of hyperbolic plasmon polaritons in WTe2. As for the far-field technique, previously we determined the OTT of plasmon polaritons in WTe2 through mapping the plasmon dispersion in the whole two-dimensional momentum space 13 , which was laborious and required numerous samples (each momentum needs a ribbon array). Fortunately, with our polarization-based far-field 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 WTe2
By leveraging Mo doping and temperature, the hyperbolic regime expands 3.1 times than that in pristine WTe2. 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 MoO3/MoO3 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 far-IR range. This expansion complements the existing hyperbolic phonon polaritons that predominantly reside in the mid-IR range.
Moreover, the far-IR range contains a multitude of intramolecular or intermolecular vibration modes (e.g., in proteins or DNA), making MoxW1-xTe2 an excellent candidate for bio-sensing and bio-imaging applications 49 . Furthermore, both the lower and higher boundaries of the hyperbolic regime, which correspond to the sigma-near-zero points along b-and a-axis, respectively, experience substantial shifts. When a material exhibits near-zero 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 epsilon-near-zero photonics 50 . Thus, MoxW1-xTe2 naturally serves as an in-plane tunable anisotropic sigma-near-zero material for functional photonic devices.
Particularly, at the lower hyperbolic boundary, the IFC comprises two nearly parallel 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 polarization-resolved extinction spectra of skew ribbons to determine the topology of IFCs, which can be of great use to investigate other anisotropic two-dimensional materials.

MoxW1-xTe2 crystal growth
MoxW1-xTe2 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 two-zone furnace. The hot zone was maintained at 850 ℃ for two weeks while the cold zone was kept at 750 ℃. The composition of the final crystal was characterized using energy dispersive spectroscopy (EDS) with a scanning electron microscope.

Sample preparation and fabrication
Single

Far-IR optical spectroscopy
For the polarized far-IR extinction spectra, we used a Bruker FTIR spectrometer (Vertex 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 max .

Fitting of IR extinction spectra and plotting pseudo color maps for plasmon spectra
The extinction spectrum is determined by the sheet optical conductivity ( ) as follows 52 : where 0 is the vacuum impedance, is the frequency of light, and s is the refractive

Characterization of the Composition of MoxW1-xTe2.
The final crystal composition was characterized using energy dispersive spectroscopy (EDS) with a scanning electron microscope. EDS measurements were taken at multiple locations on the crystal surface to obtain an average composition. Fig. S1a and b show the typical EDS at 27.8% and 50% Mo doping, respectively.

Polarization Dependence of LSPRs in Skew Ribbon Arrays.
Equation (1) in the main text fully captures the polarization dependence and we derive it in detail now. Here the Ohmic loss (real part of the conductivity) is neglected for convenience.
The conclusion, however, is general and rigorous. LSPRs can be treated as self-sustained dipoles, mimicking mechanical oscillators, which give rise to the polarization current density polar and the depolarization field depol . Note that the LSPR in the ribbons is independent of the existence of the external field, which only serves as the external stimuli to start the plasmon dipole oscillation and to compensate the damping afterwards, if there is any. Again, this is fully analogous to a mechanical damped oscillator. polar • ext accounts for the energy provided by the external field to the plasmon system. Therefore, the most efficient way to drive and feed the plasmon resonance is to let ext trace the polarization current density polar , that is, ext is parallel to polar , as displayed in Fig. S2.
The polarization current density polar is equal to . Here, P is the electric polarization caused by the depolarization field (rather than the external field) = ⃡ 0 depol ∝ ′′ ⃖���⃗ depol , where ⃡ is the polarizability tensor. The polarization ext directly generated by the external field ext is in-phase with ext , hence the associated current ext is out of phase and doesn't take energy from the external field. Note that depol is always perpendicular to the ribbon since polarization charges are along the two edges of the long ribbon, so: where ′′ and ′′ are the imaginary parts of the diagonal elements of the anisotropic conductivity tensor, and is the skew angle of the ribbon array. The polarization of the incident light for the maximal plasmon intensity is defined as max in the main text. Thus, at such polarization, we have ext = � | ext | cos max | ext | sin max �, which should be parallel to polar as discussed above. In conjunction with equation (S1), we arrive at:

The Selection of the Skew Angle .
A skew angle of -33° is chosen based on the following factors: 1), the value of the optimal polarization max at each resonance frequency ; 2), the maximal resonance frequency max we can achieve in the skew ribbon arrays with the skew angle . Note that we are discussing the absolute value here because the sign of the skew angle (or the polarization angle ) depends on our definition arbitrarily and does not affect the conclusion.
Firstly, according to equation (1) or (S2), max ( ) increases with , as shown in Fig.   S3. Thus, a large is needed to make max ( ) large enough for detection in experiments.
Secondly, max decreases with the increase of . As we know, the ribbon configuration directly determines the wavevector of the plasmon polariton 1 . Specifically, for the ribbon with width , the effective wavevector is π⁄ , and the direction of is perpendicular to the ribbon. In the elliptic regime, a smaller ribbon width gives a higher plasmon frequency, as the resonance frequency of plasmon polariton typically increases with the wavevector (e.g., the ∝ � relation in the long-wavelength limit). However, in the hyperbolic regime, the dispersion is softened due to the coupling with interband transitions. For example, the maximal resonance frequencies of WTe2 along a-and b-axis are about 632 and 429 cm -1 , respectively 2 . When the wavevector direction rotates from a-to b-axis (i.e., when increases), the maximal plasmon frequency decreases. Such effect has been confirmed by our previous work 2 ( Supplementary Figure 9 in Ref. 2). As a result, a smaller can expand the range of the plasmon resonance frequency we can achieve in the skew ribbon arrays.
At last, a moderate skew angle of -33° was chosen in our study to ensure a relatively large max ( ) and max simultaneously.

Polarization Dependence of the Drude Response in Skew Ribbon Arrays.
Free carriers are responsible for the Drude response with peak at zero frequency. Fig.   S4a exhibits the polarization dependence of the normalized Drude weight of Fig. 2b in the main text. The Drude weight is largest when the polarization is along 57°, which is precisely parallel to the ribbon (skew angle is -33°). Interestingly, this is the same as that in ribbons made from isotropic two-dimensional films, such as graphene. Here, we'll give a justification for such a scenario. For simplicity and without loss of generality, the Drude response at zero frequency or the DC response is examined. The light absorption is related to the Joule heating, namely • , where and ( = ext + depol ) are the total current density and electric field respectively. The geometrical shape of the ribbon results in the following two features: 1) the depolarization field depol is perpendicular to the ribbon due to the translation-invariance of the induced charge distribution at the edges, and 2) the current density component perpendicular to the ribbon edge ⊥ must vanish, and only ∥ survives after it reaches quasi-static state.
We can reach such a conclusion that the polarization for maximal Drude weight is always along the ribbon direction in two ways. The first way is to consider the situation when the external field is perpendicular to the ribbon. As we'll show, there is no current (either ⊥ or ∥ ) at all when it reaches the static state and no power is consumed. According to feature 1), the total electric field is perpendicular to the ribbon as well. Therefore, to meet feature 2), must be zero to get the zero value of ⊥ , i.e., the external field and depolarization field compensate each other (equal and opposite). As a result, there is no steady current when the external field is perpendicular to the ribbon, namely = 0 and no Joule heating (Drude response) at all. Since we have found the minimum, the maximal Drude response occurs at the polarization perpendicular to the minimal case, i.e., parallel to the ribbon.
The second way is to directly find the maximum for • at different polarization with constant external field amplitude | ext |. As mentioned above (feature 1), has a definitive direction which is parallel to the ribbon. According to the Ohm's law = ⃡ , where ⃡ is the conductivity tensor, where and are the diagonal elements of the conductivity tensor (real parts), and are the tilted angles of the ribbon and the total electric field respectively, as shown in Fig. S4b. Therefore, for a given conductivity tensor, the total electric field has a fixed orientation : Now the Drude response is: Obviously, the maximum of | | leads to the maximal absorption, since is fixed according to equation (S4). Because has a fixed direction and one of its components depol is always perpendicular to the ribbon, we can find its maximum geometrically. We draw a circle with radius | ext | (blue dashed circle), as shown in Fig. S4b. According to the vector summation rule (triangle rule) for = ext + depol , the external field ext parallel to the ribbon gives the maximal | |. Therefore, the Drude response is largest when the external field is applied along the ribbon direction. This rationalizes the experimental finding in Fig.   S4a.

Fitting Details and Errors of LSPRs in Skew Ribbon Arrays.
The fitting details of the spectra in Fig. 2a in the main text are plotted in Fig. S5 and listed in Table S1. The Drude weight D, the plasmon intensity p and the interband transition intensity i are fitting parameters. The plasmon frequency, the interband transition frequency, the Drude and plasmon scattering rates are fixed at 308, 800, 60 and 90 cm -1 respectively.
In our experiments, the main sources of error are associated with two parameters: the plasmon resonance frequency and the optimal polarization angle max ( ). To determine the error for the first parameter, the raw polarized spectra were fitted using different initial fitting parameters to calculate the average of the resonance frequency. As for max , the error can be attributed to two factors. Firstly, errors arise from the fabrication, such as cutting the skew angle of -33°, and the spectrum acquiring process, such as measuring the polarization angle (with errors of approximately 1°). Secondly, the fitting procedure, which involves fitting the polarization dependence of the plasmon spectral weight by cos 2 , introduces a principal error in max of about 1.5°. To obtain an overall estimation of the error in max , the contributions from these two factors were combined using the error transfer formula, as shown in Fig. 3a-c in the main text.   Table S1 | Fitting parameters of the spectra in Fig. 2a in the main text.

Simulation of Polarized Extinction Spectra in Skew Ribbon Arrays
We simulated the spectra in WTe2 skew ribbon arrays of the skew angle = −33° to verify the equation (1) Table S2 for quantitative comparison, which agree well with each other (errors originate from the fitting of the simulated spectra). In summary, the simulations verify the theory (Note 2) convincingly.

Conductivities Extracted from the Plasmon Dispersion of WTe2.
The detailed fitting process to obtain the optical conductivity of the WTe2 film from the plasmon dispersion has been discussed in the main text of Ref.
The plasmon resonance frequencies along principal crystal axes were measured in WTe2 rectangular arrays, and the corresponding wavevectors were determined by the structure size.
The loss function -Im(1⁄ ), defined by the imaginary part of the inverse of the dielectric function, was calculated to fit the plasmon dispersion. For the two-dimensional case, the non-local dielectric function is defined as follows, where env is the substrate permittivity and is the wavevector (since we consider two principal axes, the vector form of and tensor form of are omitted). The conductivities along two principal axes, which are assumed to be a Drude term plus a Lorentz term with parameters to be determined, were obtained by fitting the anisotropic plasmon dispersion using the loss function -Im(1⁄ ) and equation (S6). The obtained optical conductivities were then substituted into equation (1) (or (S2)) to draw the black solid line in Fig. 3a in the main text.

Ribbon Width Dependence of LSPRs in MoxW1-xTe2 Skew Ribbons.
In order to observe LSPRs across different frequencies and track the OTT, we changed the ribbon width. Fig. S7 displays the plasmon dispersion of MoxW1-xTe2 (x = 0.278) acquired from skew ribbon arrays ( = −33°) of different width. The thickness t of the exfoliated film of MoxW1-xTe2 in our paper ranges from 40 to 120 nm. In this thickness range, the electronic band structure remains the same as that of the bulk. The wavevector in a ribbon array 1 is determined by the ribbon width L (usually from 0.3 to 10 μm) with = π⁄ , which is further normalized to 100 nm thick sample by multiplying 100 ⁄ , since the two-dimensional conductivity is proportional to the film thickness t. As shown in Fig. S7, only at low frequencies, the plasmon dispersion follows the � relation, a characteristic feature for the two-dimensional plasmon polaritons from free carriers 3 . At higher frequencies, the dispersion softens and departs from the � relation, which is primarily due to the coupling to interband transitions (fitted by ( ( env + 0 ) ⁄ ) 1 2 ⁄ , with 0 as a parameter for the screening length ) 4,5 .

LSPRs in Relatively Large MoxW1-xTe2 Disks.
To investigate the intrinsic intraband plasmon in MoxW1-xTe2, disk arrays with larger diameters (lower wavevectors) were patterned, in which the plasmon frequencies are low enough to stay away from the interband transitions, and the dispersion follows the � relation. The thickness, disk diameter, plasmon frequencies along a-and b-axis at different doping are summarized in Table S3 for the samples. The extinction spectra and fitting curves are plotted in Fig. S8

Fitting of the Extinction Spectra of Bare Films and the Extracted Drude Weights.
We fit the extinction spectra of bare films using equations (2) and (3)  During the fitting, the Drude scattering rate was fixed at the value derived from the corresponding peak width of the low frequency LSPR.
The Drude response in the lower THz regime is beyond our measurement range, leading to possible underestimation of the Drude weights from the fitting at larger doping (particularly, the 50% doping case). For improvements, at doping composition of x = 0.5, we resorted to the plasmon spectra in Note 9, since the Drude feature dictates the plasmon feature, which shifts into our measurement range. The plasmon frequency in the small limit ( ∝ � • e e � 1 2 ⁄ , e is the sheet carrier density) at different doping is related to the Drude weight 3 ( = π 2 e e ).
Therefore, at known thickness and plasmon wavevector, 0.5 (0.5 is the composition ratio) can be calculated according to the peak frequencies of Fig. S8. Specifically, by comparing to the fitted 0 of WTe2, the Drude weight 0.5 can be obtained. By equation (3), the DC conductivity 0.5 (0) is equal to 0.5 π , which determines the extinction 0.5 at zero frequency according to equation (2). It was included to fit the extinction spectrum of the unpatterned film with x = 0.5. Finally, the fitted thickness-normalized Drude weights ( divided by the thickness t, namely, the Drude weight for the bulk) for x = 0.5 and those obtained through direct fitting of the film spectra for x = 0, and x = 0.278 are displayed in Fig.   S10.   Using the same method, the temperature dependence of the IFCs at a specific frequency (440 cm -1 ) in WTe2 is also examined, as depicted in Fig. S12. As the temperature increases, the IFCs undergo a gradual transition from a hyperbolic shape to an elliptic shape. This observation is in line with the blueshift of the energy of the OTT at higher temperatures, as illustrated in Fig. S13b.

Temperature Dependence of the OTT in WTe2.
The extinction spectra and their fitting curves (here with extended frequency range) of a WTe2 bare film (35 nm thickness), the extracted imaginary parts of the optical conductivities along both axes, and the polarization angle max as a function of the plasmon frequency at different temperatures are displayed in Fig. S13.