Abstract
Polaritons in hyperbolic van der Waals materials—where principal axes have permittivities of opposite signs—are lightmatter modes with unique properties and promising applications. Isofrequency contours of hyperbolic polaritons may undergo topological transitions from open hyperbolas to closed ellipselike curves, prompting an abrupt change in physical properties. Electronicallytunable topological transitions are especially desirable for future integrated technologies but have yet to be demonstrated. In this work, we present a dopinginduced topological transition effected by plasmonphonon hybridization in graphene/αMoO_{3} heterostructures. Scanning nearfield optical microscopy was used to image hybrid polaritons in graphene/αMoO_{3}. We demonstrate the topological transition and characterize hybrid modes, which can be tuned from surface waves to bulk waveguide modes, traversing an exceptional point arising from the anisotropic plasmonphonon coupling. Graphene/αMoO_{3} heterostructures offer the possibility to explore dynamical topological transitions and directional coupling that could inspire new nanophotonic and quantum devices.
Introduction
Hyperbolic optics first rose to prominence with the development of hyperbolic metamaterials constructed by periodically alternating metallic and dielectric media^{1}. The term “hyperbolic” refers to the shape of the isofrequency surface, which is hyperboloidal rather than ellipsoidal like in common anisotropic media, because of the opposite signs of the real dielectric permittivities of metallic and dielectric media. Hyperbolic isofrequency surfaces give rise to an increased photonic density of states, which can help engineer spontaneous^{2} and thermal^{3} emission. Furthermore, the unique properties of hyperbolic media have allowed for the development of subwavelength imaging^{4}, polarization converters^{5}, and negative refraction^{6}, among many other technologies.
In the past decade, natural van der Waals materials including hexagonal boron nitride (hBN)^{7,8,9} and αphase molybdenum trioxide (αMoO_{3})^{10,11} have been found to exhibit hyperbolicity around anisotropic phonon resonances that drive permittivity to negative values along one crystal axis or plane^{12,13,14}. Van der Waals crystals display relatively weak phonon damping, enabling propagating phonon polaritons that behave like waveguide modes with deeply subdiffractional wavelengths^{15}. Phonon polaritons in the two reststrahlen bands of uniaxial hBN are outofplane hyperbolic, meaning that the outofplane optic axis and its normal plane have oppositesigned permittivities. hBN polaritons can propagate along any inplane direction^{7,8,9}. On the other hand, αMoO_{3} is biaxial with three infrared reststrahlen bands. The upper band hosts outofplane hyperbolic modes (958–1010 cm^{−1}\(:{\varepsilon }_{x} \, > \, 0,{\varepsilon }_{y} \, > \, 0,{\varepsilon }_{z} < 0\)), while the middle (820–972 cm^{−1}\(:{\varepsilon }_{x} < 0,{\varepsilon }_{y} \, > \, 0,{\varepsilon }_{z} \, > \, 0\)) and lower (545–851 cm^{−1}: \({\varepsilon }_{x} \, > \, 0,{\varepsilon }_{y} < 0,{\varepsilon }_{z} \, > \, 0\)) bands have inplane directions with oppositesigned permittivities and support inplane hyperbolic modes with directional propagation and hyperbolic wavefronts^{10,11}.
Integrated nanophotonics requires dynamical tuning of hyperbolic polaritons, but this is challenging since hyperbolicity in the two most widely studied systems (hBN and αMoO_{3}) is rooted in crystal structure. Heterostructuring and hybridizing hyperbolic modes with other modes can modify their properties^{16,17,18,19,20,21,22}, but adjusting tuning parameters typically requires altering device design. With αMoO_{3}, for example, inplane hyperbolic phonon polaritons can be modified by fabricating doublelayer structures with a predefined twist angle as the tuning parameter^{16,17,18,19}. The isofrequency contour (IFC) characterizing the propagationdirectiondependent momenta of inplane hyperbolic modes can be tuned through a topological transition^{23} from an open hyperbola to a closed curve by sweeping twist angle. On the other hand, dynamical tuning of hBN phonon polaritons has been demonstrated by means of hybridization with graphene surface plasmons^{20}. Surface plasmonphonon polariton (SP^{3}) and hyperbolic plasmonphonon polariton (HP^{3}) hybrid modes have gatetunable dispersions enabled by the carrierdensitydependent Fermi energy \({E}_{F}\) of graphene^{24}. However, graphene/hBN admits only modest electronic tunability without topological transitions.
Here, we present a graphene on αMoO_{3} heterostructure that can be tuned electronically through a topological transition as evidenced by direct imaging of propagating polaritons. At charge neutrality, graphene does not host plasmons and the IFC is a hyperbola. Upon electronic doping, the IFC transitions to a closed curve as graphene surface plasmons hybridize with αMoO_{3} hyperbolic phonon polaritons. We demonstrate the topological transition of the polariton wavefront experimentally and tune momenta of hybrid modes by modifying E_{F} and laser frequency \(\omega\). Furthermore, calculations corroborated by experiments reveal that, by rotating the polaritonic wavevector, the hybrid mode characterized by the closed IFC is tuned from HP^{3} to SP^{3} as its plasmonphonon coupling is modified. Graphene/αMoO_{3} is thus a platform for studying the effect of anisotropic plasmonphonon coupling on properties of hybrid hyperbolic polaritons. In particular, we identify an exceptional point in the directiondependent polariton dispersion.
Results and discussion
Hybrid polariton theory
In this work, we focus on frequencies in the middle reststrahlen band of αMoO_{3} (820–972 cm^{−1}) where the topological transition occurs. Finite difference time domain simulations (Methods) show that, without doping, graphene is devoid of plasmonic response and the heterostructure is optically equivalent to bare αMoO_{3} (neglecting losses in chargeneutral graphene). In the cleavage plane of the [100] (x) and [001] (y) crystal axes (Fig. 1a), the phonon polariton wavefront is a hyperbola (Fig. 1b). The IFC, related to the wavefront by a Fourier transform, is likewise a hyperbola (Fig. 1c). Multiple hyperbolas appear: corresponding to higherorder, short wavelength hyperbolic modes. Upon doping the graphene to appreciable \({E}_{F}\), the wavefront becomes a closed curve with anisotropic polariton wavelengths as a result of plasmonphonon hybridization (Fig. 1d). The firstorder IFC undergoes a topological transition to a closed peanut shape (Fig. 1e).
The directiondependent dispersion \(q^{\prime} (\omega ,\theta )\) and dissipation \(q^{\prime\prime} (\omega ,\theta )\) of hybrid modes in graphene with optical conductivity \({\sigma }_{g}\) on a biaxial αMoO_{3} slab of thickness \(d\) at arbitrary frequency \(\omega\) can be computed by solving Equation S5 (Supplementary Note 1). Figure 1f shows that calculated IFCs at \({E}_{F}=0\) eV and 0.4 eV in the middle reststrahlen band are consistent with simulated IFCs in Fig. 1c, e, respectively. The angle \(\theta\) specifies the inplane propagation direction with respect to the [100] direction. The outofplane propagation constant inside αMoO_{3} is \({k}_{z}\approx \pm q\sqrt{{\varepsilon }_{\parallel }/{\varepsilon }_{z}}\) where \({\varepsilon }_{\parallel }={\varepsilon }_{x}{{{\cos }}}^{2}\theta +{\varepsilon }_{y}{{{\sin }}}^{2}\theta\). In the doped structure, the wave launched by the tip is a surface wave (SP^{3}: Im \({k}_{z}\) ≫ Re \({k}_{z}\)) along the [001] direction; and a waveguide mode (HP^{3}: Re \({k}_{z}\) ≫ Im \({k}_{z}\)) along the [100] direction. Along intermediate directions (Fig. 1g), the polariton is an HP^{3} up to the open angle \({\theta }_{c}={{\arctan }}\sqrt{{{{{{\rm{Re}}}}}}\,{\varepsilon }_{x}/{{{{{\rm{R}}}}}}{{{{{\rm{e}}}}}}\,{\varepsilon }_{y}}\), i.e., the asymptote of the undoped hyperbola. Above \({\theta }_{c}\), it is an SP^{3}. Note that HP^{3} and SP^{3} in this work are only defined by their k_{z}. At \({\theta }_{c}\), both real and imaginary components of \({k}_{z}\) drop precipitously. Neglecting inplane losses (dashed lines), \({k}_{z}=0\) at \({\theta }_{c}\): the polariton is an ideal plane wave propagating inplane with finite momentum.
Along the [001] direction in the middle reststrahlen band, the αMoO_{3} phonon is absent and the plasmonphonon coupling strength must be zero. Along the [100] direction, graphene plasmons and αMoO_{3} phonons hybridize due to strong coupling; that is, the plasmon mode splits into plasmonphonon polariton branches above and below the aaxis transverse optical (TO) phonon frequency \({\omega }_{{TO}}^{a}\). Tuning the inplane propagation angle \(\theta\) must then modify the plasmonphonon coupling, which transitions at some point between strong and weak coupling regimes. Prior work^{25} has recognized that the transition occurs at an exceptional point (EP), above which plasmons experience Rabi splitting from strong coupling to phonons. Below the EP, there is no mode splitting: plasmons and phonons are weakly coupled in a manner akin to electromagneticallyinduced transparency.
We now show quantitatively that tuning \(\theta\) can be expected to modify plasmonphonon coupling. Consider twodimensional αMoO_{3} with highfrequency permittivities set to unity. The total dielectric function of Drudelike graphene with electron coupling to inplane hyperbolic phonons modulated by the quantity \(\alpha (q,\theta )\equiv \frac{{qd}}{2}{{{\cos }}}^{2}\theta\) is then:
where \({\omega }_{{pl}}^{2}(q)\equiv q\left{E}_{F}\right{e}^{2}/2\pi {\hslash }^{2}{\varepsilon }_{0}\) and \({\gamma }_{g}\) and \(\gamma\) are the plasmon and phonon scattering rates, respectively (Supplementary Note 2). The real and imaginary roots of Eq. 1 are plotted in Fig. 1h, i, respectively. Larger \(\alpha\) near the [100] direction split plasmon modes at \({\omega }_{{pl}}\) into two branches above and below \({\omega }_{{TO}}^{a}\), labeled \({\omega }_{+}\) and \({\omega }_{}\), respectively (Fig. 1h). \({\omega }_{+}\) and \({\omega }_{}\) have anticorrelated scattering rates, \({\Gamma }_{+}\) and \({\Gamma }_{}\) (Fig. 1i). Splitting is reduced as \(\theta\) rotates toward the [001] direction until reaching a complex degeneracy, beyond which \({\omega }_{+}\) and \({\omega }_{}\) coalesce and \({\Gamma }_{+}\) and \({\Gamma }_{}\) split. This degeneracy is in fact the exceptional point discussed previously that marks the crossover from strong to weak coupling. The doped IFC from Fig. 1f is a curve (solid red line) on the \({\omega }_{+}\) surface, passing about the EP and traversing from strong to weak coupling (white dotted lines) in the twodimensional limit as the mode evolves from HP^{3} to SP^{3} in the finite slab.
Nanoimaging and tunability
A scatteringtype scanning nearfield optical microscope (sSNOM) was used to image polaritons and evaluate their energymomentum (\(\omega ,q\)) dispersions (Methods). An sSNOM is a tappingmode atomic force microscope with a metallized tip illuminated by laser light. The sharp metallic sSNOM tip launches propagating polaritons (Fig. 1a) that reflect off of sample edges and defects. The launched and reflected waves interfere and form standing waves, which are imaged by the scanning probe. Polariton momentum and dissipation, quantified by real and imaginary \(q={q}^{\prime}+{iq}^{\prime\prime}\), respectively, were extracted from observed wavelengths \(\lambda =\pi /q^{\prime}\) and propagation lengths \(L=1/q^{\prime\prime}\) (Methods). We used oxidized tungsten diselenide (WO_{x}) to dope graphene^{26} by a charge transfer process (Methods). Doping graphene using thin highworkfunction materials like WO_{x} or αRuCl_{3} has proven to be nondetrimental to sSNOM imaging^{27,28}. Furthermore, by stacking either monolayer or bilayer tungsten diselenide (WSe_{2}) prior to oxidation, we can tune the \({E}_{F}\) of graphene on αMoO_{3} between ~0.60 eV and 0.45 eV, respectively. Doping levels were determined by fitting (\(\omega ,q\)) data to calculated dispersions with graphene Fermi energy as a free parameter, and were corroborated by Raman spectroscopy (Supplementary Fig. 2, Supplementary Note 3).
In Fig. 2, we image polaritons in various graphene/αMoO_{3} heterostructures with and without graphene doping. In Fig. 2a, a circular void in highlydoped graphene (Supplementary Figure 3) produces a closed wavefront with the characteristic elliptical shape of hybrid plasmonphonon polaritons. [100] and [001] modes as well as modes at intermediate angles are visible. The simulated Re \({E}_{z}\) wavefront (Fig. 2b) is consistent with the experimental wavefront. For reference, we also obtained a nearfield amplitude image (Fig. 2c) and simulated wavefront (Fig. 2d) on bare αMoO_{3}. We observe the hyperbolic wavefront and absence of [001] modes consistent with previous reports^{10,11}. In Fig. 2e, we show line profiles extracted from the white dashed lines in Fig. 2a. The [100] and [001] modes have anisotropic wavelengths (compare the red and blue lines). Additionally, in Fig. 2f, we show [100] line profiles from either side of the WO_{x} edge in Fig. 2g (white dashed line) showing an increase in polariton wavelength upon doping. Fringes bend at the boundary between doped and undoped regions. The measured polariton wavelength on graphene/αMoO_{3} without WO_{x} is consistent with low to no doping, indicating that charge transfer is negligible between graphene and αMoO_{3} (Supplementary Fig. 4, Supplementary Note 3).
Nearfield amplitude images at \(\omega\)=900 cm^{−1} on monolayer and bilayer WSe_{2} samples with E_{F} ≈ 0.60 eV and E_{F} ≈ 0.45 eV are shown in Fig. 3a, b, respectively. The [001] direction, which prohibits polariton propagation in bare αMoO_{3} at this frequency, now hosts an SP^{3}. We observe a significant tunability of the [001] mode by modifying \({E}_{F}\) (Fig. 3c), while the wavelength of the [100] mode changes only slightly (compare blue and red lines). The degree of anisotropy between orthogonal directions can thus be tuned dramatically by modifying \({E}_{F}\). Both [100] and [001] modes disperse as \(\omega\) is tuned from 875–915 cm^{−1} (Fig. 3d), in agreement with the calculated dispersion (Fig. 3e). Note that SP^{3} modes merge with the firstorder hyperbolic phonon mode of the lower reststrahlen band below \({\omega }_{{LO}}^{c}\), becoming HP^{3} modes and distinguishing them from pure plasmons. We remark that [001] modes in Fig. 3b are reflected from a physical boundary of the WO_{x}/graphene layer that is 8° from the [001] direction (Supplementary Fig. 5). In Supplementary Fig. 6, we show that 8° yields a negligible correction to the dispersion calculation and we consider these modes to be effectively along the [001]. In Fig. 3e, the [100] HP^{3} has higherorder branches. Some extra fringes likely corresponding to higherorder modes are visible in 875 cm^{−1} and 885 cm^{−1} profiles (pink and purple in Fig. 3d). For reference, we also plot the polariton dispersion on bare αMoO_{3} (Fig. 3f). Modes are absent along the [001] direction for \(\omega\) = 875–915 cm^{−1}.
Topology and dissipation
We now expound on the exact topological transition point in graphene/αMoO_{3} and the role of dissipation. Figure 4a shows the IFC evolving with doping from hyperbola to peanut to extended circle. The peanutshaped IFC (blue contour in Fig. 4a) goes from concave to convex shape around the open angle \({\theta }_{c}\). Divergence of the wavevector \({q}_{c}\) at \({\theta }_{c}\) marks the topological transition, assuming no other \(q\) diverges first, since bounded and unbounded curves cannot be homeomorphic. \({q}_{c}\) increases as the graphene conductivity decreases (Fig. 4b), diverging at \({\sigma }_{g}=0\) in an approximate analytical equation neglecting inplane losses (dashed lines):
where \({q}_{p}\equiv i\omega /2\pi {\sigma }_{g}\) (Supplementary Note 1). Upon incremental doping away from zero conductivity, the IFC changes topology as momenta above \({\theta }_{c}\) gain large but finite values. In this picture, the topological transition point is exactly at \({\sigma }_{g}=0\) since even marginallyconducting IFCs will be bounded and only the hyperbola at \({\sigma }_{g}=0\) will be unbounded. Full numerical calculations (squares in Fig. 4b, Equation S5) support this trend up to low conductivities.
In practice, [001] modes will not be observable until appreciable Fermi energy \({E}_{F}\) with modest \({\sigma }_{g}\) because propagation lengths become shorter, or \(q^{\prime\prime}\) increases, as \({E}_{F}\) decreases. In Fig. 4c, \(q^{\prime\prime}\) plotted in polar coordinates is the imaginary counterpart of the IFC referred to as the loss contour. Loss contour calculations consider finite thickness αMoO_{3} with intrinsic material scattering rate γ = 4 cm^{−1} and intrabandonly graphene with dopingindependent scattering rate \({\gamma }_{g}=40\) cm^{−1}. Loss contours show \(q\hbox{''}\) increasing along all directions as \({E}_{F}\) decreases, consistent with finitedifference timedomain simulations (Supplementary Fig. 8). The exact \({E}_{F}\) at which [001] modes are first observable, initiating this practical topological transition, will thus depend on the spatial resolution and the thickness of the αMoO_{3} slab (Supplementary Fig. 9). Additionally, at very low doping, plasmons pass the interband transition threshold and are overdamped^{29}. Nonlocal and other corrections to the graphene conductivity will also become important^{30}.
Loss contours at intermediate doping (e.g., solid blue line in Fig. 4c) reveal that \(q\hbox{''}\) decreases as the wavevector rotates away from the [001] direction (imagine black arrow rotating with length matching the loss contour). Dissipation can thus be reduced relative to unhybridized graphene (dashed blue line) by changing propagation direction since \(\gamma \, < \, {\gamma }_{g}\). The same holds for the scattering rate \(\Gamma ={v}_{g}q^{\prime\prime}\), where \({v}_{g}\) is the group velocity (Fig. 4d). The scattering rate in the twodimensional approximation (red line in Fig. 1i) provides some intuition on where the excess dissipation goes. Recall that the αMoO_{3} phonon splits the graphene plasmon into two modes \({\omega }_{+}\) and \({\omega }_{}\) with respective scattering rates Γ_{+} and Γ_{−}. As the wavevector angle \(\theta\) rotates in Fig. 1i, the \({\omega }_{+}\) loss contour (red line) moves from stronger coupling where \({\Gamma }_{+} < {\Gamma }_{}\), through a degeneracy where \({\omega }_{+}\) and \({\omega }_{}\) share losses equally, to weaker coupling where \({\Gamma }_{+} \, > \, {\Gamma }_{}\). The \({\omega }_{}\) loss follows the opposing path (pink line) with \({\Gamma }_{}\) increasing when \({\Gamma }_{+}\) decreases and vice versa such that \({\Gamma }_{+}+{\Gamma }_{}={\gamma }_{g}+\gamma\) for fixed \({\gamma }_{g}\) and \(\gamma\).
In summary, we have demonstrated an electronicallyinduced topological transition and the existence of SP^{3}/HP^{3} hybrid modes in a doped graphene/αMoO_{3} heterostructure by direct imaging of polaritons using an sSNOM. Our observations are consistent with full calculations for the electrodynamics of coupled plasmonphonon modes. Also, we demonstrated the tunability of hybrid polariton modes by modifying doping level and excitation frequency. Further, our nanoimaging data augmented with calculations reveal an intimate relationship between plasmonphonon coupling and the properties of hybridized hyperbolic polaritons. Graphene/αMoO_{3} integrated into a gated structure will serve as a nanophotonic platform for engineering devices with dynamical isofrequency topologies and directional plasmonphonon coupling. Furthermore, patterning WO_{x}/graphene on αMoO_{3} could inspire new forms of photonic crystals with subwavelength periodic changes in isofrequency topology. Also, combining a graphene/αMoO_{3} device with an ultrafast laser may permit picosecond^{31} switching of isofrequency topology for timevarying metasurfaces.
Final note: the authors became aware of the following relevant works by ÁlvarezPérez et al.^{32}; Bapat et al.^{33}; Hu et al.^{34}; and Zeng et al.^{35} after completion of this work.
Methods
Sample fabrication
Highquality WO_{x}doped graphene/αMoO_{3} heterostructures were fabricated using a polycaprolactone polymerbased dry transfer technique^{36}. First, we exfoliate αMoO_{3} bulk crystals purchased from a materials supplier (HQ Graphene) onto SiO_{2}/Si substrates. A monolayer or bilayer WSe_{2} on graphene heterostructure is then transferred onto a predetermined αMoO_{3} flake. Finally, we treat samples in a UVozone generator (Jelight UVOcleaner) at 300 K to oxidize the topmost WSe_{2} monolayer into WO_{x}^{37}.
Scanning nearfield optical microscopy
A Neaspec neaSNOM nearfield microscope was used with a DRS Daylight Solutions Hedgehog tunable quantum cascade laser TLSSK41112HHG (λ=10.86–11.67 µm). Standard PtIrcoated Arrow tips with 75 kHz resonance frequencies were used with tapping amplitudes of about 40 nm in contact. The signal localized under the apex of the tip is isolated in the backscattered signal by demodulation at the 1st−5th tip tapping harmonics in a pseudoheterodyne detection scheme^{38}.
Electrodynamics simulations
Simulations of the electric field distributions on graphene/MoO_{3} heterostructures were performed with the Ansys Lumerical finite difference time domain (FDTD) solver for Maxwell’s equations (https://www.lumerical.com/products/fdtd/). We used a point dipole source positioned 20 nm above the top surface of the graphene/MoO_{3} structure and polarized along the z direction. Electric field monitors were placed on the top surface. The boundary conditions were set to the perfectly matched layer option. The NumPy fast Fourier transform library was used to compute isofrequency contours from FDTD wavefronts. In particular, we used the fft2 function followed by the fftshift function to set the zerofrequency component to the center.
Data fitting procedure
To extract polariton momenta \(q^{\prime}\) and dissipation \(q^{\prime\prime}\) from nearfield amplitude images, it is necessary to fit a damped oscillatory function to line profiles. In this work, we use a zerothorder Hankel function of the first kind \({H}_{0}^{1}\) of the following form:
where the fitting parameters are \(A,B,C,q^{\prime}\) and \({q}^{^{\prime\prime} }\). The factor of two in the argument of the Hankel function comes from the fact that we are observing interference fringes between tiplaunched and edgereflected polaritons. The \({x}_{0}\) value was fixed to match the position of the first peak, which was then cut off during fitting so that contributions from higherorder modes may be neglected.
Raman spectroscopy
Raman measurements were conducted on a commercial Bruker Senterra system using an excitation wavelength of 532 nm at 2 mW with a 60 second integration time. A grating of 1200 line/mm was used to attain an energy resolution of 3–5 cm^{−1}.
Data availability
Relevant data supporting the key findings of this study are available within the article and the Supplementary Information file. All raw data generated during the current study are available from the corresponding authors upon request.
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Acknowledgements
The authors wish to thank S.H. Park and T. Low at the University of Minnesota for helpful discussions related to exceptional points in plasmonphonon coupled systems. Research at Columbia University is solely supported as part of Programmable Quantum Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award DESC0019443. WSe_{2} synthesis was supported by the Center on PrecisionAssembled Quantum Materials, funded through the US National Science Foundation (NSF) Materials Research Science and Engineering Centers (award no. DMR2011738). D.N.B. is Moore Investigator in Quantum Materials EPIQS #9455. The Flatiron Institute is a division of the Simons Foundation.
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F.L.R. and B.S.Y.K. contributed equally to this work. F.L.R. performed the analysis and sSNOM measurements. B.S.Y.K., A.R., and S.L. fabricated the heterostructures. Z.S. developed the semianalytical model. D.J.R. and A.S.M. assisted with the analysis, and D.J.R. performed Raman measurements. A.J.M., J.C.H., and D.N.B. are the principal investigators who helped direct the course of the research.
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Ruta, F.L., Kim, B.S.Y., Sun, Z. et al. Surface plasmons induce topological transition in graphene/αMoO_{3} heterostructures. Nat Commun 13, 3719 (2022). https://doi.org/10.1038/s4146702231477z
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DOI: https://doi.org/10.1038/s4146702231477z
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