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# Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers

## Abstract

Twisted two-dimensional bilayer materials exhibit many exotic electronic phenomena. Manipulating the ‘twist angle’ between the two layers enables fine control of the electronic band structure, resulting in magic-angle flat-band superconductivity1,2, the formation of moiré excitons3,4,5,6,7,8 and interlayer magnetism9. However, there are limited demonstrations of such concepts for photons. Here we show how analogous principles, combined with extreme anisotropy, enable control and manipulation of the photonic dispersion of phonon polaritons in van der Waals bilayers. We experimentally observe tunable topological transitions from open (hyperbolic) to closed (elliptical) dispersion contours in bilayers of α-phase molybdenum trioxide (α-MoO3), arising when the rotation between the layers is at a photonic magic twist angle. These transitions are induced by polariton hybridization and are controlled by a topological quantity. At the transitions the bilayer dispersion flattens, exhibiting low-loss tunable polariton canalization and diffractionless propagation with a resolution of less than λ0/40, where λ0 is the free-space wavelength. Our findings extend twistronics10 and moiré physics to nanophotonics and polaritonics, with potential applications in nanoimaging, nanoscale light propagation, energy transfer and quantum physics.

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## Data availability

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

## Change history

• ### 08 July 2020

The online publication date in the printed version of this article was listed incorrectly as 10 June 2020; the date was correct online.

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## Acknowledgements

We thank Z. Q. Xu for discussions and help with the sample fabrication. This work was performed at the CUNY Advanced Science Research Center (ASRC), National University of Singapore (NUS) and the Melbourne Centre for Nanofabrication (MCN) in the Victorian Node of the Australian National Fabrication Facility (ANFF). G.H., A.K., Y.M. and A.A. acknowledge support from the Air Force Office of Scientific Research (MURI grant no. FA9550-18-1-0379), the Department of Defense under the Vannevar Bush Fellowship program, the Office of Naval Research (grant no. N00014-19-1-2011), the Simons Foundation and the National Science Foundation. Q.O. acknowledges support from ARC Centre of Excellence in Future Low-Energy Electronics Technologies (FLEET). J.W. acknowledges the A*STAR Career Development Award Funding from the Science and Engineering Research Council (grant no. A1820g0087). Z.D. acknowledges the National Natural Science Foundation of China (grant no. 51601131). Q.B. acknowledges support from the Australian Research Council (ARC, FT150100450, CE170100039 and IH150100006). C.-W.Q. acknowledges financial support from A*STAR Pharos Program (grant number 15270 00014, with project number R-263-000-B91-305).

## Author information

Authors

### Contributions

G.H. and A.A. conceived the idea. G.H., A.A., Y.M., A.K. and C.-W.Q. developed the theory. G.H. performed the simulations and advised in the experimental designs. Q.O. and Q.B. led the experiments. Q.O. designed the structures, and performed the optical measurements. G.S. and Q.O. fabricated the defects and edges on the sample. Y.W. contributed to material synthesis. G.H., Q.O., J.W., Z.D., Q.Z., Q.B., C.-W.Q. and A.A. analysed the data and all authors discussed the results. G.H., A.A., Q.O., Q.B. and C.-W.Q. wrote the manuscript, with input and comments from all authors. A.A., Q.B. and C.-W.Q. supervised the project.

### Corresponding authors

Correspondence to Qiaoliang Bao, Cheng-Wei Qiu or Andrea Alù.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Zhe Fei, Vincenzo Giannini and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Properties of a single α-MoO3 layer.

a, Permittivity of α-MoO3. ε11 and ε22 are two permittivity tensor components along the [100] and [001] crystal directions, respectively. The permittivity is obtained by fitting the obtained data (refs. 24,25,40) with the Lorentzian model. b, In-plane open angle (β) of a single layer hyperbolic α-MoO3 flake. c, Isofrequency dispersion of a single α-MoO3 layer (with a thickness of 100 nm) in free space at 925.9 cm−1. The red line is the theoretical dispersion, the dashed black lines are the asymptotic line of the hyperbolic dispersion and the green arrow shows the group velocity direction at large wavevector. The open angle therefore illustrates the behaviour of the asymptotic dispersion line as well as the group velocity directions.

### Extended Data Fig. 2 Rotation-induced dispersion engineering of tBL α-MoO3.

a, b, The crossing of two dispersion lines of the top and bottom layer upon anticlockwise (a) and clockwise (b) rotation. The black line is the dispersion of the bottom layer; the yellow, purple, green, blue and red lines correspond to the dispersion of the top layer at twist angles of ±30°, ±45°, ±70°, ±80° and ±90°, respectively. cf, The dispersion of a tBL α-MoO3 sample (d1 = d2 = 100 nm, $${\varepsilon }_{d}=1$$) at twist angles of ±30° (c), ±70° (d), ±80° (e) and ±90° (f). The plotted dispersion lines correspond to the sample of the bottom layer without rotation (solid black lines), the dispersion of the top layer (solid purple lines), the analytical dispersion for tBL α-MoO3 with ds = 1 nm (solid red lines), and the analytical dispersion for tBL α-MoO3 with ds = 0.1 nm (dashed blue lines), ds = 10 nm (dashed green lines) and ds = 0 nm (dashed cyan lines). gj, The dispersion of a tBL α-MoO3 sample (d1 = d2 = 100 nm, ds = 1 nm). The plotted dispersion lines correspond to the sample of the bottom layer without rotation (solid black lines), the dispersion of the top layer (solid purple lines) and the analytical dispersion for tBL α-MoO3 with εd = 1 (solid red lines), εd = 2 (dashed green lines), εd = 3 (dashed blue lines), εd = 4 (dashed cyan lines), εd = 5 (dashed dark yellow lines) and εd = 10 (dashed orange lines).

### Extended Data Fig. 3 Numerical field distributions and dispersions for a single layer α-MoO3 at ω = 925.93 cm−1.

aj, The numerical field distribution (Re(Ez); top), the numerically obtained dispersions (FFT(Re(Ez)); contours, bottom) and the analytical dispersions (red dashed lines, bottom), at thicknesses (H) of 40 nm (a), 50 nm (b), 60 nm (c), 80 nm (d), 100 nm (e), 200 nm (f), 300 nm (g), 400 nm (h), 500 nm (i) and 600 nm (j).

### Extended Data Fig. 4 Numerical field distributions and dispersions of twisted bilayer α-MoO3 at a fixed thickness of d2 = 200 nm.

ae, The numerical field distribution (Re(Ez); top), the numerically obtained dispersions (FFT(Re(Ez)); contours, bottom) and the analytical dispersions (red dashed lines, bottom) for top-layer thicknesses (d1) of 40 nm (a), 100 nm (b), 200 nm (c), 300 nm (d) and 400 nm (e) at ω = 925.93 cm−1. The dashed light magenta lines, dashed cyan lines and solid red lines are the analytical dispersions of the individual bottom layer, individual top layers and twisted bilayers, respectively. The twist angle is 30° in all cases.

### Extended Data Fig. 5 Numerical result for a top layer of fixed thickness of d1 = 200 nm.

ae, The numerical field distribution (Re(Ez); top), the numerically obtained dispersions (FFT(Re(Ez)); contours, bottom) and the analytical dispersions (red dashed lines, bottom) for bottom-layer thicknesses (d2) of 40 nm (a), 100 nm (b), 200 nm (c), 300 nm (d) and 400 nm (e) at ω = 925.93 cm−1. The dashed light magenta lines, dashed cyan lines and solid red lines are the analytical dispersions of individual bottom layers, individual top layers and twisted bilayers, respectively. The twist angle is 30° in all cases.

### Extended Data Fig. 6 Mechanism of defect-enabled observation of PhPs.

The metallic AFM tip in the s-SNOM setup could scatter the incident field and launch the highly confined PhPs (referred to as tip-launched PhPs and shown as red/blue solid lines) via near-field light–matter interactions. For case A, the AFM tip is close to the point defect in the sense that the tip-launched PhPs could mostly reach and then be scattered by the point defect (referred to as reflected PhPs and shown as red and blue dashed lines). Thus the wave could be reflected, and strong signals of total PhPs would be expected. For case B, the AFM tip is far from the point defect in the sense that the tip-launched PhPs cannot reach the point defect or the reflected signals cannot be collected by the AFM tip before it is mostly decayed. Therefore, no signal or only weak signals are expected from the reflected PhPs. Note that the optical path should be twice the distance between the tip and point defect, leading to the measured field $${{\rm{e}}}^{-{\rm{i}}{\vec{k}}_{{\rm{P}}{\rm{h}}{\rm{P}}}\cdot 2(\vec{r}-{\vec{r}}_{{\rm{P}}{\rm{D}}})}$$. Therefore, when performing the spatial Fourier transform, the obtained spectrum should be of $$2{\vec{k}}_{{\rm{PhP}}}$$, that is, the value of the dispersion lines should be multiplied by 2.

### Extended Data Fig. 7 Comparison of experimental and numerical results of the field distribution and the dispersion at ω = 903.8 cm−1.

a, Experimentally measured field distribution (left) and dispersion (right) of a single α-MoO3 layer of thickness 125 nm (the same as in Fig. 2e, i). b, Numerical field distribution (left) launched by the dipole and the obtained dispersion (right) of a single α-MoO3 layer of thickness 125 nm. c, Experimentally measured field distribution (left) and the dispersion (right) of the tBL α-MoO3 flake with Δθ = −44°, d1 = d2 = 128 nm (the same as in Fig. 2f, j). d, Numerical field distribution (left) launched by the dipole and the obtained dispersion (right) of the tBL α-MoO3 flake with Δθ = −44°, d1 = d2 = 128 nm. e, Experimentally measured field distribution (left) and dispersion (right) of the tBL α-MoO3 flake with Δθ = 65°, d1 = 120 nm and d2 = 235 nm (the same as in Fig. 2g, k). f, Numerical field distribution (left) launched by the dipole and the obtained dispersion (right) of the tBL α-MoO3 flake with Δθ = 65°, d1 = 120 nm and d2 = 235 nm. g, Experimentally measured field distribution (left) and dispersion (right) of the tBL α-MoO3 flake with Δθ = −77°, d1 = 125 nm, and d2 = 210 nm (the same as in Fig. 2h, l). h, Numerical field distribution (left) launched by the dipole and the obtained dispersion (right) of the tBL α-MoO3 flake with Δθ = −77°, d1 = 125 nm and d2 = 210 nm. In the dispersion plot, the red lines correspond to the analytical dispersions (kx, ky) and the white lines correspond to the analytical dispersion with a factor 2—that is, (2kx, 2ky). The experimentally obtained dispersion fits better for the white-coloured dispersion (2kx, 2ky) owing to doubled optical path via the collection of field by the tip, whereas the numerical one fits better for the dashed red dispersion line (kx, ky) as it is propagating wave.

### Extended Data Fig. 8 Observation of the topological transition at ω = 925.9 cm−1.

ai, Optical images (a, d, g), experimentally measured field distributions (b, e, h) and the obtained dispersions (c, f, i) of tBL α-MoO3 flakes. For ac, Δθ = −33°, d1 = 224 nm and d2 = 280 nm. For df, Δθ = 65°, d1 = 120 nm and d2 = 235 nm. For gi, Δθ = −77°, d1 = 125 nm and d2 = 210 nm. These results show that for small twist angles (|Δθ| = 33°, 65°; that is, <72°), the obtained dispersions are hyperbolic, and for large twist angles (|Δθ| = 77° > 72°), the dispersion is elliptical.

### Extended Data Fig. 9 The canalization angle at ω = 903.8 cm−1.

a, The near-field images near the point defect in Fig. 4a. b, The cut-line plots of the PhP signal at different propagation distances along the group velocity $${\vec{v}}_{{\rm{g}}}$$ from the point defect. c, The extracted FWHM from different line profiles at different propagation lengths, along the group velocity $${\vec{v}}_{{\rm{g}}}$$. The reference of zero distance is the black line.

### Extended Data Fig. 10 Field canalization and extreme anisotropy at ω = 903.8 cm−1.

a, The field distribution (|Ez|) of a single α-MoO3 flake with a thickness of 100 nm. The permittivity is artificially set as ε11 = −100 + 100i, ε22 = 1.1 and εzz = 8.9. The energy mostly propagates along the x axis, which is caused by a flattened band at the extreme anisotropy. b, The line plot of the field along the dashed blacked line in a. The red dots are the numerical value and the blue line is the fitting. This plot gives the exponential decay $${k}_{{\rm{i}}}^{-1}=0.35\,{\rm{\mu }}{\rm{m}}$$. c, The decay length with respect to εi (the imaginary part of ε11), as ε11 = −100 + εi × i, while the other permittivity components do not change.

## Supplementary information

### Supplementary Information

This file contains Supplementary Discussions 1–4.

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Hu, G., Ou, Q., Si, G. et al. Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers. Nature 582, 209–213 (2020). https://doi.org/10.1038/s41586-020-2359-9

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• DOI: https://doi.org/10.1038/s41586-020-2359-9

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