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Probing topological phase transitions using high-harmonic generation


The prediction and realization of topological insulators have sparked great interest in experimental approaches to the classification of materials1,2,3. The phase transition between non-trivial and trivial topological states is important, not only for basic materials science but also for next-generation technology, such as dissipation-free electronics4. It is therefore crucial to develop advanced probes that are suitable for a wide range of samples and environments. Here we demonstrate that circularly polarized laser-field-driven high-harmonic generation is distinctly sensitive to the non-trivial and trivial topological phases in the prototypical three-dimensional topological insulator bismuth selenide5. The phase transition is chemically initiated by reducing the spin–orbit interaction strength through the substitution of bismuth with indium atoms6,7. We find strikingly different high-harmonic responses of trivial and non-trivial topological surface states that manifest themselves as a conversion efficiency and elliptical dichroism that depend both on the driving laser ellipticity and the crystal orientation. The origins of the anomalous high-harmonic response are corroborated by calculations using the semiconductor optical Bloch equations with pairs of surface and bulk bands. As a purely optical approach, this method offers sensitivity to the electronic structure of the material, including its nonlinear response, and is compatible with a wide range of samples and sample environments.

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Fig. 1: Probing topological phase transitions using HHG.
Fig. 2: Ellipticity-dependent HHG in topologically trivial and non-trivial materials.
Fig. 3: Intensity- and helicity-dependent HHG generation for the topologically trivial and non-trivial samples.
Fig. 4: Elliptical dichroism.

Data availability

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon request.


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This work is primarily supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division through the AMOS program. Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. C.H. acknowledges support from the Humboldt Fellowship and the W. M. Keck Foundation. Y.K. acknowledges the Urbanek–Chorodow Fellowship from Stanford University. D.R.B. acknowledges support from the Swiss National Science Foundation (SNSF) through projects P2EZP2_184255 and P400P2_194343. D.J. and S.O. are supported by National Science Foundation’s DMR2004125 and MURI W911NF2020166. J.A.S., M.H. and P.S.K. were supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering. Theoretical calculations were performed using the Sherlock HPC cluster at Stanford University. We thank Alexis Chacón and Fang Liu for fruitful discussions.

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Authors and Affiliations



C.H., Y.K. and S.G. conceived the project. C.H. and Y.K. designed the setup. C.H. and Y.K. performed the experiments and analysed the data. T.F.H., D.A.R. and S.G. supervised the project. D.R.B. developed the theoretical model and performed the numerical calculations. D.J. and S.O. synthesized and characterized the samples. C.H., M.H. and J.A.S. performed the ARPES measurements at SSRL. All authors contributed to the interpretation of data.

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Correspondence to Christian Heide or Shambhu Ghimire.

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Nature Nanotechnology thanks Dieter Bauer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 ARPES measurements.

ARPES measurements of Bi2Se3 for 5 different indium concentrations.

Extended Data Fig. 2 Topological phase transition.

Ratio of HH yield for circular vs. linear excitation for various indium concentrations x. Measurements are performed at peak field E0 = 0.15 V/nm.

Extended Data Fig. 3 Transition from ordinary to anomalous ellipticity dependence.

a, b, c, Simulations of crystal orientation and MIR ellipticity dependent HH yield for three different field strength E0 = 0.06, 0.14, 0.25 V/nm. d, The ratio between HH obtained under circular and linear excitation for Φ= 0 is shown for various field strengths. The blue filled dots are the experimentally obtained values, the red line represents values from the simulation. The threshold field strength E0,threshold is defined when this ratio becomes 1. e, The threshold field scales inversely with the wavelengths.

Supplementary information

Supplementary Information

Supplementary Sections 1–7 and Figs. 1–5.

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Heide, C., Kobayashi, Y., Baykusheva, D.R. et al. Probing topological phase transitions using high-harmonic generation. Nat. Photon. 16, 620–624 (2022).

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