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Tunable non-integer high-harmonic generation in a topological insulator


When intense lightwaves accelerate electrons through a solid, the emerging high-order harmonic (HH) radiation offers key insights into the material1,2,3,4,5,6,7,8,9,10,11. Sub-optical-cycle dynamics—such as dynamical Bloch oscillations2,3,4,5, quasiparticle collisions6,12, valley pseudospin switching13 and heating of Dirac gases10—leave fingerprints in the HH spectra of conventional solids. Topologically non-trivial matter14,15 with invariants that are robust against imperfections has been predicted to support unconventional HH generation16,17,18,19,20. Here we experimentally demonstrate HH generation in a three-dimensional topological insulator—bismuth telluride. The frequency of the terahertz driving field sharply discriminates between HH generation from the bulk and from the topological surface, where the unique combination of long scattering times owing to spin–momentum locking17 and the quasi-relativistic dispersion enables unusually efficient HH generation. Intriguingly, all observed orders can be continuously shifted to arbitrary non-integer multiples of the driving frequency by varying the carrier-envelope phase of the driving field—in line with quantum theory. The anomalous Berry curvature warranted by the non-trivial topology enforces meandering ballistic trajectories of the Dirac fermions, causing a hallmark polarization pattern of the HH emission. Our study provides a platform to explore topology and relativistic quantum physics in strong-field control, and could lead to non-dissipative topological electronics at infrared frequencies.

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Fig. 1: HH emission from a TI.
Fig. 2: Dependence of the HH emission on the CEP of the driving field.
Fig. 3: Microscopic origin of high-efficiency HHG from the TSS.
Fig. 4: Tracing geometric phase effects in HH radiation from the TSS.

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Data availability

The data supporting the findings of this study are available from the corresponding authors upon request. Source data are provided with this paper.

Code availability

The in-house program package CUED that was used to solve the SBE is freely available from GitHub (


  1. Chin, A. H., Calderón, O. G. & Kono, J. Extreme midinfrared nonlinear optics in semiconductors. Phys. Rev. Lett. 86, 3292–3295 (2001).

    Article  ADS  CAS  PubMed  Google Scholar 

  2. Ghimire, S. et al. Observation of high-order harmonic generation in a bulk crystal. Nat. Phys. 7, 138–141 (2011).

    Article  CAS  Google Scholar 

  3. Schubert, O. et al. Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations. Nat. Photon. 8, 119–123 (2014).

    Article  ADS  CAS  Google Scholar 

  4. Hohenleutner, M. et al. Real-time observation of interfering crystal electrons in high-harmonic generation. Nature 523, 572–575 (2015).

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Luu, T. T. et al. Extreme ultraviolet high-harmonic spectroscopy of solids. Nature 521, 498–502 (2015).

    Article  ADS  CAS  PubMed  Google Scholar 

  6. Vampa, G. et al. Linking high harmonics from gases and solids. Nature 522, 462–464 (2015); corrigendum 542, 260 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  7. Garg, M. et al. Multi-petahertz electronic metrology. Nature 538, 359–363 (2016).

    Article  ADS  CAS  PubMed  Google Scholar 

  8. Yoshikawa, N., Tamaya, T. & Tanaka, K. High-harmonic generation in graphene enhanced by elliptically polarized light excitation. Science 356, 736–738 (2017).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  9. Sivis, M. et al. Tailored semiconductors for high-harmonic optoelectronics. Science 357, 303–306 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  10. Hafez, H. A. et al. Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions. Nature 561, 507–511 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  11. Floss, I. et al. Ab initio multiscale simulation of high-order harmonic generation in solids. Phys. Rev A. 97, 011401(R) (2018).

    Article  ADS  Google Scholar 

  12. Langer, F. et al. Lightwave-driven quasiparticle collisions on a subcycle timescale. Nature 533, 225–229 (2016).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  13. Langer, F. et al. Lightwave valleytronics in a monolayer of tungsten diselenide. Nature 557, 76–80 (2018).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  14. Chen, Y. L. et al. Experimental realization of a three-dimensional topological insulator, Bi2Te3. Science 325, 178–181 (2009).

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Giorgianni, F. et al. Strong nonlinear terahertz response induced by Dirac surface states in Bi2Se3 topological insulator. Nat. Commun. 7, 11421 (2016).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  16. Bauer, D. & Hansen, K. K. High-harmonic generation in solids with and without topological edge states. Phys. Rev. Lett. 120, 177401 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  17. Reimann, J. et al. Subcycle observation of lightwave-driven Dirac currents in a topological surface band. Nature 562, 396–400 (2018).

    Article  ADS  CAS  PubMed  Google Scholar 

  18. Silva, R. E. F., Jiménez-Galán, A., Amorim, B., Smirnova, O. & Ivanov, M. Topological strong field physics on sub-laser cycle timescale. Nat. Photon. 13, 849–854 (2019).

    Article  ADS  CAS  Google Scholar 

  19. Baykusheva, D. et al. Strong-field physics in three-dimensional topological insulators. Phys. Rev. A 103, 023101 (2021).

    Article  ADS  CAS  Google Scholar 

  20. Wilhelm, J. et al. Semiconductor-Bloch formalism: derivation and application to high-harmonic generation from Dirac fermions. Phys. Rev. B 103, 125419 (2021).

    Article  ADS  CAS  Google Scholar 

  21. Higuchi, T., Heide, C., Ullmann, K., Weber, H. B. & Hommelhoff, P. Light-field-driven currents in graphene. Nature 550, 224–228 (2017).

    Article  ADS  PubMed  Google Scholar 

  22. McIver, J. W. et al. Light-induced anomalous Hall effect in graphene. Nat. Phys. 16, 38–41 (2020).

    Article  CAS  PubMed  Google Scholar 

  23. Kuroda, K., Reimann, J., Güdde, J. & Höfer, U. Generation of transient photocurrents in the topological surface state of Sb2Te3 by direct optical excitation with midinfrared pulses. Phys. Rev. Lett. 116, 076801 (2016).

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Wu, L. et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science 354, 1124–1127 (2016).

    Article  ADS  MathSciNet  CAS  PubMed  MATH  Google Scholar 

  25. Mahmood, F. et al. Selective scattering between Floquet–Bloch and Volkov states in a topological insulator. Nat. Phys. 12, 306–310 (2016).

    Article  CAS  Google Scholar 

  26. Braun, L. et al. Ultrafast photocurrents at the surface of the three-dimensional topological insulator Bi2Se3. Nat. Commun. 7, 13259 (2016).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  27. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Luu, T. T. & Wörner, H. J. Measurement of the Berry curvature of solids using high-harmonic spectroscopy. Nat. Commun. 9, 916 (2018).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  29. Banks, H. B. et al. Dynamical birefringence: electron–hole recollisions as probes of Berry curvature. Phys. Rev. X 7, 041042 (2017).

    Google Scholar 

  30. Cheng, B. et al. Efficient terahertz harmonic generation with coherent acceleration of electrons in the Dirac semimetal Cd3As2. Phys. Rev. Lett. 124, 117402 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  31. Sell, A., Leitenstorfer, A. & Huber, R. Phase-locked generation and field-resolved detection of widely tunable terahertz pulses with amplitudes exceeding 100 MV/cm. Opt. Lett. 33, 2767–2769 (2008).

    Article  ADS  CAS  PubMed  Google Scholar 

  32. Michiardi, M. et al. Bulk band structure of Bi2Te3. Phys. Rev. B 90, 075105 (2014).

    Article  ADS  CAS  Google Scholar 

  33. Austin. I. G. The optical properties of bismuth telluride. Proc. Phys. Soc. 72, 545–552 (1958).

    Article  ADS  CAS  Google Scholar 

  34. Kokh, K. A. et al. Melt growth of bulk Bi2Te3 crystals with a natural p–n junction. CrystEngComm 16, 581–584 (2014).

    Article  CAS  Google Scholar 

  35. Liu, C.-X. et al. Model Hamiltonian for topological insulators. Phys. Rev. B 82, 045122 (2010).

    Article  ADS  Google Scholar 

  36. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    Article  ADS  MathSciNet  CAS  MATH  Google Scholar 

  37. Gradhand, M. et al. First-principle calculations of the Berry curvature of Bloch states for charge and spin transport of electrons. J. Phys. Condens. Matter 24, 213202 (2012).

    Article  ADS  CAS  PubMed  Google Scholar 

  38. Kane, E. O. Zener tunneling in semiconductors. J. Phys. Chem. Solids 12, 181–188 (1960).

    Article  ADS  CAS  Google Scholar 

  39. Lange, C. et al. Extremely nonperturbative nonlinearities in GaAs driven by atomically strong terahertz fields in gold metamaterials. Phys. Rev. Lett. 113, 227401 (2014).

    Article  ADS  CAS  PubMed  Google Scholar 

  40. Junginger, F. et al. Nonperturbative interband response of a bulk InSb semiconductor driven off resonantly by terahertz electromagnetic few-cycle pulses. Phys. Rev. Lett. 109, 147403 (2012).

    Article  ADS  CAS  PubMed  Google Scholar 

  41. Kira, M. & Koch, S. W. Semiconductor Quantum Optics (Cambridge Univ. Press, 2012).

  42. Mikhailov, S. A. Non-linear electromagnetic response of graphene. Europhys. Lett. 79, 27002 (2007).

    Article  ADS  Google Scholar 

  43. Huard, S. Polarization of Light (Wiley, 1997).

  44. Junk, V., Reck, P., Gorini, C. & Richter, K. Floquet oscillations in periodically driven Dirac systems. Phys. Rev. B 101, 134302 (2020).

    Article  ADS  CAS  Google Scholar 

  45. Krückl, V. Wave Packets in Mesoscopic Systems: From Time-dependent Dynamics to Transport Phenomena in Graphene and Topological Insulators. PhD thesis, Univ. Regensburg (2013).

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We thank P. Merkl, J. Freudenstein, C. Lange, D. E. Kim, M. Nitsch and I. Floss for helpful discussions. The work in Regensburg has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project ID 422 314695032-SFB 1277 (Subprojects A03, A05 and A07) as well as project HU1598/8. Work in Marburg has been supported by the Deutsche Forschungsgemeinschaft (DFG) through Project ID 223848855-SFB 1083 and grant number GU 495/2. O.E.T. and K.A.K. have been supported by the Russian Science Foundation (project number 17-12-01047) and the state assignment of IGM SB RAS and ISP SB RAS. The work of J.C. was supported by the NSF (National Science Foundation) DMR-1828489.

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Authors and Affiliations



F.E., J.W., K.R., U.H. and R.H. conceived the study. K.A.K. and O.E.T. provided the high-quality Bi2Te3 samples and performed transport measurements. C.P.S., L.W., S.S., S.I., M.M., N.H., D.A. and J.G. set up and carried out the optical experiments and characterized the sample orientation. C.P.S., L.W., P.G., J.C., F.E. and J.W. developed and carried out the semiconductor Bloch equations simulations. V.J., C.G. and K.R. set up and carried out the quantum mechanical wave-packet simulation. C.P.S., L.W. and P.G. implemented the semiclassical calculations. All authors analysed the data and discussed the results. C.P.S. and R.H. wrote the manuscript with contributions from all authors.

Corresponding authors

Correspondence to J. Wilhelm, K. Richter or R. Huber.

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Peer review information Nature thanks Olga Smirnova, Ryusuke Matsunaga and Alexander Kemper for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Crystallographic orientation of the Bi2Te3 sample.

a, Low-energy electron diffraction of the Bi2Te3 sample measured with an electron energy of 78 eV. The white spots mark the reciprocal lattice vectors. b, Schematic of the reciprocal lattice vectors (blue) overlaid with the surface Brillouin zone (red) and the corresponding high-symmetry directions.

Extended Data Fig. 2 Carrier-injection into bulk states and comparison of bulk- and surface-state HHG.

a, Calculated carrier density injected, within one THz half cycle, into the bulk conduction band by Zener tunnelling, as a function of the peak THz electric field. Red horizontal line: carrier density, nbulk, injected for ETHz = 3 MV cm−1. b, HH spectra, IHH, calculated by Boltzmann equations for massive electrons in the bulk crystal described by a cosine-shaped band structure (blue curve), and by Boltzmann equation for Dirac electrons in the surface state of a topological insulator. Inset: corresponding band structures for bulk (blue curve) and surface state electrons (red curve).

Source Data

Extended Data Fig. 3 Comparison of the intensity of bulk and surface HHG.

a, HH intensity, IHH, for two select driving frequencies. Above-bandgap excitation at νTHz = 37 THz (red curve) allows for strong bulk contributions. For νTHz = 28 THz (blue curve) resonant interband transitions in the bulk are not possible and the peak electric THz field (about 3 MV cm−1) is too low for efficient non-resonant interband excitation. Therefore, the bulk contribution drops by orders of magnitude and the observed spectrum is dominated by HHG from the metallic TSS. In a direct comparison of the two spectra, this contribution is reduced with respect to the above-bandgap bulk HH intensity by only one order of magnitude. b, A direct comparison of the same spectra IHH as a function of the harmonic order, n, instead of the harmonic frequency, even reveals a slight enhancement of HHG in the TSS (νTHz = 28 THz) with respect to the above-bandgap bulk HHG (νTHz = 37 THz). Considering the low effective thickness of the TSS of about 1 nm compared with the optical penetration depth of about 30 nm to 100 nm over which bulk HHG is collected, this comparison attests to the strong nonlinearity of Dirac electrons.

Source Data

Extended Data Fig. 4 CEP dependence for HHG in the TSS.

HH intensity, IHH, generated in the TSS as a function of the CEP, φCEP, along the black dotted line in the inset. The intensity of the emitted HHs monotonically increases with increasing φCEP.

Source Data

Extended Data Fig. 5 SBE simulation of CEP-dependent HHG from TSS without interband transitions.

Numerical simulation of IHH from the TSS with the SBEs, as in Fig. 2e, but deactivated interband transitions. This calculation is equivalent to the semiclassical solution using the Boltzmann equation, which accounts only for intraband dynamics. The results reproduce both the CEP dependence observed in the experiment of Fig. 2b and the full SBE results of Fig. 2e.

Source Data

Extended Data Fig. 6 Momentum-space origin of HHG in the TSS.

a, Calculated HH spectra, IHH, (black curve) for two test charges placed at the wave vector ky = ±0.001 Å−1 (kx = 0), as obtained from a semiclassical solution of the equations of motion (νTHz = 25 THz, ETHz = 0.1 MV cm−1). b, HH intensity (colour scale) of order n = 15 (see arrow and red dotted area in a) as a function of the starting point (kxky) of the test charges in momentum space.

Source Data

Extended Data Fig. 7 Quantum mechanical wave-packet motion in the TSS.

a, Top: normalized vector potential, ATHz, of the driving multi-THz waveform (frequency νTHz = 25 THz; peak electric field ETHz = 1 MV cm−1). Dashed lines highlight the zero crossings of the vector potential and the momentum space trajectories. Bottom: group velocity components of the electrons in the TSS parallel (vx, blue) and perpendicular (vy, red) to the THz driving field calculated by solving the full time-dependent Schrödinger equation. Both components reverse sign during zero crossings of the momentum space trajectories. b, Real space trajectory of lightwave-driven Dirac electrons calculated by the velocities in a.

Source Data

Extended Data Fig. 8 Extended analysis of the polarimetry measurements.

Extracted orientation angle, α, ellipticity angle, γ, and degree of polarization, σ, as a function of the harmonic order, n. Although α shows an alternating behaviour for even and odd orders, the ellipticity remains relatively small for all orders. The degree of polarization, σ, decreases with increasing order, but still remains sufficiently high to guarantee a reliable extraction of α and γ.

Source Data

Extended Data Fig. 9 Zitterbewegung in a topological surface state.

Left: three-dimensional scheme of the Dirac-like electron dispersion of the TSS. The blue arrow highlights the quantum interference of different branches of the Dirac system. Right: high-frequency oscillations (blue waveform) indicative of Zitterbewegung depend on the energy separation of the interfering states residing at different energy branches in our quantum mechanical calculations. The black waveform represents the driving THz field ETHz.

Source Data

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Schmid, C.P., Weigl, L., Grössing, P. et al. Tunable non-integer high-harmonic generation in a topological insulator. Nature 593, 385–390 (2021).

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