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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.

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 (


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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.

Author information




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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The authors declare no competing interests.

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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.

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 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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