Lightwave control of topological properties in 2D materials for sub-cycle and non-resonant valley manipulation


Modern light generation technology offers extraordinary capabilities for sculpting light pulses, with full control over individual electric field oscillations within each laser cycle1,2,3. These capabilities are at the core of lightwave electronics—the dream of ultrafast lightwave control over electron dynamics in solids on a sub-cycle timescale, aiming at information processing at petahertz rates4,5,6,7,8. Here, bringing the frequency-domain concept of topological Floquet systems9,10 to the few-femtosecond time domain, we develop a theoretical method that can be implemented with existing technology, to control the topological properties of two-dimensional materials on few-femtosecond timescales by controlling the sub-cycle structure of non-resonant driving fields. We use this method to propose an all-optical, non-element-specific technique, physically transparent in real space, to coherently write, manipulate and read selective valley excitation using fields carried in a wide range of frequencies and on timescales that are orders of magnitude shorter than the valley lifetime, crucial for the implementation of valleytronic devices11,12.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Light-induced modification of the band structure with tailored field.
Fig. 2: Selective valley excitation in strong fields.
Fig. 3: Strong-field manipulation and optical reading of valley polarization.
Fig. 4: Time-dependent AHC in hBN induced by the bicircular field.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.


  1. 1.

    Wirth, A. et al. Synthesized light transients. Science 334, 195–200 (2011).

    ADS  Article  Google Scholar 

  2. 2.

    Kfir, O. et al. Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics. Nat. Photon. 9, 99–105 (2014).

    ADS  Article  Google Scholar 

  3. 3.

    Eckle, P. et al. Attosecond angular streaking. Nat. Phys. 4, 565–570 (2008).

    Article  Google Scholar 

  4. 4.

    Goulielmakis, E. et al. Attosecond control and measurement: lightwave electronics. Science 317, 769–775 (2007).

    ADS  Article  Google Scholar 

  5. 5.

    Krausz, F. & Ivanov, M. Attosecond physics. Rev. Mod. Phys. 81, 163–234 (2009).

    ADS  Article  Google Scholar 

  6. 6.

    Krausz, F. & Stockman, M. I. Attosecond metrology: from electron capture to future signal processing. Nat. Photon. 8, 205–213 (2014).

    ADS  Article  Google Scholar 

  7. 7.

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

    ADS  Article  Google Scholar 

  8. 8.

    Wolter, B. et al. Strong-field physics with mid-IR fields. Phys. Rev. X 5, 021034 (2015).

    Google Scholar 

  9. 9.

    Oka, T. & Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406 (2009).

    ADS  Article  Google Scholar 

  10. 10.

    Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).

    Article  Google Scholar 

  11. 11.

    Schaibley, J. R. et al. Valleytronics in 2D materials. Nat. Rev. Mater. 1, 16055 (2016).

    ADS  Article  Google Scholar 

  12. 12.

    Vitale, S. A. et al. Valleytronics: opportunities, challenges and paths forward. Small 14, 1801483 (2018).

    Article  Google Scholar 

  13. 13.

    Mak, K. F., He, K., Shan, J. & Heinz, T. F. Control of valley polarization in monolayer MoS2 by optical helicity. Nat. Nanotechnol. 7, 494–498 (2012).

    ADS  Article  Google Scholar 

  14. 14.

    Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

    ADS  Article  Google Scholar 

  15. 15.

    Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).

    ADS  Article  Google Scholar 

  16. 16.

    Oliaei Motlagh, S. A., Wu, J.-S., Apalkov, V. & Stockman, M. I. Femtosecond valley polarization and topological resonances in transition metal dichalcogenides. Phys. Rev. B 98, 081406 (2018).

    ADS  Article  Google Scholar 

  17. 17.

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

    ADS  Article  Google Scholar 

  18. 18.

    Ghimire, S. et al. Observation of high-order harmonic generation in a bulk crystal. Nat. Phys. 7, 138–141 (2010).

    Article  Google Scholar 

  19. 19.

    Schiffrin, A. et al. Optical-field-induced current in dielectrics. Nature 493, 70–74 (2012).

    ADS  Article  Google Scholar 

  20. 20.

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

    ADS  Article  Google Scholar 

  21. 21.

    Liu, H. et al. High-harmonic generation from an atomically thin semiconductor. Nat. Phys. 13, 262–265 (2016).

    Article  Google Scholar 

  22. 22.

    Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    ADS  Article  Google Scholar 

  23. 23.

    Dutreix, C., Stepanov, E. A. & Katsnelson, M. I. Laser-induced topological transitions in phosphorene with inversion symmetry. Phys. Rev. B 93, 241404 (2016).

    ADS  Article  Google Scholar 

  24. 24.

    Eichmann, H. et al. Polarization-dependent high-order two-color mixing. Phys. Rev. A 51, R3414–R3417 (1995).

    ADS  Article  Google Scholar 

  25. 25.

    Pisanty, E. & Jiménez-Galán, Á. Strong-field approximation in a rotating frame: high-order harmonic emission from p states in bicircular fields. Phys. Rev. A 96, 63401 (2017).

    ADS  Article  Google Scholar 

  26. 26.

    Nag, T., Slager, R.-J., Higuchi, T. & Oka, T. Dynamical synchronization transition in interacting electron systems. Phys. Rev. B 100, 134301 (2019).

    ADS  Article  Google Scholar 

  27. 27.

    Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    ADS  Article  Google Scholar 

  28. 28.

    Tancogne-Dejean, N. & Rubio, A. Atomic-like high-harmonic generation from two-dimensional materials.Sci. Adv. 4, eaao5207 (2018).

    ADS  Article  Google Scholar 

  29. 29.

    Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, 1489–1492 (2014).

    ADS  Article  Google Scholar 

  30. 30.

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

    ADS  Article  Google Scholar 

  31. 31.

    Barth, I. & Smirnova, O. Spin-polarized electrons produced by strong-field ionization. Phys. Rev. A 88, 013401 (2013).

    ADS  Article  Google Scholar 

  32. 32.

    Silva, R. E. F., Martín, F. & Ivanov, M. High harmonic generation in crystals using maximally localized Wannier functions. Phys. Rev. B 100, 195201 (2019).

    ADS  Article  Google Scholar 

  33. 33.

    Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).

    Article  Google Scholar 

  34. 34.

    Mostofi, A. A. et al. wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 178, 685–699 (2008).

    ADS  Article  Google Scholar 

  35. 35.

    Cayssol, J., Dóra, B., Simon, F. & Moessner, R. Floquet topological insulators. Phys. Status Solidi RRL 7, 101–108 (2013).

    Article  Google Scholar 

  36. 36.

    Yudin, D., Eriksson, O. & Katsnelson, M. I. Dynamics of quasiparticles in graphene under intense circularly polarized light. Phys. Rev. B 91, 075419 (2015).

    ADS  Article  Google Scholar 

Download references


Á.J.-G. and M.I. acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) Quantum Dynamics in Tailored Intense Fields (QUTIF) grant IV 152/6-1. R.E.F.S. and M.I. acknowledge support from the Engineering and Physical Sciences Research Council/Defence Science and Technology Laboratory (EPSRC/DSTL) Multidisciplinary University Research Initiative (MURI) grant no. EP/N018680/1. R.E.F.S. acknowledges support from the European Research Council Starting Grant (ERC-2016-STG714870). O.S. acknowledges support from the DFG Schwerpunktprogramm 1840 Quantum Dynamics in Tailored Intense Fields project SM 292/5-1 and Molecular Electron Dynamics Investigated by Intense Fields and Attosecond Pulses (MEDEA) project, which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant no. 641789.

Author information




All authors developed the idea. Á.J.-G. and R.E.F.S. developed and performed the numerical calculations and analysed the data. M.I. and O.S. developed the analytical treatment. Á.J.-G. and M.I. wrote the main part of the manuscript, which was discussed by all authors.

Corresponding authors

Correspondence to Á. Jiménez-Galán or M. Ivanov.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Supplementary information

Supplementary Information

Supplementary discussion and Figs. 1–6.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jiménez-Galán, Á., Silva, R.E.F., Smirnova, O. et al. Lightwave control of topological properties in 2D materials for sub-cycle and non-resonant valley manipulation. Nat. Photonics 14, 728–732 (2020).

Download citation

Further reading


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing