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arxiv: 2604.06051 · v1 · submitted 2026-04-07 · ❄️ cond-mat.mes-hall · cond-mat.other· physics.atom-ph· physics.optics

Disentangling High Harmonic Generation from Surface and Bulk States of a Topological Insulator

Sha Li , Wenyi Zhou , Kazi A. Imroz , Yaguo Tang , Tiana A. Townsend , Vyacheslav Leshchenko , Larissa Boie , Pierre Agostini
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Alexandra S. Landsman Roland K. Kawakami Lun Yue Louis F. DiMauro
This is my paper

Pith reviewed 2026-05-10 18:42 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.otherphysics.atom-phphysics.optics
keywords high harmonic generationtopological insulatorsBi2Se3surface statesbulk statesBerry curvatureshift vectorthin films
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The pith

Tuning the thickness of topological insulator films and applying a terahertz field separates high harmonic generation from surface and bulk states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates high harmonic generation in the topological insulator Bi2Se3. It shows that ultrathin films of about 6 nm thickness make the harmonic emission come mostly from the topological surface states, while thicker 50 nm films are dominated by the bulk states. A quasi-static terahertz perturbing field is used to further disentangle these contributions and to highlight the role of the surface states' shift vector and Berry curvature in the process. This provides a practical way to isolate the responses of the topological surface states, addressing questions about detecting topological features through high harmonic generation.

Core claim

We demonstrate that the contributions of bulk and surface states to the harmonic emission can be controlled by tuning the thickness of thin film samples. An ultrathin (6 nm) film substantially enhances HHG from the surface states, while the bulk states dominate HHG in a thicker (50 nm) film. By applying a quasi-static terahertz perturbing field, we disentangle the bulk and surface responses and reveal the significant impact of the surface states' shift vector and Berry curvature on HHG.

What carries the argument

Controlling high harmonic generation contributions through sample thickness variation and a quasi-static terahertz perturbing field, which isolates the effects of surface states' shift vector and Berry curvature.

If this is right

  • Contributions to harmonic emission can be controlled by tuning thin film thickness.
  • Ultrathin films enhance surface state high harmonic generation.
  • Thicker films have high harmonic generation dominated by bulk states.
  • Quasi-static terahertz fields disentangle bulk and surface responses.
  • The surface states' shift vector and Berry curvature have significant impact on high harmonic generation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This method of thickness tuning may allow similar isolation of surface state signals in other topological insulator materials.
  • Resolving whether topological signatures can be extracted from HHG this way could enable new optical probes of Berry curvature in condensed matter systems.

Load-bearing premise

The thickness of the film and the quasi-static terahertz perturbing field cleanly separate the surface and bulk state contributions to high harmonic generation without other thickness-dependent effects interfering.

What would settle it

Experimental observation that high harmonic generation does not vary with film thickness between 6 nm and 50 nm films or that the terahertz field does not produce distinct changes in the emission spectra would falsify the ability to disentangle the states.

Figures

Figures reproduced from arXiv: 2604.06051 by Alexandra S. Landsman, Kazi A. Imroz, Larissa Boie, Louis F. DiMauro, Lun Yue, Pierre Agostini, Roland K. Kawakami, Sha Li, Tiana A. Townsend, Vyacheslav Leshchenko, Wenyi Zhou, Yaguo Tang.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
read the original abstract

The discovery of topological phases has introduced a new dimension to materials science. Three-dimensional (3D) topological insulators (TIs) are a remarkable class of matter that is insulating in the bulk while hosting conductive topological surface states (TSSs) with unique charge and spin properties. High-order harmonic generation (HHG) has emerged as a powerful tool to probe condensed matter systems by providing insights into their electronic structure and dynamic behavior. Here, we investigate HHG in the prototype 3D-TI Bi$_2$Se$_3$. We demonstrate that the contributions of bulk and surface states to the harmonic emission can be controlled by tuning the thickness of thin film samples. An ultrathin (6 nm) film substantially enhances HHG from the surface states, while the bulk states dominate HHG in a thicker (50 nm) film. By applying a quasi-static terahertz perturbing field, we disentangle the bulk and surface responses and reveal the significant impact of the surface states' shift vector and Berry curvature on HHG. Our study provides effective methods for isolating the optical responses of TSSs from those of the bulk, which opens the door to resolving an ongoing debate regarding whether it is possible to reliably extract topological signatures in HHG.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports an experimental investigation of high-harmonic generation (HHG) in Bi₂Se₃ thin films. It claims that film thickness can be used to control the relative contributions of topological surface states (TSS) versus bulk states to the HHG emission, with an ultrathin 6 nm film enhancing surface-state HHG while a 50 nm film is bulk-dominated. Application of a quasi-static terahertz perturbing field is used to disentangle the two responses, allowing the authors to attribute observed differences to the shift vector and Berry curvature of the surface states. The work concludes by proposing these controls as methods to isolate TSS optical responses and address debates on extracting topological signatures from HHG.

Significance. If the thickness-based separation and THz disentanglement are shown to be robust against confounding mechanisms, the result would provide a practical experimental route to isolating topological surface-state contributions in nonlinear optics of 3D TIs. This could help resolve questions about whether Berry curvature or shift-vector effects produce identifiable signatures in HHG spectra, and the approach might generalize to other topological systems.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (thickness-dependent HHG data): the central claim that 6 nm films are surface-dominated while 50 nm films are bulk-dominated rests on the assumption that thickness variation achieves orthogonal isolation. At ~6 nm (a few quintuple layers), hybridization between top and bottom surface states is expected to open a gap and renormalize the Berry curvature and shift vector; the manuscript must supply either ARPES data, tight-binding calculations, or explicit checks showing that these effects do not alter the HHG attribution in the ultrathin limit.
  2. [§4] §4 (THz perturbation and disentanglement procedure): the quasi-static THz field is asserted to selectively modulate one channel, enabling subtraction. The paper should demonstrate that this perturbation does not introduce comparable intraband acceleration, heating, or new interband pathways in both films; quantitative field-strength dependence, polarization-resolved spectra, or a model of the subtraction protocol (with error propagation) are needed to establish that the extracted surface response is not a linear combination of multiple thickness-dependent mechanisms.
minor comments (2)
  1. [Methods and Figure captions] Figure captions and methods section: include sample characterization (XRD, AFM, transport) and substrate details to allow assessment of possible band-bending or defect contributions that scale with thickness but are not surface-specific.
  2. [Introduction and Theory] Notation: define the shift vector and Berry curvature explicitly in the context of the HHG process (e.g., which harmonic orders are most sensitive) rather than assuming reader familiarity with the prior literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us improve the clarity and rigor of our presentation. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (thickness-dependent HHG data): the central claim that 6 nm films are surface-dominated while 50 nm films are bulk-dominated rests on the assumption that thickness variation achieves orthogonal isolation. At ~6 nm (a few quintuple layers), hybridization between top and bottom surface states is expected to open a gap and renormalize the Berry curvature and shift vector; the manuscript must supply either ARPES data, tight-binding calculations, or explicit checks showing that these effects do not alter the HHG attribution in the ultrathin limit.

    Authors: We agree that hybridization in the ultrathin limit must be carefully addressed to support the thickness-based separation. Our data show a pronounced enhancement of higher-order harmonics in the 6 nm films relative to thicker ones, consistent with surface-state dominance. In the revised manuscript we have added a dedicated paragraph in §3 that references established tight-binding calculations for Bi₂Se₃ films of comparable thickness (6 QL), which indicate that any hybridization gap remains small (~few–10 meV) and does not substantially renormalize the shift vector or Berry curvature in the energy window probed by our HHG measurements. We also include an explicit scaling analysis of HHG yield versus thickness that matches expectations for surface versus bulk contributions without invoking strong hybridization corrections. While ARPES on the precise samples used here is not available, the consistency with prior ARPES and transport studies on similar films supports our attribution. We view this as a partial but substantive clarification. revision: partial

  2. Referee: [§4] §4 (THz perturbation and disentanglement procedure): the quasi-static THz field is asserted to selectively modulate one channel, enabling subtraction. The paper should demonstrate that this perturbation does not introduce comparable intraband acceleration, heating, or new interband pathways in both films; quantitative field-strength dependence, polarization-resolved spectra, or a model of the subtraction protocol (with error propagation) are needed to establish that the extracted surface response is not a linear combination of multiple thickness-dependent mechanisms.

    Authors: We appreciate the need for quantitative validation of the THz disentanglement. In the revised §4 and supplementary information we now present field-strength dependence data for both 6 nm and 50 nm films, showing that the THz modulation remains perturbative (linear response regime) with no detectable heating (no thermal broadening or irreversible changes) or additional interband channels (polarization dependence preserved). We have added a step-by-step description of the subtraction protocol together with propagated uncertainties, demonstrating that the extracted surface spectrum is robust across a range of THz amplitudes and is not a spurious linear combination of thickness-dependent effects. These additions confirm the selectivity of the perturbation to the surface channel. revision: yes

Circularity Check

0 steps flagged

No circularity; experimental demonstration without derivations or fitted predictions

full rationale

The paper is an experimental study demonstrating control of HHG contributions via film thickness (6 nm vs 50 nm) and THz perturbation to isolate surface vs bulk responses in Bi2Se3. No equations, derivations, parameter fits, or theoretical reductions appear in the abstract or described methods; claims rest on observed differences in harmonic emission and their attribution to shift vector/Berry curvature effects. The work contains no self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce results to inputs by construction. This is a standard experimental isolation technique with independent content from measurements, yielding a self-contained result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is experimental; the abstract does not introduce new free parameters, axioms, or invented entities beyond standard assumptions in nonlinear optics and topological materials. Specific details on any fitting or background assumptions are absent.

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Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages

  1. [1]

    +” and “−

    Tight-binding model We consider a tight-binding model for the TSSs in Bi 2Se3 [20, 52–54] previously derived in Ref. [20], which captures the correct symmetries of the TSSs and their properties such as spin-momentum locking and hexagonal warping [55] at large crystal momenta. The Hamiltonian of the TSSs can be parametrized to the form Hk = 3X i=0 Bk i σi,...

    work page

  2. [2]

    Gauge-invariant dynamics We implemented a gauge-invariant formulation of the semiconductor Bloch equations (GI-SBEs) [39] to simulate the dynamics of the TSSs driven by intense, linearly polarized electric fields. In such a formalism, each term in the GI-SBEs is gauge invariant, which allows us to turn on/off certain properties such as dipole magnitudes a...

    work page

  3. [3]

    & Pepper, M

    von Klitzing, K., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance.Phys. Rev. Lett.45, 494-497 (1980)

    work page 1980

  4. [4]

    Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with frac- tionally charged excitations.Phys. Rev. Lett.50, 1395-1398 (1983)

    work page 1983

  5. [5]

    J., Kohmoto, M., Nightingale, M

    Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential.Phys. Rev. Lett.49, 405-408 (1982)

    work page 1982

  6. [6]

    Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators.Rev. Mod. Phys.82, 3045- 3067 (2010)

    work page 2010

  7. [7]

    & Zhang, S.-C

    Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors.Rev. Mod. Phys.83, 1057-1110 (2011)

    work page 2011

  8. [9]

    Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase.Nature452, 970-974 (2008)

    work page 2008

  9. [10]

    Chen, Y. L. et al. Experimental realization of a three-dimensional topological insulator, Bi2Te3.Science325, 178-181 (2009)

    work page 2009

  10. [11]

    Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface.Nat. Phys.5, 398-402 (2009)

    work page 2009

  11. [12]

    K¨ onig, M. et al. Quantum spin Hall insulator state in HgTe quantum wells.Science318, 766-770 (2007). 14

    work page 2007

  12. [13]

    Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator.Science340, 167-170 (2013)

    work page 2013

  13. [14]

    Wang, Y. H. et al. Observation of a warped helical spin texture in Bi 2Se3 from circular dichroism angle-resolved photoemission spectroscopy.Phys. Rev. Lett.107, 207602 (2011)

    work page 2011

  14. [18]

    Ferray, M. et al. Multiple-harmonic conversion of 1064 nm radiation in rare gases.J. Phys. B: At. Mol. Opt. Phys.21, L31 (1988)

    work page 1988

  15. [19]

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

    work page 2011

  16. [20]

    & Reis, D

    Ghimire, S. & Reis, D. A. High-harmonic generation from solids.Nat. Phys.15, 10-16 (2019)

    work page 2019

  17. [21]

    Reimann, J. et al. Subcycle observation of lightwave-driven Dirac currents in a topological surface band.Nature562, 396-400 (2018)

    work page 2018

  18. [22]

    Baykusheva, D. et al. Strong-field physics in three-dimensional topological insulators.Phys. Rev. A103, 023101 (2021)

    work page 2021

  19. [23]

    & Hansen, K

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

    work page 2018

  20. [24]

    Silva, R. E. F., Jim´ enez-Gal´ an, ´A., Amorim, B., Smirnova, O. & Ivanov, M. Topological strong-field physics on sub-laser-cycle timescale.Nat. Photon.13, 849-854 (2019)

    work page 2019

  21. [25]

    Influence of chirp and carrier- envelope phase on noninteger high-harmonic generation.Phys

    Graml, M., Nitsch, M., Seith, A., Evers, F & Wilhelm, J. Influence of chirp and carrier- envelope phase on noninteger high-harmonic generation.Phys. Rev. B107, 054305 (2023)

    work page 2023

  22. [26]

    S., Reis, D

    You, Y. S., Reis, D. A. & Ghimire, S. Anisotropic high-harmonic generation in bulk crystals. Nat. Phys.13, 345-349 (2017)

    work page 2017

  23. [27]

    Saito, N. et al. Observation of selection rules for circularly polarized fields in high-harmonic generation from a crystalline solid.Optica4, 1333-1336 (2017)

    work page 2017

  24. [28]

    & Tanaka, K

    Yoshikawa, N., Tamaya, T. & Tanaka, K. High-harmonic generation in graphene enhanced by elliptically polarized light excitation.Science356, 736-738 (2017). 15

    work page 2017

  25. [29]

    & Rubio, A

    Neufeld, O., Tancogne-Dejean, N., H¨ ubener, H., De Giovannini, U. & Rubio, A. Are there universal signatures of topological phases in high-harmonic generation? Probably not.Phys. Rev. X13, 031011 (2019)

    work page 2019

  26. [31]

    & Fang, Z

    Zhang, W., Yu, R., Zhang, H.-J., Dai, X. & Fang, Z. First-principles studies of the three- dimensional strong topological insulators Bi2Te3, Bi2Se3 and Sb2Te3.New J. Phys.12, 065013 (2010)

    work page 2010

  27. [32]

    & Cohen, O

    Neufeld, O., Podolsky, D. & Cohen, O. Floquet group theory and its application to selection rules in harmonic generation.Nat. Commun.10, 405 (2019)

    work page 2019

  28. [33]

    & Beswick, A

    Ben-Tal, N., Moiseyev, N. & Beswick, A. The effect of Hamiltonian symmetry on generation of odd and even harmonics.J. Phys. B: At. Mol. Opt. Phys.26, 3017-3024 (1993)

    work page 1993

  29. [34]

    Rotational Coefficient

    Hall, E. H. XVIII. On the “Rotational Coefficient” in nickel and cobalt.Philos. Mag.12, 157-172 (1881)

    work page

  30. [35]

    Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys.82, 1539-1592 (2010)

    work page 2010

  31. [36]

    & Kraut, W

    von Baltz, R. & Kraut, W. Theory of the bulk photovoltaic effect in pure crystals.Phys. Rev. B23, 5590-5596 (1981)

    work page 1981

  32. [37]

    Sipe, J. E. & Shkrebtii, A. I. Second-order optical response in semiconductors.Phys. Rev. B 61, 5337-5352 (2000)

    work page 2000

  33. [38]

    Young, S. M. & Rappe, A. M. First principles calculation of the shift current photovoltaic effect in ferroelectrics.Phys. Rev. Lett.109, 116601 (2012)

    work page 2012

  34. [39]

    & Gaarde, M

    Yue, L. & Gaarde, M. B. Characterizing anomalous high-harmonic generation in solids.Phys. Rev. Lett.130, 166903 (2023)

    work page 2023

  35. [40]

    Qian, C. et al. Role of shift vector in high-harmonic generation from noncentrosymmetric topological insulators under strong laser fields.Phys. Rev. X12, 021030 (2022)

    work page 2022

  36. [41]

    M., Moloney, J

    Parks, A. M., Moloney, J. V. & Brabec, T. Gauge invariant formulation of the semiconductor Bloch equations.Phys. Rev. Lett.131, 236902 (2023)

    work page 2023

  37. [42]

    Luu, T. T. & W¨ orner, H. J. Measurement of the Berry curvature of solids using high-harmonic spectroscopy.Nat. Commun.9, 916 (2018). 16

    work page 2018

  38. [43]

    Lv, Y.-Y. et al. High-harmonic generation in Weyl semimetalβ-WP 2 crystals.Nat. Commun. 12, 6437 (2021)

    work page 2021

  39. [44]

    Faeyrman, L. et al. Revealing the Berry phase under the tunneling barrier. Preprint at https://arxiv.org/abs/2408.03105 (2024)

    work page arXiv 2024

  40. [45]

    Bai, Y. et al. Probing Berry phase effect in topological surface states.Phys. Rev. Lett.133, 243801 (2024)

    work page 2024

  41. [46]

    Zhang, Y. et al. Crossover of the three-dimensional topological insulator Bi 2Se3 to the two- dimensional limit.Nat. Phys.6, 584-588 (2010)

    work page 2010

  42. [51]

    W., Maker, P

    Terhune, R. W., Maker, P. D. & Savage, C. M. Optical harmonic generation in calcite.Phys. Rev. Lett.8, 404-406 (1962)

    work page 1962

  43. [52]

    Brahlek, M. et al. Topological-metal to band-insulator transition in (Bi 1−xInx)2Se3 thin films. Phys. Rev. Lett.109, 186403 (2012)

    work page 2012

  44. [53]

    A., Shafique, S

    Levy, I., Garcia, T. A., Shafique, S. & Tamargo, M. C. Reduced twinning and surface roughness of Bi2Se3 and Bi2Te3 layers grown by molecular beam epitaxy on sapphire substrates.J. Vac. Sci. Technol. B36, 02D107 (2018)

    work page 2018

  45. [54]

    Zhang, H. et al. Topological insulators in Bi 2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface.Nat. Phys.5, 438-442 (2009)

    work page 2009

  46. [55]

    & Shen, S.-Q

    Shan, W.-Y., Lu, H.-Z. & Shen, S.-Q. Effective continuous model for surface states and thin films of three-dimensional topological insulators.New J. Phys.12, 043048 (2010)

    work page 2010

  47. [56]

    Liu, C.-X. et al. Model Hamiltonian for topological insulators.Phys. Rev. B82, 045122 (2010)

    work page 2010

  48. [57]

    Hexagonal warping effects in the surface states of the topological insulator Bi 2Te3

    Fu, L. Hexagonal warping effects in the surface states of the topological insulator Bi 2Te3. Phys. Rev. Lett.103, 266801 (2009). 17

    work page 2009

  49. [58]

    Chac´ on, A. et al. Circular dichroism in higher-order harmonic generation: Heralding topolog- ical phases and transitions in Chern insulators.Phys. Rev. B102, 134115 (2020)

    work page 2020

  50. [59]

    & Gaarde, M

    Yue, L. & Gaarde, M. B. Introduction to theory of high-harmonic generation in solids: tutorial. J. Opt. Soc. Am. B39, 535–555 (2022)

    work page 2022

  51. [60]

    visualized

    Houston, W. V. Acceleration of electrons in a crystal lattice.Phys. Rev.57, 184-186 (1940). ACKNOWLEDGMENTS This material is based upon work supported by the US Air Force Office of Scientific Research under Grant No. FA9550-2-110415 and US Department of Energy BES under Grant No. DE-FG02-04ER15614. A. S. L. and L. Y. acknowledge support from the Center fo...

    work page 1940

  52. [61]

    Thickness-dependent efficiency of surface HHG in Bi 2Se3 thin films 2

    work page

  53. [62]

    Additional measurements: 5 th-order harmonic generation in the 6 nm film 7

    work page

  54. [63]

    Roles of Berry curvature and shift vector in HHG 8

    work page

  55. [64]

    HHG by the MIR driver alone: Roles of shift vector and Berry curvature 11 4.2

    Simulation results of HHG in TSSs of Bi 2Se3 10 4.1. HHG by the MIR driver alone: Roles of shift vector and Berry curvature 11 4.2. THz-assisted HHG: Coupling between intrinsic and THz-field-induced symmetry breaking 12 References 15 1

    work page

  56. [65]

    In an ideal, free-standing Bi 2Se3 film crystal with slab geometry, the two surfaces—top and bottom—are structurally identical

    Thickness-dependent efficiency of surface HHG in Bi 2Se3 thin films In this section, we present a detailed explanation of the thickness-dependent surface HHG in Bi2Se3 and the substantial enhancement observed in ultrathin films below 10 nm. In an ideal, free-standing Bi 2Se3 film crystal with slab geometry, the two surfaces—top and bottom—are structurally...

    work page

  57. [66]

    This result is counterintuitive for an ultrathin (6 nm) film with minimum bulk volume

    Additional measurements: 5 th-order harmonic generation in the 6 nm film In the main text, we have shown that at a fixed MIR-THz time delay, the contrast value for the 5th-order harmonic is extremely small (main text Figure 5c), indicating that it is bulk- dominated. This result is counterintuitive for an ultrathin (6 nm) film with minimum bulk volume. To...

    work page

  58. [67]

    +” and “−

    Roles of Berry curvature and shift vector in HHG In this section, we briefly discuss the roles of Berry curvature (Ω) and shift vector (R) in strong-field HHG. Recent theoretical studies have shown the importance of including the Berry connection (A) and transition dipole phase (ϕ) in the semiconductor Bloch equations (SBEs) to cal- culate HHG in systems ...

    work page

  59. [68]

    TSSs model and HHG simulation methods

    Simulation results of HHG in TSSs of Bi 2Se3 In this section, we present the simulation results of HHG in TSSs of Bi 2Se3. The simu- lation details are provided in the “TSSs model and HHG simulation methods” subsection of theMethodssection in the main text. Our aim is to capture the essential features of how the shift vector and Berry curvature affect HHG...

    work page

  60. [69]

    & Guo, H

    Zhao, Y., Hu, Y., Liu, L., Zhu, Y. & Guo, H. Helical states of topological insulator Bi 2Se3. Nano Lett.11, 2088-2091 (2011)

    work page 2088

  61. [70]

    Kim, S. et al. Surface scattering via bulk continuum states in the 3D topological insulator Bi2Se3.Phys. Rev. Lett.107, 056803 (2011)

    work page 2011

  62. [71]

    Wang, Y. H. et al. Measurement of intrinsic Dirac Fermion cooling on the surface of the topo- logical insulator Bi 2Se3 using time-resolved and angle-resolved photoemission spectroscopy. Phys. Rev. Lett.109, 127401 (2012)

    work page 2012

  63. [72]

    Wang, Z. et al. The roles of photo-carrier doping and driving wavelength in high harmonic generation from a semiconductor.Nat. Commun.8, 1686 (2017)

    work page 2017

  64. [73]

    J., Nie, Z., de Keijzer, B

    van Essen, P. J., Nie, Z., de Keijzer, B. & Kraus, P. M. Toward complete all-optical intensity modulation of high-harmonic generation from solids.ACS Photon.11, 1832-1843 (2024)

    work page 2024

  65. [74]

    Huml´ ıˇ cek, J. et al. Raman and interband optical spectra of epitaxial layers of the topological insulators Bi2Te3 and Bi2Te3 on BaF2 substrates.Phys. Scr.2014, 014007 (2014)

    work page 2014

  66. [75]

    Schmid, C. P. et al. Tunable non-integer high-harmonic generation in a topological insulator. Nature593, 385-390 (2021)

    work page 2021

  67. [76]

    Bai, Y. et al. High-harmonic generation from topological surface states.Nat. Phys.17, 311-315 (2021)

    work page 2021

  68. [77]

    Heide, C. et al. Probing topological phase transitions using high-harmonic generation.Nat. Photon.16, 620-624 (2022)

    work page 2022

  69. [78]

    He, L. et al. Surface-dominated conduction in a 6 nm thick Bi 2Se3 thin film.Nano Lett.12, 1486-1490 (2012)

    work page 2012

  70. [79]

    Heide, C. et al. High-harmonic generation from artificially stacked 2D crystals.Nanophotonics 12, 255-261 (2023)

    work page 2023

  71. [80]

    Jiang, S. et al. Role of the transition dipole amplitude and phase on the generation of odd and even high-order harmonics in crystals.Phys. Rev. Lett.120, 253201 (2018)

    work page 2018

  72. [81]

    Li, J. et al. Phase invariance of the semiconductor Bloch equations.Phys. Rev. A100, 043404 (2019)

    work page 2019

  73. [82]

    & Gaarde, M

    Yue, L. & Gaarde, M. B. Structure gauges and laser gauges for the semiconductor Bloch equations in high-order harmonic generation in solids.Phys. Rev. A101, 053411 (2020). 15

    work page 2020

  74. [83]

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

    work page 1984

  75. [84]

    & Morimoto, T

    Kitamura, S., Nagaosa, N. & Morimoto, T. Nonreciprocal Landau–Zener tunneling.Commun. Phys.3, 63 (2020)

    work page 2020

  76. [85]

    Li, S. et al. High-order harmonic generation from a thin film crystal perturbed by a quasi-static terahertz field.Nat. Commun.14, 2603 (2023)

    work page 2023

  77. [86]

    Yang, B. C. & Robicheaux, F. Field-ionization threshold and its induced ionization-window phenomenon for Rydberg atoms in a short single-cycle pulse.Phys. Rev. A90, 063413 (2014)

    work page 2014

  78. [87]

    A., Hirori, H., Sanari, Y., Kanemitsu, Y

    Sato, S. A., Hirori, H., Sanari, Y., Kanemitsu, Y. & Rubio, A. High-order harmonic generation in graphene: Nonlinear coupling of intraband and interband transitions.Phys. Rev. B103, L041408 (2021)

    work page 2021

  79. [88]

    & Gaarde, M

    Yue, L. & Gaarde, M. B. Characterizing anomalous high-harmonic generation in solids.Phys. Rev. Lett.130, 166903 (2023). 16

    work page 2023