Second harmonic generation and third harmonic generation in topological insulator-based van der Waals metamaterials

Alessandra Di Gaspare; Craig Knox; Edmund H. Linfield; Joshua Freeman; Miriam S. Vitiello; Sara Ghayeb

arxiv: 2604.27864 · v1 · submitted 2026-04-30 · ⚛️ physics.optics · cond-mat.mtrl-sci

Second harmonic generation and third harmonic generation in topological insulator-based van der Waals metamaterials

Alessandra Di Gaspare , Sara Ghayeb , Craig Knox , Edmund H. Linfield , Joshua Freeman , Miriam S. Vitiello This is my paper

Pith reviewed 2026-05-07 05:46 UTC · model grok-4.3

classification ⚛️ physics.optics cond-mat.mtrl-sci

keywords topological insulatorssecond harmonic generationthird harmonic generationvan der Waals heterostructuresmetamaterialsterahertznonlinear opticssurface states

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The pith

Topological insulator van der Waals heterostructures embedded in split-ring resonators produce both even and odd high-harmonic generation in the 6.4-9.7 THz range.

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

The paper shows that topological insulators can generate both even and odd harmonics when their symmetry properties are combined with strong local field enhancement from metamaterial resonators. Bulk states remain centro-symmetric and yield only odd multiples, while the conducting surface states break inversion symmetry and add even multiples. This separation is observed through second-harmonic signals at 6.4 THz and third-harmonic signals at 9.7 THz. A reader would care because it supplies a practical route to solid-state THz frequency up-conversion in a spectral window that conventional lasers struggle to reach. The approach also offers an experimental handle on distinguishing surface from bulk contributions in these materials.

Core claim

In topological insulators the insulating bulk preserves inversion symmetry and therefore supports only odd-order harmonics, whereas the gapless surface states break that symmetry and permit even-order harmonics as well. By placing thin films or heterostructures of Bi2Se3 and related compounds inside arrays of single or double split-ring resonators, the local THz electric field is strongly amplified, driving observable up-conversion. The resulting spectrum contains both even harmonics (originating at the surfaces) and odd harmonics (originating in the bulk), directly confirming the predicted dual contribution.

What carries the argument

Split-ring resonator arrays that concentrate the driving THz field inside van der Waals topological insulator films, allowing the distinct symmetry properties of bulk centro-symmetric states (odd harmonics only) and surface symmetry-broken states (odd and even harmonics) to be read out separately.

If this is right

Compact solid-state sources become available for even-order THz frequencies that are otherwise difficult to reach.
The same resonator-plus-TI platform can be tuned across a range of driving frequencies to map the transition between surface-dominated and bulk-dominated regimes.
Nonlinear THz spectroscopy gains a direct probe of surface-state inversion-symmetry breaking without requiring surface-sensitive techniques such as ARPES.
Van der Waals heterostructures can be engineered to control the relative strength of even versus odd harmonics by adjusting layer thickness or composition.
The demonstrated field-enhancement geometry extends to other layered materials whose bulk and interface symmetries differ.

Where Pith is reading between the lines

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

The same symmetry-based separation could be used to test whether surface states remain topologically protected under intense THz driving.
Integration with gate-tunable devices might allow electrical control of the even-harmonic output, turning the structure into a THz modulator.
Because even harmonics require surface inversion breaking, their absence in a given sample could serve as a rapid diagnostic for surface degradation or contamination.
The approach is compatible with wafer-scale van der Waals stacking, suggesting a route to arrays of phase-locked THz sources.

Load-bearing premise

Even harmonics are produced exclusively by symmetry breaking at the topological surface states rather than by any fabrication defects or unintended asymmetries in the resonator array.

What would settle it

Even harmonics should vanish in an otherwise identical resonator array that contains a non-topological or inversion-symmetric material instead of the Bi2Se3 film, or when the surface states are deliberately suppressed while the bulk remains unchanged.

read the original abstract

High-order harmonic generation (HHG) in solids - the frequency up-conversion of an optical signal - is governed by symmetries. At terahertz (THz) frequencies, HHG is a key technology to access high frequency spectral windows that are usually difficult to cover using conventional solid state laser technologies. This effect has been recently exploited in graphene where HHG has been demonstrated, albeit only at odd multiples of the driving frequency owing to its inherent centro-symmetry. In topological insulators (TIs), the combination of spin-orbit interaction and time-reversal symmetry create an insulating bulk state with an inverted band order, inseparably connected with conducting surface states. TIs have been predicted to support unconventional high harmonic generation from the bulk and topological surface, which are usually difficult to be distinguished. However, no experimental results have been provided, so far. Here, we exploit the strong optical field amplification provided by an array of single or double split ring resonators, with embedded Bi2Se3 or (InxBi(1-x))2Se3/ Bi2Se3 van der Waals heterostructures, to achieve up-conversion in the 6.4 (even) - 9.7 (odd) THz frequency range. This results from bulk centro-symmetry (odd states) and symmetry breaking in the topological surface states (odd and even).

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.

Desk Editor's Note private letter to a colleague

The paper reports the first experimental THz HHG separating even and odd harmonics in TI-metamaterial devices, but the even-harmonic assignment to surface states is not isolated from the resonators' own symmetry breaking.

read the letter

The key point is that this work measures both even (6.4 THz) and odd (9.7 THz) harmonics from Bi2Se3 and In-doped heterostructures embedded in split-ring resonator arrays, attributing the even ones to symmetry breaking at the topological surface states and the odd ones to the bulk centro-symmetry. The abstract notes that no prior experiments had demonstrated this separation, so the result is new on that narrow point. The metamaterial provides the field enhancement needed to reach detectable signals in this frequency window, which is a practical engineering step for THz upconversion using van der Waals layers. The fabrication approach with both single and double resonators and different TI compositions shows some systematic variation in the setup. The main weakness is that split-ring resonators are chosen precisely because they break inversion symmetry to boost the local field. That same geometric asymmetry can produce even-order nonlinearities at THz frequencies on its own, without any contribution from the embedded material. The abstract gives no sign of control samples such as bare resonators, non-TI dielectrics, or thickness-dependent TI layers that would isolate the surface-state signal. If the full paper lacks those controls or equivalent checks, the central attribution remains ambiguous and alternative explanations like fabrication artifacts or interface effects are not ruled out. This paper is aimed at researchers working on nonlinear THz sources or optical probes of topological surface states. Someone building metamaterial-enhanced devices or looking for new ways to access even harmonics would find the experimental combination worth examining, even if the interpretation needs tightening. It deserves peer review because the combination of materials and the frequency range are fresh enough for referees to assess the data directly and request the missing controls if they are absent.

Referee Report

2 major / 1 minor

Summary. The manuscript reports experimental observation of high-order harmonic generation at THz frequencies in metamaterials formed by embedding Bi2Se3 or (InxBi(1-x))2Se3/Bi2Se3 van der Waals heterostructures within arrays of single or double split-ring resonators. The authors attribute the even harmonic (6.4 THz) to symmetry breaking at the topological surface states and the odd harmonic (9.7 THz) to the centro-symmetric bulk, enabled by resonator-induced field amplification.

Significance. If the central attribution is experimentally isolated and reproducible, the result would be significant: it would furnish the first direct experimental separation of bulk versus surface contributions to HHG in topological insulators, confirming long-standing theoretical predictions and offering a route to symmetry-engineered THz up-conversion sources. The metamaterial field-enhancement strategy itself is a practical strength.

major comments (2)

[Abstract] Abstract: The assignment of the even-order (6.4 THz) harmonic exclusively to symmetry breaking in the topological surface states lacks any reported control experiments (bare resonators, non-TI dielectrics, or thickness-dependent TI layers). Because the split-ring geometry is deliberately inversion-asymmetric to produce field enhancement, this structural feature can itself generate even-order nonlinearities at THz frequencies, rendering the surface-state attribution ambiguous. This is load-bearing for the paper's central claim.
[Abstract] Abstract: No experimental methods, driving frequency, incident power, detection scheme, spectral data, error bars, or conversion-efficiency values are supplied to substantiate the reported frequencies or their even/odd classification. Without these, the claim that the observed signals arise from the stated symmetries cannot be evaluated.

minor comments (1)

[Abstract] Abstract: The driving frequency should be stated explicitly so that the reader can verify that 6.4 THz and 9.7 THz correspond to the second and third harmonics, respectively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's insightful comments, which have helped us identify areas for improvement in the manuscript. We address each major comment below and have made revisions to clarify the experimental details and strengthen the attribution of the observed harmonics.

read point-by-point responses

Referee: [Abstract] Abstract: The assignment of the even-order (6.4 THz) harmonic exclusively to symmetry breaking in the topological surface states lacks any reported control experiments (bare resonators, non-TI dielectrics, or thickness-dependent TI layers). Because the split-ring geometry is deliberately inversion-asymmetric to produce field enhancement, this structural feature can itself generate even-order nonlinearities at THz frequencies, rendering the surface-state attribution ambiguous. This is load-bearing for the paper's central claim.

Authors: We agree that the split-ring geometry's inversion asymmetry could contribute to even harmonics, making the attribution to surface states potentially ambiguous without controls. To address this, we will add control experiments with bare resonators and non-TI dielectrics in the revised manuscript, along with a discussion of how the observed even harmonic at 6.4 THz is specifically linked to the topological surface states through symmetry breaking. We will also include thickness-dependent measurements to further isolate the surface contribution. revision: yes
Referee: [Abstract] Abstract: No experimental methods, driving frequency, incident power, detection scheme, spectral data, error bars, or conversion-efficiency values are supplied to substantiate the reported frequencies or their even/odd classification. Without these, the claim that the observed signals arise from the stated symmetries cannot be evaluated.

Authors: We note that abstracts are by nature concise and do not typically contain full methodological details. However, we will revise the abstract to include the driving frequency of 3.2 THz, typical incident powers, and a brief description of the detection scheme. The complete experimental methods, spectral data, error bars, and conversion efficiencies are detailed in the main text and supplementary materials. We will update the abstract to better reference these elements and substantiate the even/odd classification based on the symmetries. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental report with direct observations, no derivation chain

full rationale

The paper is an experimental report on high-harmonic generation in metamaterial structures embedding topological insulators. It describes fabrication of split-ring resonator arrays with Bi2Se3 or heterostructure layers, field enhancement, and observation of even (6.4 THz) and odd (9.7 THz) harmonics. No mathematical model, equations, fitted parameters, or derivation chain is presented that could reduce to its own inputs by construction. Claims rest on measured frequencies and symmetry arguments (bulk centro-symmetry for odd, surface-state breaking for even), but these are interpretive attributions of experimental data rather than self-referential predictions or self-definitional steps. Self-citations, if present, are not load-bearing for any calculation. The potential ambiguity in source of even harmonics (surface states vs. resonator geometry) is an experimental-control issue, not circularity. This matches the default non-circular outcome for papers without closed-loop derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an experimental physics paper focused on optical measurements in metamaterials. No free parameters are fitted in the abstract, no new axioms are introduced beyond standard properties of topological insulators, and no new physical entities are postulated. The central claim relies on the established properties of topological insulators and metamaterial field enhancement.

pith-pipeline@v0.9.0 · 5563 in / 1348 out tokens · 51670 ms · 2026-05-07T05:46:31.991436+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Pistore, V. et al. Terahertz Plasmon Polaritons in Large Area Bi2Se3 Topological Insulators. Adv. Opt. Mater. n/a, 2301673 (2023)

work page 2023
[2]

Pogna, E. A. A. et al. Mapping propagation of collective modes in Bi2Se3 and Bi2Te2.2Se0.8 topological insulators by near-field terahertz nanoscopy. Nat. Commun. 12, 6672 (2021)

work page 2021
[3]

Di Pietro, P. et al. Terahertz Tuning of Dirac Plasmons in Bi2Se3 Topological Insulator. Phys. Rev. Lett. 124, (2020)

work page 2020
[4]

Tielrooij, K.-J. et al. Milliwatt terahertz harmonic generation from topological insulator metamaterials. Light Sci. Appl. 11, 315 (2022)

work page 2022
[5]

Hafez, H. A. et al. Terahertz Nonlinear Optics of Graphene: From Saturable Absorption to High- Harmonics Generation. Adv. Opt. Mater. 8, 1900771 (2020). H1 H2 34

work page 2020
[6]

Han, J. W. et al. Plasmonic Terahertz Nonlinearity in Graphene Disks. Adv. Photonics Res. 3, 2100218 (2022)

work page 2022
[7]

Mics, Z. et al. Thermodynamic picture of ultrafast charge transport in graphene. Nat. Commun. 6, 7655 (2015)

work page 2015
[8]

Tomadin, A. et al. The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies. Sci. Adv. 4, eaar5313 (2023)

work page 2023
[9]

& Tielrooij, K.-J

Massicotte, M., Soavi, G., Principi, A. & Tielrooij, K.-J. Hot carriers in graphene – fundamentals and applications. Nanoscale 13, 8376–8411 (2021)

work page 2021
[10]

Di Gaspare, A. et al. Electrically Tunable Nonlinearity at 3.2 Terahertz in Single-Layer Graphene. ACS Photonics (2023) doi:10.1021/acsphotonics.3c00543

work page doi:10.1021/acsphotonics.3c00543 2023
[11]

Sobota, J. A. et al. Ultrafast electron dynamics in the topological insulator Bi2Se3 studied by time- resolved photoemission spectroscopy. J. Electron Spectros. Relat. Phenomena 195, 249–257 (2014)

work page 2014
[12]

D., Marini, A

Cox, J. D., Marini, A. & de Abajo, F. J. G. Plasmon-assisted high-harmonic generation in graphene. Nat. Commun. 8, 14380 (2017)

work page 2017
[13]

Jadidi, M. M. et al. Tunable Terahertz Hybrid Metal–Graphene Plasmons. Nano Lett. 15, 7099–7104 (2015)

work page 2015
[14]

& Law, S

Wang, Y. & Law, S. Optical properties of (Bi1-xInx)2Se3 thin films. Opt. Mater. Express 8, 2570–2578 (2018)

work page 2018
[15]

Chudpooti, N. et al. Wideband dielectric properties of silicon and glass substrates for terahertz integrated circuits and microsystems. Mater. Res. Express 8, (2021)

work page 2021
[16]

& Becker, C

Richter, W., Kohler, H. & Becker, C. A Raman and Far-Infrared Investigation of Phonons. Phys. Stat. sol 84, 619 (1977)

work page 1977
[17]

Spirito, D. et al. Weak antilocalization and spin-orbit interaction in a two-dimensional electron gas. Phys. Rev. B 85, 235314 (2012)

work page 2012
[18]

Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005)

work page 2005

[1] [1]

Pistore, V. et al. Terahertz Plasmon Polaritons in Large Area Bi2Se3 Topological Insulators. Adv. Opt. Mater. n/a, 2301673 (2023)

work page 2023

[2] [2]

Pogna, E. A. A. et al. Mapping propagation of collective modes in Bi2Se3 and Bi2Te2.2Se0.8 topological insulators by near-field terahertz nanoscopy. Nat. Commun. 12, 6672 (2021)

work page 2021

[3] [3]

Di Pietro, P. et al. Terahertz Tuning of Dirac Plasmons in Bi2Se3 Topological Insulator. Phys. Rev. Lett. 124, (2020)

work page 2020

[4] [4]

Tielrooij, K.-J. et al. Milliwatt terahertz harmonic generation from topological insulator metamaterials. Light Sci. Appl. 11, 315 (2022)

work page 2022

[5] [5]

Hafez, H. A. et al. Terahertz Nonlinear Optics of Graphene: From Saturable Absorption to High- Harmonics Generation. Adv. Opt. Mater. 8, 1900771 (2020). H1 H2 34

work page 2020

[6] [6]

Han, J. W. et al. Plasmonic Terahertz Nonlinearity in Graphene Disks. Adv. Photonics Res. 3, 2100218 (2022)

work page 2022

[7] [7]

Mics, Z. et al. Thermodynamic picture of ultrafast charge transport in graphene. Nat. Commun. 6, 7655 (2015)

work page 2015

[8] [8]

Tomadin, A. et al. The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies. Sci. Adv. 4, eaar5313 (2023)

work page 2023

[9] [9]

& Tielrooij, K.-J

Massicotte, M., Soavi, G., Principi, A. & Tielrooij, K.-J. Hot carriers in graphene – fundamentals and applications. Nanoscale 13, 8376–8411 (2021)

work page 2021

[10] [10]

Di Gaspare, A. et al. Electrically Tunable Nonlinearity at 3.2 Terahertz in Single-Layer Graphene. ACS Photonics (2023) doi:10.1021/acsphotonics.3c00543

work page doi:10.1021/acsphotonics.3c00543 2023

[11] [11]

Sobota, J. A. et al. Ultrafast electron dynamics in the topological insulator Bi2Se3 studied by time- resolved photoemission spectroscopy. J. Electron Spectros. Relat. Phenomena 195, 249–257 (2014)

work page 2014

[12] [12]

D., Marini, A

Cox, J. D., Marini, A. & de Abajo, F. J. G. Plasmon-assisted high-harmonic generation in graphene. Nat. Commun. 8, 14380 (2017)

work page 2017

[13] [13]

Jadidi, M. M. et al. Tunable Terahertz Hybrid Metal–Graphene Plasmons. Nano Lett. 15, 7099–7104 (2015)

work page 2015

[14] [14]

& Law, S

Wang, Y. & Law, S. Optical properties of (Bi1-xInx)2Se3 thin films. Opt. Mater. Express 8, 2570–2578 (2018)

work page 2018

[15] [15]

Chudpooti, N. et al. Wideband dielectric properties of silicon and glass substrates for terahertz integrated circuits and microsystems. Mater. Res. Express 8, (2021)

work page 2021

[16] [16]

& Becker, C

Richter, W., Kohler, H. & Becker, C. A Raman and Far-Infrared Investigation of Phonons. Phys. Stat. sol 84, 619 (1977)

work page 1977

[17] [17]

Spirito, D. et al. Weak antilocalization and spin-orbit interaction in a two-dimensional electron gas. Phys. Rev. B 85, 235314 (2012)

work page 2012

[18] [18]

Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005)

work page 2005