Disentangling high harmonic generation from surface and bulk states of a topological insulator

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 Bi₂Se₃. 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.

Solid-state high-harmonic generation (HHG) provides a way to probe material structure. Here, the authors study HHG in the prototype three-dimensional topological insulator Bi2Se3, showing continuous control of band structure and unveiling concepts like shift vectors & Berry curvature on topological surface states.

Introduction

In the 1980s, foundational concepts related to topological phenomena, particularly in the context of the quantum Hall effect, were established^1–3, leading to a new paradigm for classifying materials based on their topological order. Three-dimensional (3D) topological insulators (TIs)^4,5 are a quantum phase of matter that is insulating in the bulk but conductive on the surface. The metallic surface arises from the presence of gapless Dirac cone topological surface states (TSSs), which are inherently tied to the nontrivial topology of the bulk band structure that experiences a band inversion under time-reversal symmetry (TRS) and strong spin-orbit coupling (SOC). The TSSs are robust against time-reversal-preserving (e.g., non-magnetic) perturbations and exhibit unique properties^4,5: they are spin-polarized, and the spin and momentum of the charge-carriers are locked at a right angle (spin-momentum locking). In addition, their helical spin texture supports chiral spin current, the spin-momentum locking prevents back-scattering, allowing for dissipationless charge and spin transport, and the spin vortex induces a π Berry phase⁶ in the wavefunction when looping around the Dirac point. These properties make 3D-TIs of substantial fundamental and practical interest.

Methods for exploring TIs include (spin and) angular resolved photoelectron spectroscopy (s-ARPES)^7–9, measurement of the (quantum) Hall effect^10,11, and circular dichroism examination¹², which are employed to probe intrinsic properties of TIs or their perturbative interactions with light. Recently, high-order harmonic generation (HHG) has been observed in 3D-TIs^13–15, offering an all-optical probe of them interacting with strong fields. HHG is a frequency up-conversion process that occurs when a strong laser field interacts with matter, typically atomic or molecular gases¹⁶. It is a hallmark of ultrafast physics and a key method for generating extreme-ultraviolet (XUV) and X-ray photons as well as attosecond pulses. The realization of HHG in solid-state systems¹⁷ has launched extensive efforts to uncover ultrafast charge-carrier dynamics in strong-field-driven solids and to exploit high-harmonic spectroscopy as a probe of condensed matter properties^18–20.

The ability of HHG to probe topology has recently become a topic of animated debate. On one hand, a number of influential works have highlighted intriguing features in the HHG spectra of materials with nontrivial topology. For instance, the carrier-envelope-phase-dependent, non-integer “harmonic” generation from TSSs has been explained by the ballistic acceleration of Dirac fermions through the vicinity of the Dirac point^14,21. The “anomalous” (non-gas-like) dependence of the harmonic yield on the driving laser ellipticity has been attributed to the vortex spin texture and the hexagonal warping of the TSSs^15,22. Calculations of modeled systems have predicted orders of magnitude enhancement in harmonic conversion efficiency²³ and sign flipping of harmonic light helicity²⁴ between the nontrivial and trivial topological phases. On the other hand, a subsequent work has suggested that the non-integer harmonics were due to the chirp of the driving pulse²⁵. Anomalous HHG driven by elliptically or circularly polarized light has also been observed in many topologically trivial materials^26–28 and hence alone may not reflect the system’s topology. Furthermore, a numerical study cast doubt on the reliability of extracting topological phases via HHG from prior works²⁹.

A serious challenge in probing TSSs via HHG is suppressing the responses of bulk states. Due to the very small bulk band gap of TIs, charge-carrier excitation and harmonic generation in the bulk can easily be triggered by a strong laser field. In addition, typically, mid-infrared (MIR) and long-wavelength infrared (LWIR) pulses are used to drive HHG in TIs, producing harmonic light in the visible to infrared wavelength regime. The absorption lengths of the fundamental and harmonic light are respectively >100 nm and ~20–30 nm³⁰, while the TSSs only have a penetration depth of less than 2 nm³¹. Therefore, even in reflection geometry, the contribution of bulk states to the harmonic emission can be significant, particularly when only in-plane responses are considered.

Nevertheless, experiments can be carefully designed to selectively probe HHG from the surface states. For the Bi₂Se₃ family of TIs, the bulk belongs to the inversion symmetric D_3d point group while the surface symmetry reduces to C_3v. As a result, even-order harmonics arising from electric-dipole transitions are permitted only from the surface, with their relative efficiency serving as a potential indicator of the contribution of surface states to HHG. Schmid et al.¹⁴ suggest that by using a weak below-band gap driving field, HHG in the bulk of Bi₂Te₃ can be strongly suppressed. The spectrum consisting of commensurate odd- and even-order harmonic light has been attributed to the TSSs. However, for Bi₂Se₃ and BiSbTeSe₂ (BSTS), most previous studies have reported the yields of even-order harmonics that are 2–3 orders of magnitude weaker than those of the odd orders^13,15, suggesting that harmonic emission from the surface states is weak.

While most previous studies either assumed that HHG originates solely from the surface or lacked solid evidence to support this claim—particularly for odd-order harmonics, which are allowed from both the bulk and surface states—this manuscript presents a novel approach for studying HHG in the prototype 3D-TI Bi₂Se₃ and addresses the following key question: How can we suppress/enhance harmonic emission from the bulk/surface states and effectively distinguish between these contributions? We demonstrate: (1) The contributions of bulk and surface states to the MIR-driven HHG 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. (2) Introducing a quasi-static terahertz (THz) perturbing field, i.e., a MIR-THz two-color field scheme³², provides an unambiguous method to differentiate between the bulk and surface HHG responses, applicable to both the even- and odd-order harmonic light. Our study provides effective methods for isolating TSSs from the bulk, paving the way for further exploration of the role of nontrivial topology in HHG.

Symmetry is important throughout our discussion. Henceforth, we will use $\mathcal{P}$- and $\mathcal{T}$- to denote parity (inversion) and time-reversal symmetry, respectively. For convenience, the real-space crystal orientation is described using the corresponding reciprocal-space symmetry directions ΓM or ΓK (see Fig. 1).

Fig. 1. The experimental design.

Polarization-resolved harmonic selection rules when the driving fields are polarized along the real-space directions corresponding to the a ΓM or b ΓK directions in reciprocal space. Black hexagons indicate the Bi₂Se₃ surface Brillouin zone, colored triangles with dots show the real-space lattice structure projected onto the (111) plane. Symmetry breaking of the TSSs is “visualized” by the gray arrows and color scale bar, which represent local directions of the in-plane shift vector R^k,y along ΓM and amplitude of the out-of-plane (z) Berry curvature ${\mathbf{\Omega }}_z^{\mathbf{k}}$ along ΓK, respectively. Reciprocal space two-dimensional distributions of R and Ω can be found in Fig. 2. Black arrows indicate the MIR and (instantaneous) THz field directions. Purple arrows show the HHG polarization directions. a When the driving fields are polarized along the ΓM direction, the differential HHG yield between the cases where the instantaneous THz field and the TSSs' shift vector are parallel (+, same direction, as shown in (a)) or anti-parallel (−, opposite directions, 180° crystal orientation of (a)), defined as $\mathrm{\Delta }{Y}^{\pm }\equiv Y\left(F_{\mathrm{THz}\parallel \mathrm{\Gamma }{\mathrm{M}}_{+}}\right)-Y\left(F_{\mathrm{THz}\parallel \mathrm{\Gamma }{\mathrm{M}}_{-}}\right)$, serves to distinguish harmonic emission from the bulk (R = 0, ΔY = 0) and surface (R ≠ 0, ΔY ≠ 0) states. b When the driving fields are polarized along the ΓK direction, and for the harmonic emission perpendicular to the driving fields polarization, the HHG yield difference with and without the THz field, defined as ΔY_⊥ = Y_⊥(F_THz≠0) − Y_⊥(F_THz = 0), probes an exclusive surface states HHG response (Bulk: Ω = 0, ΔY_⊥ = 0; Surface: Ω ≠ 0, ΔY_⊥ ≠ 0).

Results and discussion

The experiments are conducted with normal incident light in transmission geometry. A strong 60 fs, 3.6 μm MIR pulse drives HHG in the (111) plane of Bi₂Se₃ thin-film crystal grown on (0001) Al2O3 substrate. A weak ~2 ps period single-cycle THz pulse is applied to perturb or control the HHG process. The THz field is considered quasi-static since its period is much longer than the MIR pulse duration. Both the MIR and THz fields are linearly polarized along the vertical direction in the laboratory frame, and the crystal can be oriented with respect to the laser polarization. Harmonic light is dispersed by a monochromator and detected by an ICCD camera. The harmonic polarization is analyzed using a Rochon prism. A detailed experimental setup can be found in the “Methods” section.

Experimental concept

According to Floquet group theory and electric dipole selection rules, the forbidden even-order HHG in a $\mathcal{P}$-invariant medium driven by a monochromatic laser field arises from the H(r, t) = H( − r, t + T₀/2) dynamical symmetry of the Hamiltonian^33,34, here T₀ is the laser period. An intrinsic $\mathcal{P}$-violation of the medium or an asymmetric driving field F(t) ≠ − F(t + T₀/2), both can break this symmetry and promote even-order harmonic emission. Our experiments exploit this principle, we introduce an asymmetry in the driving field by applying a quasi-static THz perturbation, and demonstrate that the coupling between intrinsic and THz-field-induced symmetry breaking enables the differentiation of surface ($\mathcal{P}$-violated) and bulk ($\mathcal{P}$-invariant) HHG responses in Bi₂Se₃.

Intrinsic $\mathcal{P}$-violation that gives rise to even-order optical responses is often characterized by vector quantities. For instance, nonzero Berry curvature (Ω, pseudovector) and shift vector (R, polar vector) due to $\mathcal{P}$-violation are crucial for the generation of even-order harmonics. Figure 2 shows Ω and R for the TSSs of Bi₂Se₃. Phenomenologically, they can be understood as vectors that define the directions of symmetry breaking in the crystal frame, similar to the spin of spin-polarized molecules or the polar axis of polar molecules. In condensed matter physics, Ω plays an essential role in phenomena such as the anomalous Hall effect^35,36, while R contributes to the shift current photovoltaic effect^37–39. The roles of Ω and R in strong-field HHG have been actively studied recently^40–42, and a brief discussion is provided in Supplementary Information (SI) Section 3. Quantitative analysis of HHG with nonzero Ω and R not only offers a means to extract intrinsic material properties but also deepens our understanding of the HHG dynamics in solids^43–46.

Fig. 2. Reciprocal space inversion symmetry breaking in the (111) plane of Bi₂Se₃ TSSs.

a Band structure E^k and b amplitude of the out-of-plane Berry curvature ${\mathbf{\Omega }}_z^{\mathbf{k}}$, for the Dirac cone upper ("+'', top panel) and lower ("−”, bottom panel) states. c Shift vector ${\mathbf{R}}^{\mathbf{k},x}\equiv \left(R_{xx}^{\mathbf{k}},R_{yx}^{\mathbf{k}}\right)$ (top panel) and ${\mathbf{R}}^{\mathbf{k},y}\equiv \left(R_{xy}^{\mathbf{k}},R_{yy}^{\mathbf{k}}\right)$ (bottom panel); the color scale and white arrows indicate the magnitude and local direction of the shift vector, respectively. $\mathcal{T}$-invariance requires a symmetric band structure E^k = E^−k. The surface states $\mathcal{P}$-violation is restored by the nonzero Ω and R, which are odd and even functions of k, respectively: Ω^k = − Ω^−k and R^k = R^−k. See Eqs. (4–6) and (9) in “Methods” for the mathematical definitions, and Section 3 in SI for detailed physical meanings, of Ω and R.

In this study, we demonstrate that the bulk (Ω = 0, R = 0) and surface (Ω ≠ 0, R ≠ 0) states of Bi₂Se₃ produce distinct harmonic spectra, and respond differently to the THz perturbation. Moreover, the interplay between intrinsic and THz-field-induced symmetry breaking in TSSs enables us to probe the roles of the shift vector and Berry curvature in HHG. The expected experimental outcomes are summarized in Fig. 1. To support the key features of our experimental observations, we perform time-dependent strong-field simulations using a tight-binding model for the TSSs combined with a gauge-invariant formulation of the semiconductor Bloch equations (see “Methods” for simulation details and SI Section 4 for simulation results).

Control HHG from the bulk and surface states by thin film thickness

The experiments that compare the bulk- and surface-dominated HHG responses are realized with high-quality thin film samples of varying thicknesses. Given the extremely short penetration depth (~2 nm) of the TSSs, we fabricate an ultrathin film with a thickness of 6 nm ( ≈ 1 nm per layer) such that it primarily consists of two (top and bottom) surfaces while the bulk volume is minimized. This thickness is chosen because studies have shown that when the film is thinner than 5–6 nm, the interaction between top and bottom TSSs results in re-opening of the band gap, and the material may lose its nontrivial topology⁴⁷. For comparison, we also prepare a thick film of 50 nm thickness to represent a sample with maximized effective bulk volume. Films thicker than 50 nm do not further enhance the bulk HHG due to the short absorption length of the harmonic light.

Figure 3 shows the harmonic spectra for the 6 and 50 nm samples. Due to the longer effective interaction length for HHG, the total harmonic yield for the 50 nm sample is a few times stronger than that for the 6 nm sample. However, the even-order harmonics, which can only be generated from the $\mathcal{P}$-violated surface, are nearly two orders of magnitude stronger in the 6 nm sample compared to the 50 nm sample, suggesting that the surface states’ HHG is significantly enhanced in the ultrathin film. When the MIR driving field is polarized along the ΓK direction, the yields of adjacent odd and even harmonic orders are commensurate, similar to the LWIR-driven HHG in Bi₂Te₃ in reflection geometry¹⁴. Such efficient relative even-order harmonic conversion strongly indicates that in the ultrathin film, HHG is dominated by the surface states.

Fig. 3. MIR-driven high-order (5th to 11th) harmonic spectra from the Bi₂Se₃ thin film crystals.

The 50 nm (shaded blue) and 6 nm (solid red) samples are oriented with a ΓM and b ΓK along the MIR polarization direction, respectively. All four spectra are scaled by the same factor. Top: Schematics of the polarization-resolved HHG measurements using a Rochon prism: harmonic components polarized parallel and perpendicular to the fundamental field are spatially separated along the vertical direction and simultaneously detected by the spectrometer camera. Gray arrows indicate the polarization direction of the light.

To investigate how HHG from the bulk and surface states evolves with the film thickness, we measure the harmonic yields for samples of varying thickness. The results are shown in Fig. 4. The total harmonic yield increases with the film thickness and peaks at about 20 nm. This saturation behavior is attributed to the short absorption lengths of both the fundamental and harmonic light, as well as thin film interference and self-phase modulation of the fundamental light, which may affect the laser peak intensity. In contrast, the total yield of surface-exclusive even-order harmonics increases as the film thickness decreases, with a steep rise below 10 nm. This sharp transition and significant increase of surface HHG in films thinner than 10 nm cannot be explained by light absorption alone. It indicates that the reduced bulk volume plays a central role in enhancing surface HHG and points to a mechanism in which the bulk and surface of a TI are not fully isolated. We attribute this mechanism to bulk-surface electron scattering^48,49, and provide a detailed discussion in SI Section 1. In brief, the photoexcited bulk electrons scatter into TSSs, incoherently generating excess hot electrons and increasing the electron-electron scattering within the surface during HHG, thereby suppressing the efficiency of surface HHG^50,51. Because bulk states are delocalized throughout the crystal while TSSs are confined within only ~2 nm near the surface, only bulk electrons sufficiently close to the surface scatter efficiently into TSSs. The observed transition near 10 nm therefore suggests an upper bound of ~5 nm for the bulk-surface coupling length (Supplementary Fig. 1).

Fig. 4. Evolution of the harmonic yield (Y) and MIR transmission as a function of the thin film thickness.

Total harmonic yield ${\sum }_{q=5}^{11}{\mathrm{Y}}_q$ (open orange squares), total odd-order harmonic yield (open blue circles), 100  × total even-order harmonic yield (open red circles). All three curves are normalized by the same factor (the total harmonic yield maximum). The MIR driving laser transmission is shown as purple crosses. Despite of large ( > 5) index of refraction, the MIR transmission is high due to thin film interference.

Disentangling HHG from the bulk and surface states with a terahertz field

In this section, we confirm the dominant contributions of bulk and surface states to the HHG from the 50 nm and 6 nm samples, respectively, and verify the experimental concept introduced earlier: by applying a quasi-static terahertz perturbing field, the bulk and surface HHG responses are disentangled, allowing us to examine the influence of TSSs’ shift vector (R) and Berry curvature (Ω) on HHG.

As shown in Fig. 2, along the ΓM direction, Ω vanishes and R exhibits only a longitudinal component; the polarity of R characterizes the direction of symmetry breaking in the crystal frame. We define M₊ and M₋ such that R^k,y points from M₋ to M₊ along ΓM. When the driving field is polarized along ΓM, dynamical symmetry restricts all the harmonic light to be polarized parallel to the driver (see Fig. 3a, top for measurement and Supplementary Fig. 5 for simulation). Due to $\mathcal{T}$-invariance, the band structure is symmetric regardless of bulk or surface states, therefore, ΓM₊ and ΓM₋ are indistinguishable from the harmonic yield with a monochromatic (bipolar) driver. However, for the TSSs, the polarity of R can be referenced by applying a polar THz perturbing field, which serves as a direct knob to tune the effect of R on HHG. In the experiments, we orient the crystal to compare HHG in the two cases where R^k,y points towards (ΓM₊) or opposite (ΓM₋) to the instantaneous THz field direction, as shown in Fig. 5a. To the lowest order nonlinearity, this investigates the coupling between intrinsic and electric field-induced second harmonic generation (i-SHG and EFISHG⁵²). The difference between the two cases can be understood through perturbative nonlinear optics: $P_{2\omega }^{\pm }=\left({\chi }^{\left(2\right)}\mp {\chi }^{\left(3\right)}{F}_{\mathrm{THz}}\right)F_{\omega }^2$, here P, F, and χ are absolute values of the polarization, electric field, and susceptibility, respectively. Only the $\mathcal{P}$-violated (χ⁽²⁾ ≠ 0) surface states produce a difference in the harmonic yield ( ∝ P²) between the two orientation configurations.

Fig. 5. Probing the influence of the shift vector on HHG by the THz field.

a Experimental configuration: The MIR and THz fields are polarized along the ΓM direction. In the crystal frame, M₊ and M₋ are defined by the polarity of the TSSs' shift vector R^k,y along ΓM, which points from M₋ to M₊. In the laboratory frame, the polarity of the first main half-cycle of the THz field (see inset c) defines the “up” direction. b For the 6 nm (left panels) and 50 nm (right panels) samples, the 6th- (upper panels) and 7th- (lower panels) order harmonic yields are measured as a function of the MIR-THz time delay, for the two crystal orientations shown in a when ΓM₊ (red) and ΓM₋ (blue) point upwards. All yields are scaled by the same factor. Time zero is defined at the field zero-crossing of the THz main cycle, and large negative time delays correspond to MIR leading. The middle panels show a typical THz waveform at focus in air, measured separately. c At a fixed MIR-THz time delay of about −250 fs when the MIR pulse overlaps with the crest of the first main half-cycle of the THz field (see inset), the contrast between ΓM₊ and ΓM₋ crystal orientations, defined as the harmonic-yield difference divided by their sum, (Y₊ − Y₋)/(Y₊ + Y₋), is shown as a function of the harmonic order. Error bars are standard deviations of ten measurements.

Figure 5b shows the 6th- and 7th-order harmonic yields as a function of the MIR-THz time delay. For the 6 nm sample, both harmonics exhibit pronounced differences between the ΓM₊ and ΓM₋ crystal orientations, while the differences are minimal for the 50 nm sample, indicating that these two harmonics are dominated by the surface and bulk states for the 6 and 50 nm samples, respectively. At a fixed MIR-THz time delay, and from the 5th- to 10th-order harmonics, Fig. 5c plots the contrast between the two crystal orientations, defined as the harmonic-yield difference divided by their sum, (Y₊ − Y₋)/(Y₊ + Y₋). The contrast is a parameter reflecting the strength of surface states HHG, as pure bulk responses yield a zero contrast. For the 50 nm sample, the contrast values remain very low across all the harmonic orders, suggesting a bulk-dominated HHG. For the 6 nm sample, large contrasts, a signature of surface-dominated HHG, are observed except for the 5th-order harmonic.

Our measurements demonstrate that the influence of R on HHG can be probed by introducing a weak THz perturbation. Importantly, a recent theoretical study of HHG in ZnO driven by a single-color field concluded that R has little effect on odd-order harmonics⁴². In contrast, our results highlight the crucial role of the THz perturbation in revealing the otherwise hidden impact of R on odd-order harmonic emission. Our simulations qualitatively reproduce the strong contrasts observed for the low-order harmonics (Supplementary Fig. 7), while for harmonics above the 9th-order the simulated contrasts become much smaller, indicating our current model does not fully capture the higher-order responses.

The extremely small contrast of the 5th-order harmonic indicates that, under our experimental conditions, it is bulk-dominated even in an ultrathin (6 nm) film with minimum bulk volume. Additional HHG measurements by varying the MIR-THz time delay and by rotating the crystal orientation, presented in Supplementary Fig. 3, further support this conclusion. This counterintuitive behavior is likely due to strong bulk charge-carrier excitation induced by the relatively high fundamental photon energy (~0.34 eV) above the bulk band gap (~0.3 eV), which produces strong low-order harmonic emission from the bulk.

In two-dimensional and topological materials, the Berry-curvature-induced perpendicular HHG (∣j_⊥,Ω∣ ∝ F × Ω) can be significant^14,40. Our approach provides a promising avenue to explore the Berry curvature effect on HHG by considering the case when the MIR driving field is polarized along the ΓK direction, where the longitudinal component of R vanishes and symmetry breaking is characterized by the out-of-plane Ω (Fig. 2). In this case, the Hamiltonian possesses a ${\widehat{Z}}_{\mathrm{M}}={\widehat{\tau }}_2\cdot {\widehat{\sigma }}_v^{\mathrm{M}}$ reflection dynamical symmetry, restricting the odd- and even- (if allowed) order harmonic light to be polarized parallel and perpendicular to the fundamental driving field, respectively (see Fig. 3b, top for measurement, and Supplementary Fig. 6 for simulation)³³. Here ${\widehat{\tau }}_2$ and ${\widehat{\sigma }}_v^{\mathrm{M}}$ are the time translation (t → t + T₀/2) and mirror reflection (off the vertical mirror plane parallel to ΓM) operators. When a parallel THz field is applied, the ${\widehat{Z}}_{\mathrm{M}}$ symmetry is lifted, affecting the HHG in both dimensions. However, the harmonic modulation in the perpendicular direction reflects an exclusive surface state response. We measure this effect in the 6 and 50 nm samples, as shown in Fig. 6. In the perpendicular direction, only even-order harmonics are produced with the MIR driving field alone, applying the THz field not only suppresses the yields of even-order harmonics but also enables the generation of odd-order harmonics. Remarkably, for the surface-dominated 6 nm sample, the perpendicular harmonic yields exhibit pronounced modulations across the full spectrum, which are accurately captured by the simulated HHG spectra (Supplementary Fig. 8), demonstrating the effectiveness of this method for probing HHG from isolated surface states.

Fig. 6. Probing the influence of Berry curvature on HHG by the THz field.

The MIR and THz fields are polarized along the ΓK direction, and only the harmonic light polarized perpendicular to the driving fields is measured. For the a 6 nm and b 50 nm samples, perpendicular harmonic spectra without the THz field (shaded blue) and with the THz field (solid red) at a fixed MIR-THz time delay of about −250 fs (see Fig. 5c, inset), are shown. All four spectra are scaled by the same factor. a (inset) For the 6 nm sample, the perpendicular 6th- (red) and (10 × ) 7th- (blue) order harmonic yield as a function of the MIR-THz field time delay. b (inset) Crystal orientation relative to the laser fields. The color scale bar demonstrates the amplitude of the out-of-plane Berry curvature along ΓK.

In conclusion, we have demonstrated that the MIR-driven HHG from bulk and surface states of 3D-TI Bi₂Se₃ can be controlled by varying the thickness of thin film samples. The substantial enhancement of surface HHG in ultrathin films (<10 nm) is attributed to reduced bulk-surface electron scattering. To probe TIs via high-harmonic spectroscopy, it is crucial to disentangle the bulk and surface responses. We achieve this by employing a MIR-THz two-color field scheme. The observed effects of the TSSs’ shift vector and Berry curvature on HHG, probed by the THz field, are significant, making it intriguing to explore the potential role of nontrivial topology. Future experiments using ultrathin films with topologically trivial and nontrivial surface states (controllable through doping⁵³), in comparison with theoretical calculations, could provide deeper insights. In addition, the concepts introduced in this work—controlling the ratio of surface area to bulk volume by varying the thin film thickness, and exploiting the coupling between intrinsic and field-induced symmetry breaking to distinguish surface and bulk optical responses—are broadly applicable and will provide valuable guidance for future experimental designs of probing TSSs beyond HHG studies. Finally, we note that whether the nontrivial topological features of the bulk states, such as the band inversion, produce distinctive HHG signatures has largely been overlooked and merits future investigations.

Methods

Experimental setup

In the experiments, ~60 fs (intensity FWHM), 3.6 μm (83.3 THz, 0.34 eV) MIR pulses are generated from an optical parametric amplifier based on KTiOAsO4 (KTA) crystals. Single-cycle ~2 ps period (600 μm, 0.5 THz, 2.1 meV) THz pulses are generated via tilted-pulse-front-pumping (TPFP) optical rectification in a LiNbO₃ crystal. Both the MIR and the THz generations are pumped by laser pulses delivered from the same Ti:sapphire system (80 fs, 800 nm, 5.5 mJ, 1 kHz). The THz pulse characteristics (central frequency, asymmetry between positive and negative half-cycles, small satellite cycles before/after the main cycle) may have small day-to-day variations; however, these do not affect our study in this manuscript. The MIR and THz fields are linearly polarized in the laboratory frame vertical direction. The MIR intensity and polarization are controlled via a half-waveplate followed by a nano-particle polarizer (Thorlabs, extinction ratio (ER) >10⁶), and the THz intensity and polarization are controlled by a pair of wire grid polarizers (Specac, ER  > 10⁹). Harmonic light is collected by a visible-vacuum ultraviolet monochromator (VIS-VUV, 120–900 nm) and detected by a cooled ICCD camera (Princeton Instruments, 16 bits, noise level  < 5). The intensity standard deviation of the harmonic light is about 1%. The harmonic polarization is analyzed by an ultrabroad-bandwidth (130–7000 nm) Rochon prism (Edmund optics, ER >10⁵). The highly linearly polarized MIR and THz fields, high ER Rochon prism, and high dynamic range detector allow for high precision polarization measurements.

The experiments are conducted with normal incident light in transmission geometry in ambient air. The samples are mounted frontwards (the lasers hit the thin film first and then the substrate). The focal size (1/e² intensity diameter) of the MIR and THz beams is ~220 μm and ~1.5 mm, respectively. In the experiments, the maximum applied MIR and THz peak intensities (in air) are ~80 GW cm⁻² and ~0.48 GW cm⁻², respectively. We measure from the 5th- to 11th-order (327–720 nm) harmonic light. When pumping slightly above the crystal damage threshold, we are able to detect up to the 13th-order (277 nm) harmonic in Bi₂Se₃.

In this study, we only focus on the THz-field-induced symmetry breaking. However, we also notice that the extremely low-frequency THz pulse heats the crystal through incoherent charge-carrier acceleration. This heating lasts for several hundred picoseconds (verified by MIR-THz time delay scan), and globally reduces the HHG efficiency.

Sample preparation

The highly-crystalline (111) Bi₂Se₃ thin films are deposited in a Veeco 930 molecular beam epitaxy (MBE) system with a base pressure of 2 × 10⁻¹⁰ Torr. The 0.5-mm-thick (0001) Al₂O₃ substrates (MTI Corporation), chosen for their favorable lattice and symmetry match, are sonicated in acetone, methanol, and isopropyl alcohol for five min each, and annealed in air at 1000 °C for 3 h before loading into the MBE chamber. The substrates are then brought up to 800 °C in the ultra-high vacuum chamber to degas for an hour prior to growth. For synthesis, the films are grown by co-depositing Bi (99.95%, Alfa Aesar) and Se (99.999%, Puratronic) with an atomic flux ratio of ~1:30. The substrate is first heated up to 260 °C to grow Bi₂Se₃ for 2 min, then annealed at 550 °C to evaporate it off. Substrate is then cooled down to 260 °C to deposit Bi₂Se₃ buffer layer for 2 min again before being heated up to 360 °C to deposit until the desired thickness is reached. The pre-growth procedure of buffer layers is aimed to reduce twinning domains and surface roughness⁵⁴. All Bi₂Se₃ samples are capped with a 5 nm CaF₂ protection layer to prevent oxidation and degradation.

TSSs model and HHG simulation methods

Tight-binding model

We consider a tight-binding model for the TSSs in Bi₂Se₃^22,55–57 previously derived in ref. ²², which captures the correct symmetries of the TSSs and their properties such as spin-momentum locking and hexagonal warping⁵⁸ at large crystal momenta.

The Hamiltonian of the TSSs can be parametrized in the form

$$H^{\mathbf{k}}={\sum }_{i=0}^3{B}_i^{\mathbf{k}}{\sigma }_i,$$ 1

with k = (k_x, k_y) the in-plane crystal momenta, σ₀ ≡ I the identity matrix, {σ_i, i = 1, 2, 3} the Pauli matrices, and $\left\{B_i^{\mathbf{k}},i=1,2,3\right\}$ are functions defined in the tight-binding model²². As usual, it is convenient to consider the system as an analog to a spin-$\frac{1}{2}$ particle in a pseudo-magnetic field, and define polar and azimuthal angles as⁵⁹$\cos {\theta }_{\mathbf{k}}=B_3^{\mathbf{k}}/\sqrt{{\sum }_{i=1}^3{B}_i^{\mathbf{k}}}$ and ${\varphi }_{\mathbf{k}}=\arg \left(B_1^{\mathbf{k}}+i{B}_2^{\mathbf{k}}\right)$, where $B^{\mathbf{k}}\equiv \sqrt{{\sum }_{i=1}^3{B}_i^{\mathbf{k}}}$. The eigenstates can then be written in a particular gauge as

$$∣u_{+}^{\mathbf{k}}⟩=\left(\begin{array}{c}\hfill \cos \frac{{\theta }_{\mathbf{k}}}{2}\hfill \\ \hfill e^{i{\varphi }_{\mathbf{k}}}\sin \frac{{\theta }_{\mathbf{k}}}{2}\hfill \end{array}\right),∣u_{-}^{\mathbf{k}}⟩=\left(\begin{array}{c}\hfill e^{-i{\varphi }_{\mathbf{k}}}\sin \frac{{\theta }_{\mathbf{k}}}{2}\hfill \\ \hfill -\cos \frac{{\theta }_{\mathbf{k}}}{2}\hfill \end{array}\right).$$ 2

with the eigenenergies

$$E_{\pm }^{\mathbf{k}}=B_0^{\mathbf{k}}\pm B^{\mathbf{k}}.$$ 3

Here, “+” and “−” correspond to the Dirac cone upper and lower states, respectively. We obtain analytical forms of gauge-invariant quantities such as the absolute value of the transition dipoles $\mid d_a^{\mathbf{k}}\mid$, the (out-of-plane) Berry curvature ${\mathrm{\Omega }}_{\pm }^{\mathbf{k}}$ and the shift vectors $R_{ab}^{\mathbf{k}}$ (a = x, y and b = x, y specify a particular Cartesian vector component)

$$\mid d_a^{\mathbf{k}}\mid =∣⟨u_{+}^{\mathbf{k}}\mid {\partial }_a\mid u_{-}^{\mathbf{k}}⟩∣=\frac{1}{4}\sqrt{{\left({\partial }_a{\varphi }_{\mathbf{k}}\right)}^2{\sin }^2{\theta }_{\mathbf{k}}+{\left({\partial }_a{\theta }_{\mathbf{k}}\right)}^2}$$ 4

$${\mathrm{\Omega }}_{\pm }^{\mathbf{k}}=-2\mathrm{Im}⟨{\partial }_x{u}_{\pm }^{\mathbf{k}}⟩{\partial }_y{u}_{\pm }^{\mathbf{k}}=\pm \frac{1}{2}\sin {\theta }_{\mathbf{k}}\left[{\partial }_x{\varphi }_{\mathbf{k}}{\partial }_y{\theta }_{\mathbf{k}}-{\partial }_x{\theta }_{\mathbf{k}}{\partial }_y{\varphi }_{\mathbf{k}}\right]$$ 5

$$R_{ab}^{\mathbf{k}}=\mathrm{\Delta }{\mathcal{A}}_a^{\mathbf{k}}-{\partial }_a\arg \left(d_b^{\mathbf{k}}\right)={\partial }_a{\varphi }_{\mathbf{k}}\cos {\theta }_{\mathbf{k}}-{\partial }_a\arg \left[{\partial }_b{\varphi }_{\mathbf{k}}\sin {\theta }_{\mathbf{k}}+i{\partial }_b{\theta }_{\mathbf{k}}\right],$$ 6

where we have used the notation ∂_a ≡ ∂/∂k_a, a = x, y for the partial derivates. For brevity, here we have not fully expanded Eq. (6).

Gauge-invariant dynamics

We implemented a gauge-invariant formulation of the semiconductor Bloch equations (GI-SBEs)⁴² 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 and shift vectors to observe their contributions to the dynamics. We work in the dipole approximation, using a vector potential in the form $\mathbf{A}\left(t\right)=A\left(t\right)\widehat{\mathbf{n}}=\left[A_{\mathrm{MIR}}\left(t\right)+A_{\mathrm{THz}}\left(t\right)\right]\widehat{\mathbf{n}}$, with $\widehat{\mathbf{n}}$ the polarization unit vector. The electric field is then obtained by $\mathbf{F}\left(t\right)=-\stackrel{°}{\mathbf{A}}\left(t\right)=F\left(t\right)\widehat{\mathbf{n}}$. In a frame moving with the vector potential A(t), or equivalently, in a basis spanned by the Houston states^60,61, the two-band GI-SBEs take on the form

$${\stackrel{°}{f}}_{\pm }^{\mathbf{K}}\left(t\right)=\mp 2F\left(t\right)\mid d^{\mathbf{K}+\mathbf{A}\left(t\right)}\mid \mathrm{Im}\left[P^{\mathbf{K}}\left(t\right)\right]-\left[f_{\pm }^{\mathbf{K}}\left(t\right)-f_0^{\mathbf{K}}\right]/T_1$$ 7

$$\begin{array}{ccc}\hfill {\stackrel{°}{P}}_{\left(t\right)}^{\mathbf{K}}& \hfill =\hfill & -i\left[\mathrm{\Delta }{E}^{\mathbf{K}+\mathbf{A}\left(t\right)}+\mathbf{F}\left(t\right)\cdot {\mathbf{R}}^{\mathbf{K}+\mathbf{A}\left(t\right)}\right]P^{\mathbf{K}}\left(t\right)\hfill \\ \hfill & \hfill \hfill & +iF\left(t\right)\mid d^{\mathbf{K}+\mathbf{A}\left(t\right)}\mid \mathrm{\Delta }{f}^{\mathbf{K}}\left(t\right)-P^{\mathbf{K}}/T_2\hfill \end{array}$$ 8

with $f_{\pm }^{\mathbf{K}}$ the band populations (diagonal elements of the density matrix $f_{\pm }^{\mathbf{K}}\equiv {\rho }_{\pm \pm }^{\mathbf{K}}$), P^K the band coherences (off-diagonal parts of the density matrix $P^{\mathbf{K}}\equiv {\rho }_{+-}^{\mathbf{K}}$), $\mathrm{\Delta }{E}^{\mathbf{k}}\equiv E_{+}^{\mathbf{k}}-E_{-}^{\mathbf{k}}$ the band gap, $\mathrm{\Delta }{f}^{\mathbf{K}}\equiv f_{+}^{\mathbf{K}}-f_{-}^{\mathbf{K}}$ the population difference, $d^{\mathbf{k}}\equiv \widehat{\mathbf{n}}\cdot {\mathbf{d}}^{\mathbf{k}}$ the dipole along $\widehat{\mathbf{n}}$, R^k the shift vector “along” $\widehat{\mathbf{n}}$, i.e.,

$${\mathbf{R}}^{\mathbf{k}}=\left\{\begin{array}{c}{\mathbf{R}}^{\mathbf{k},x}\equiv \left(R_{xx}^{\mathbf{k}},R_{yx}^{\mathbf{k}}\right)\mathrm{for}\widehat{\mathbf{n}}=\widehat{\mathbf{x}}\hfill \\ {\mathbf{R}}^{\mathbf{k},y}\equiv \left(R_{xy}^{\mathbf{k}},R_{yy}^{\mathbf{k}}\right)\mathrm{for}\widehat{\mathbf{n}}=\widehat{\mathbf{y}}.\hfill \end{array}\right)$$ 9

Phenomenological dephasing and decay times within the relaxation-time approximation, T₂ = 10 fs and T₁ = 1 ps, are included to mimic elastic and inelastic electron scattering²¹. The initial condition is chosen as P^K(0) = 0 and $f_{\pm }^{\mathbf{K}}\left(0\right)=f_0^{\mathbf{K}}$, where $f_0^{\mathbf{K}}$ is the Fermi–Dirac distribution at 300 K and at a Fermi level of 0.25 eV above the Dirac point.

The total current along the Cartesian direction a is

$$j_a\left(t\right)=-{\sum }_{\mathbf{K}}{\partial }_a{E}_{+}^{\mathbf{K}+\mathbf{A}\left(t\right)}{f}_{+}^{\mathbf{K}}\left(t\right)-{\partial }_a{E}_{-}^{\mathbf{K}+\mathbf{A}\left(t\right)}{f}_{-}^{\mathbf{K}}\left(t\right)-2{\sum }_{\mathbf{K}}\mathrm{\Delta }{E}^{\mathbf{K}+\mathbf{A}\left(t\right)}\mid d_a^{\mathbf{K}+\mathbf{A}\left(t\right)}\mid \mathrm{Im}{P}^{\mathbf{K}}{e}^{-i\left[\arg \left(d_a^{\mathbf{K}+\mathbf{A}\left(t\right)}\right)-\arg \left(d^{\mathbf{K}+\mathbf{A}\left(t\right)}\right)\right]}$$ 10

and the harmonic intensity along a specific polarization $\widehat{\mathbf{s}}$ is then calculated as $S_{\widehat{\mathbf{s}}}\left(\omega \right)\propto {\omega }^2\mid \tilde{\mathbf{j}}\left(\omega \right)\cdot \widehat{\mathbf{s}}\mid$, where $\tilde{\mathbf{j}}\left(\omega \right)$ is the Fourier transform of the current j(t) weighted with a ${\cos }^2$ mask. In addition, the contribution from the anomalous current depending on the Berry curvature can be calculated as

$${\mathbf{j}}^{\mathrm{anom}}\left(t\right)=-\mathbf{F}\left(t\right)\times {\sum }_{\mathbf{K}}\left[{\mathbf{\Omega }}_{-}^{\mathbf{K}+\mathbf{A}\left(t\right)}{f}_{-}^{\mathbf{K}}\left(t\right)+{\mathbf{\Omega }}_{+}^{\mathbf{K}+\mathbf{A}\left(t\right)}{f}_{+}^{\mathbf{K}}\left(t\right)\right].$$ 11

For convenience, in the main text, when the Berry curvature and shift vector are introduced conceptually, we omit their explicit k-dependence and denote them as Ω and R, provided no ambiguity arises.

Author contributions

S.L., L.Y., and W.Z. conceived the study. S.L., T.A.T., and L.B. performed the experiment. L.Y. performed the simulations. K.A.I. and W.Z. prepared and characterized the samples. S.L., Y.T., and V.L. developed the laser and light source systems. Y.T. developed the data acquisition software packages. P.A., R.K.K., A.S.L., and L.F.D. supervised the study. All authors contributed to the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

 Source data are provided with this paper.

Code availability

The codes for calculating HHG from topological surface states of Bi₂Se₃ are available upon request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-74310-7.