Rashba-induced intersubband interference in high-harmonic generation

Abstract

High-harmonic spectroscopy is exquisitely sensitive to the electronic structure and dynamics of quantum materials, yet quantum interference between spin-orbit-split bands has remained experimentally elusive. We observe this interference between Rashba-split bands in bismuth tellurium iodide (BiTeI) driven by strong mid-infrared laser fields. The sixth harmonic exhibits intensity-scaling kinks and an M-shaped ellipticity dependence. Theoretical simulations attribute these features to spin-orbit–mediated multichannel quantum interference, which arises from Landau-Zener-Stückelberg–type intersubband tunneling between Rashba-split bands. The suppression of harmonic yield under linearly polarized pulses arises from destructive interference among electron-hole recombination pathways, whereas increasing ellipticity modifies their relative phase and induces a distinct spectral redshift. These results establish high-harmonic spectroscopy as a probe of spin-textured band dynamics and ultrafast spin-polarized currents.

INTRODUCTION

High-harmonic generation (HHG) under strong laser fields is a powerful tool for probing ultrafast dynamics in solids (1, 2) and is highly sensitive to electronic band topology (3–5), attosecond carrier dynamics (6, 7), and lattice potential and vibrations (8–11). This extreme sensitivity establishes high-harmonic spectroscopy (HHS) as a powerful probe for quantum dynamics, capable of revealing phenomena such as multiband interference (12, 13), intervalley transport (14), spin-polarized topological states (15–18), and Berry curvature– and Berry phase–mediated nonlinear responses (19–23). Quasiparticle dynamics in spin-textured states is central to the study of strong-field HHG. Recent research has demonstrated that HHG is closely associated with electronic spin dynamics, as spin-dependent electronic information can be directly encoded in the HHS (24–26). To date, however, spin-dependent quantum interference has not yet been identified in experiments, and their underlying microscopic mechanisms remain unresolved.

Spin-orbit coupling (SOC) fundamentally links the orbital motion of an electron to its spin, acting as an effective momentum-dependent magnetic field that locks spin orientation to electron momentum (27, 28). In Rashba materials with broken inversion symmetry, this coupling creates a spin momentum–locked texture, providing a key platform for studying electric field–driven spin dynamics (29, 30). SOC underpins a range of phenomena, including spin-charge conversion (the Edelstein effect) (31), the quantum spin Hall and quantum anomalous Hall effects (32), and nonreciprocal magnetoelectric responses (33, 34), while offering a foundational framework for engineering topological phases (35, 36). Beyond magnetic field control, SOC also enables pathways for optically driven spin dynamics through the interaction of intense laser fields with spin-split states, directly manifesting in nonperturbative harmonic emission. Notably, in topological Dirac semimetals, SOC-induced modifications to the harmonic response affect both electron dynamics and time-delayed emission (24).

Among materials with strong SOC, BiTeI (trigonal space group P3m1) is an ideal candidate for HHG studies due to its exceptionally strong bulk Rashba spin splitting at room temperature (37, 38). Originating from its noncentrosymmetric crystal structure, which consists of alternating Bi-Te-I layers (see inset in Fig. 1A), a strong intrinsic out-of-plane electric polarization is generated (39, 40), resulting in substantial spin-polarized bands near the Fermi level. Recent study has confirmed that the pronounced shift vector in BiTeI critically governs HHG and enables the probing of topological phase transitions (41).

Fig. 1. HHG from the layered polar semiconductor BiTeI.

(A) Spectra measured under linearly polarized excitation (ε = 0) at E_MIR = 3.3 and 4.7 MV/cm show harmonics up to H10. Inset: Schematic of the experimental setup (MIR field, red; HHG signal, purple) in reflection geometry, alongside the crystal structure of BiTeI (see details in fig. S1). (B) Intensity scaling of the even-order harmonics H4, H6, and H8 versus driving field strength for linear (ε = 0) and elliptical (ε = 0.37) polarization. The insets illustrate the polarization of the pump field (red) and the even-order harmonics (purple). During measurements, the polarization orientation of the driving field remains fixed and parallel to the ΓK direction (see fig. S2).

In this Letter, we investigate HHG in BiTeI crystals and observe pronounced kinks and suppressed harmonic yield within a specific range of driving field strength, with the suppression diminishing as the laser ellipticity increases. On the basis of a multiband tight-binding model and semiconductor Bloch equations (SBEs), we reproduce the observations and identify multichannel harmonic interference involving spin-split subbands near the Fermi level as the key mechanism. Our findings show that SOC, by entangling spin and orbital degrees of freedom, enables intersubband transitions to dominate electron excitation under strong laser field driving.

RESULTS

In our experiments, a polarization-tunable mid-infrared (MIR) laser field (wavelength, 3.8 μm; photon energy, 0.326 eV; peak field amplitude, 5.5 MV/cm) was directed onto the pristine surface of a BiTeI crystal at an incidence angle of 3°. HHG were collected and detected in a reflection geometry (see inset of Fig. 1A and fig. S1). Figure 1A presents HHG spectra from BiTeI under linearly polarized (ε = 0) light field oriented perpendicular to the mirror plane. The spectrum contains both even- and odd-order harmonics, with well-resolved peaks ranging from the 4th harmonic (H4; 1.3 eV) to H10 (3.26 eV). Even-order harmonics exhibit intensities comparable to those of adjacent odd-order harmonics (e.g., H6 and H7), a characteristic arising from the noncentrosymmetric crystal structure of BiTeI.

We focus on even-order harmonic spectral, as it arises directly from the broken inversion symmetry induced by the Janus-type (Te-Bi-I) polar structure of BiTeI (37). Figure 1B shows the intensity scaling of even-order harmonics with driving field strength under linear (ε = 0) and elliptical (ε = 0.37) polarization, with the polarization principal axis along Γ-K direction (fig. S2). Under linear driving, H6 exhibits a clear turning point, which is a kink feature with reduced intensity growth, that markedly deviates from perturbative scaling at moderate MIR fields (2 to 3 MV/cm). Beyond 3 MV/cm, it reverts to the initial perturbative trend ($\propto E_{\text{MIR}}^{2n}$, n = 6). In contrast, H4 and H8 maintain a monotonic increase across the entire driving field range. At pump ellipticity ε = 0.37, the kink in H6 disappears, and its signal grows monotonically, while H4 and H8 retain scaling characteristics similar to those at ε = 0.

To further investigate ellipticity-dependent responses, we systematically recorded harmonic intensities as a function of MIR field strength at five ellipticities (see Fig. 2, A and B). Below a driving field threshold of E_MIR ≈ 4 MV/cm, the H6 kink weakens with increasing ellipticity, accompanied by an ellipticity-dependent rise in harmonic yield. Above this threshold, H6 follows distinct scaling under linear polarization, resulting in faster intensity growth and higher total yield than under elliptical driving. In contrast, H4 intensity remains largely unaffected by ellipticity and increases monotonically throughout.

Fig. 2. Intensity scaling and ellipticity dependence of HHG in BiTeI.

(A and B) Intensities of H4 (A) and H6 (B) as functions of the driving laser field ellipticity ε, which varies from 0 to 0.37. (C and D) Measured ellipticity-dependent responses of the x components of H4 (C) and H6 (D), where ε ranges from −0.95 (right-handed circular polarization) to 0 (linear polarization) to 0.95 (left-handed circular polarization). (E and F) Simulated ellipticity-dependent responses of H4 (E) and H6 (F) arising from interband contributions, calculated under the same laser parameters as in (C) and (D) (wavelength, 3.8 μm; pulse duration, 50 fs).

Figure 2 (C and D) presents the x-component intensities of H4 and H6 as a function of the driving field ellipticity for five pump-field amplitudes (see fig. S3 for the y components). Notably, H6 exhibits a distinct M-shaped dependence on E_MIR for values below 4 MV/cm, with intensity peaking at an ellipticity of approximately 0.37. As ellipticity increases further, the H6 signal decreases rapidly and is completely quenched under circular polarization. This behavior is consistent with optical selection rules constrained by the C_3v symmetry of BiTeI (42, 43). At higher driving fields (E_MIR ≥ 4 MV/cm), an Λ-shaped profile emerges, with the maximum intensity shifting to linear driving field. As a comparison, H4 and H8 show no such anomalous behavior and remain observable even under circular driving (Fig. 2C and fig. S4).

Ellipticity-dependent harmonic emission is strongly correlated with specific material band structures (44), such as linear dispersion in graphene (45, 46), ionic potential–modulated interband transitions in MgO (9, 47, 48), and band topology in Bi₂Se₃ (5,17). Our experimental results shed light on the strong-field dynamics of Rashba spin-split bands in BiTeI. We therefore constructed a 10-band tight-binding model comprising six valence bands (VBs) and four conduction bands (CBs) to simulate HHG in BiTeI (see fig. S6). The simulated high-harmonic spectrum spans from H4 to H10, in agreement with experiment, and successfully reproduces the distinctive nonmonotonic (kink-shaped) intensity dependence observed in H6 (see figs. S7 and S8). For even-order harmonics such as H6, interband polarization dominates the generation mechanism, which originates from SOC-driven Rashba spin-split bands (see section S3 for simulations with and without SOC). We also examined the field dependence of odd-order harmonics; H5 exhibits a weak kink attributed to inter- and intraband interference (fig. S5), distinct from H6. A recent study in monolayer WS₂ has identified similar kinks in the H7 intensity near the exciton resonance, also arising from quantum interference between intra- and interband channels (49). In BiTeI, however, we find that these even-order features are specifically governed by interband coherence within Rashba-split bands and arise directly from strong SOC effects.

Figure 2 (E and F) shows the simulated ellipticity-dependent responses of the x components of H4 and H6 under varying field strengths, revealing the M-shaped pump ellipticity dependence of H6 for E_MIR < 4 MV/cm. Beyond this field range, the maximum harmonic intensity shifts to linear polarization, confirming the reliability of our model in simulating the harmonic response in BiTeI. To gain deeper insight, we disentangled the contributions from various interband transition pathways (see fig. S9). This analysis identifies the four bands nearest to the Fermi level, specifically, two VBs (V₁ and V₂) and two CBs (C₁ and C₂), as the dominant contributors to the observed harmonic generation. The M-shaped ellipticity dependence arises from the interference between the C₁ → V₁ and C₂ → V₂ transition channels.

On this basis, we constructed a minimal model comprising the four spin-split bands to elucidate the underlying interband interference mechanism, as illustrated in Fig. 3A. This model captures the essential transition channels between these bands (50). We investigated interband transitions by calculating electron populations and transition dipole moments (Fig. 3B). Figure 3 (C to F) presents k-space–resolved electron populations in C₁ and C₂ at two time instants (t = −37.5 and 0 fs). Electrons in C₁ predominantly originate from V₁ → C₁ transitions, as evidenced by similarities with the transition dipole moment ${\mathbf{d}}_{{\mathrm{V}}_1{\mathrm{C}}_1}^{\mathbf{k}}$ (see Fig. 3B).

Fig. 3. Transition channels and electron population dynamics in Rashba-split bands.

(A) Band dispersion of BiTeI near the Fermi level at k_z = π, showing V₁, V₂, C₁, and C₂. Left inset: C₁ and C₂ electron spins in the k_x-k_y plane at ∼200 meV above the Dirac point. Right inset: Schematic band dispersion near the Dirac point along k_y at k_x = 0.024 Å⁻¹ (vertical plane). The small bandgap (C₁-C₂ and V₁-V₂) enables LZS-type tunneling (Tun.) and supports transitions of electrons (C₁ ↔ C₂) and holes (V₁ ↔ V₂). Upward black arrow indicates direct V₁ → C₁ excitation (Exc.) near the band edges. Downward arrows mark the C₁ → V₁ (black) and C₂ → V₂ (purple) interband recombination (Rec.) channels. LZS-type intersubband tunneling provides an indirect C₁ → V₁ recombination path (path 2, purple curved arrows) as the accelerated (Acc.) electrons/holes pass through the small bandgap region, complementing the direct path (path 1, black curved arrows). (B) Transition dipole amplitudes (y component) for V₁ → C₁ interband (left) and the C₁ → C₂ intersubband (right) transitions. (C to F) Electron populations in spin-split subbands C₁ and C₂ are compared at two time instants: t = −37.5 fs [(C) for C₁ and (E) for C₂] and t = 0 fs [(D) for C₁ and (F) for C₂], corresponding to times before and at the driving field peak. The insets show the driving laser field (E_MIR = 2.3 MV/cm), with red dots marking the corresponding time snapshots. (C) to (F) share the same color scale.

In contrast, the population in C₂ originates primarily from intersubband dipole transitions from C₁ rather than from direct V₂ → C₂ excitation (see Fig. 3, E and F). The C₂ population profile highlights the combined role of the ${\mathbf{d}}_{{\mathrm{V}}_1{\mathrm{C}}_1}^{\mathbf{k}}$ and intersubband ${\mathbf{d}}_{{\mathrm{C}}_1{\mathrm{C}}_2}^{\mathbf{k}}$ dipoles. Notably, there is a clear mismatch between the contours of these two dipoles, indicating that efficient population transfer from C₁ to C₂ requires laser-driven transport to bridge the momentum space gap. The C₁ → V₁ transition occurs predominantly where the V₁-C₁ dipole is largest, near the minimal gap region; peak excitation thus coincides with the crest of the MIR field oscillation. Before the crest, the C₁ population distribution matches the dipole profile (Fig. 3C). After excitation, the electron distribution moves toward the Dirac point region due to laser-driven acceleration (see fig. S12), a hallmark of the strong-field regime. Intersubband transitions are activated in this region, and, consequently, the C₂ population is greatly enhanced (Fig. 3F). Suppressing these intersubband transitions strongly reduces the C₂ population (see fig. S13), confirming their key contribution. These interband excitations, driven by strong SOC, simultaneously elevate the electron energy and flip its spin (Fig. 3A). Consequently, the harmonic emission is dominated by interband recombination through the C₁ → V₁ and C₂ → V₂ channels, which have opposite spin polarization textures.

The ellipticity-dependent response in H6 is reproducible with our simplified four-band model. Figure 4A shows that the M-shaped H6 intensity arises primarily from C₂ → V₂ and C₁ → V₁. Under linearly polarized excitation, the C₂ → V₂ contribution is approximately five times larger than that of C₁ → V₁. Intuitively, destructive interference between these two channels accounts for the reduced overall harmonic intensity (fig. S9). These results indicate that interband transitions between spin-split subbands play an indispensable role in producing H6.

Fig. 4. Ellipticity dependence of H6 mediated by intersubband transitions.

(A) H6 intensity decomposition in the four-band model at E_MIR = 2.85 MV/cm: contributions from dominant channels C₁ → V₁ (orange), C₂ → V₂ (red), and the total signal (black) including all allowed interband transitions (C₁ → V₁, C₂ → V₂, C₂ → C₁, V₁ → V₂, C₁ → V₂, C₂ → V₁). Inset: Rashba-split band structure showing valence subbands V₁ and V₂ and conduction subbands C₁ and C₂, with intersubband transition dipoles marked by shaded regions. (B and C) Dominant channel contributions under selective momentum-space suppression of ${\mathbf{d}}_{{\mathrm{C}}_1{\mathrm{C}}_2}^{\mathbf{k}}$ and ${\mathbf{d}}_{{\mathrm{V}}_1{\mathrm{V}}_2}^{\mathbf{k}}$ dipoles: (B) suppressed for |k| ≤ 0.04 Å⁻¹; (C) suppressed for |k| > 0.04 Å⁻¹ (gray-shaded regions in the inset).

To pinpoint the role of momentum-space localization in intersubband transitions, we systematically suppressed the transition dipoles within selected k-regions near the Dirac point and simulated the corresponding ellipticity dependence of H6. When only transitions very close to the Dirac point are excluded, the anomalous ellipticity dependence remain observable, although its contrast is reduced (see fig. S14 for |k| ≤ 0.035 Å⁻¹). As the suppressed k-region is progressively enlarged, the characteristic M-shaped profile gradually vanishes, and the harmonic intensity shows a sharp drop (see Fig. 4B for |k| ≤ 0.04 Å⁻¹ and fig. S14 for other k region). This disappearance coincides with a pronounced decrease in the C₂ population (fig. S13), indicating that the anomalous response originates predominantly from the intersubband transitions within 0.02 Å⁻¹ < |k| ≤ 0.04 Å⁻¹. Conversely, the anomalous ellipticity-dependent response of H6 reemerges when only transitions within |k| < 0.04 Å⁻¹ are allowed (Fig. 4C). Partial suppression of intersubband transitions notably reduces harmonic yield, with larger regions resulting in lower intensity (fig. S14), highlighting their essential role in HHG. Together, these results demonstrate that the anomalous ellipticity dependence stems from SOC-induced intersubband coupling, which establishes a momentum-localized interband coherence responsible for modulating the nonperturbative harmonic response.

The anomalous scaling and ellipticity-dependent responses of H6 are directly attributed to spectral interference between the C₁ → V₁ and C₂ → V₂ recombination channels. Our experimental and numerical results further confirm this interference process between these two dominant transition channels. Figure 5A reveals a pronounced red shift in the measured H6 spectrum with varying pump ellipticity, a feature that is also reproduced in our theoretical calculations (see Fig. 5B). Figure 5C separately extracted the H6 spectra of the two channels and their spectral phase difference for three pump ellipticities (ε = 0, 0.2, and 0.4).

Fig. 5. Spectral shift of H6 induced by multichannel interference.

(A) Measured H6 spectrum versus driving field ellipticity at E_MIR = 2.85 MV/cm. (B) The simulated spectrum at the same peak field strength as (A), including multichannel interference. In (A) and (B), spectra are shown on logarithmic scales, with each spectrum shifted according to its ellipticity. The vertical dashed lines indicate the harmonic spectral peaks at ε = 0 and ε = 0.4, clearly marking the spectral redshift. (C) Independent contributions of the C₁ → V₁ (orange) and C₂ → V₂ (red) transition channels, together with their phase difference (gray circles) and coherent sum (blue), for three pump ellipticities: ε = 0, 0.2, and 0.4. Vertical dashed lines at 1.96 and 1.98 eV mark the spectral region where ellipticity-driven destructive-to-constructive interference occurs predominantly on the low-energy side of the spectral peak, causing the observed redshift.

Under weak driving fields, the spectral phase of the C₁ → V₁ channel exhibits a pronounced chirp that varies with driving ellipticity, whereas the phase of the C₂ → V₂ channel remains largely unaffected (see fig. S15, A and B). As a result, the phase difference Δφ between the two channels is close to π (destructive interference) on the low-frequency side of the spectrum and close to 0 or 2π (constructive interference) on the high-frequency side (fig. S15C). For example, at 1.96 eV, Δφ increases markedly from 1.27π to 2.3π, while near the peak center (1.98 eV), it increases from 1.5π to 2.34π. Therefore, increasing the ellipticity gradually converts the interference condition, causing an ellipticity-dependent redshift of the spectral peak and an increase in H6 intensity (Fig. 5C).

The spectral chirp in the C₁ → V₁ channel originates from the indirect pathway (C₁ → C₂ → C₁ → V₁) introduced by intersubband transitions (see Fig. 3A and fig. S15D). This pathway is active only near the Dirac point, where the C₁-C₂ gap closes and intersubband coupling acts as a coherent beam splitter—similar to Landau-Zener-Stückelberg (LZS) interference in graphene (51). The ellipticity-sensitive chirp arises from the finite-momentum region near the Dirac point (fig. S17A), where partial tunneling enables coherent superposition of direct and indirect pathways. The indirect trajectory acquires an ellipticity-sensitive phase shift, yielding a tunable spectral chirp. Control calculations that suppress the transition dipole near the Dirac point confirm the existence of the indirect pathway: When it is removed, only the direct pathway (C₁ → V₁) remains, and the chirp disappears (fig. S15, E and F). The momentum-selective suppression in Fig. 4 (B and C) pinpoints the active region for LZS tunneling in the H6 pathways to 0.02 Å⁻¹ < |k| ≤ 0.04 Å⁻¹, confirming that this momentum zone controls the beam-splitting and the resulting chirp. At 5.2 MV/cm, carriers are driven away from the Dirac point, suppressing the indirect pathway and rendering the C₁ → V₁/C₂ → V₂ phase difference nearly independent of ellipticity. This results in stable interference, a monotonic ellipticity dependence, and an unchanged center frequency (fig. S16). In contrast, the H8 x component lacks the M-shaped response (fig. S4), confirming that the H6 anomaly is specifically tied to the SOC-induced indirect pathway near the Dirac point (section S5).

DISCUSSION

In summary, the kink feature in the intensity scaling of HHG and the M-shaped ellipticity dependence directly reveal the strong field–driven motion of spin-polarized electrons in BiTeI. We identify SOC-induced multichannel quantum interference as the key mechanism underlying the observed harmonic spectral shifts and intensity suppression. In general, this strong-field mechanism can occur if (i) strong SOC and broken inversion symmetry produce nonzero intersubband dipoles between opposite-spin subbands; (ii) a sufficiently strong laser field drives the system into the nonperturbative regime, enabling carrier transport to active intersubband coupling regions; and (iii) a closed transition loop among spin degenerate–lifted bands allows multichannel interference. This finding provides a universal theoretical framework applicable to various strong SOC systems, elucidating interactions between intense laser fields and spin-textured electronic states through the revealed quantum interference between spin-polarized bands. These results offer fundamental insights for advancing strong-field physics and developing opto-spintronic devices with ultrafast spin control.

MATERIALS AND METHODS

Experimental setup

MIR driving pulses were generated by combining a dual optical parametric amplifier with a difference frequency generation system, pumped by a commercial Ti:sapphire amplifier (Coherent Legend Elite Duo; 8 mJ, 40 fs, 800 nm, 1 kHz), as illustrated in fig. S1. The output pulse was centered at 3.8 μm (79 THz, 326 meV). The MIR pulse power was controlled using a half-wave plate (HWP) in conjunction with a wire-grid polarizer (LP, extinction ratio ∼ 10⁵:1), followed by fine adjustment of the polarization ellipse via a second HWP and a quarter-wave plate (QWP). The MIR pulses were focused onto the sample at an incident angle of ∼3°, yielding a focal spot with a full width at half maximum of ∼130 μm. This corresponds to a peak intensity of 48 GW/cm² and a maximum field strength of 6 MV/cm.

The polarization of the 3.8-μm driving field was continuously adjustable from linear to circular using the HWP and QWP specifically designed for this wavelength, as shown in fig. S2. The HWP and QWP were rotated synchronously through different angles to ensure that the major axis of the driving field remained aligned with the predefined crystallographic direction of the sample. Using this configuration, the polarization state of the driving field was varied, and the power distribution was measured as a function of angle with a linear polarizer and a power meter. We achieved control over the polarization of the driving field from left-handed to right-handed, with the |ε| from 0 to 0.95, as shown in fig. S2. The orientation of the major axis remained stable throughout the tuning process, with deviations kept within ±1° from the reference crystallographic direction for |ε| < 0.6.

The ellipticity was defined as $\mid \mathrm{\epsilon }\mid =\sqrt{I_{\text{min}}/I_{\text{max}}}$, and the major-axis orientation was given by $\mathrm{\varphi }=\text{arctan}(I_y/I_x)$. The values of $I_{\text{min}}$, $I_{\text{max}}$, $I_x$, and $I_y$ were determined by fitting the angular dependence of the measured power to a theoretical model for polarization analysis. Specifically, the power $P\left(\mathrm{\theta }\right)$ as a function of the polarizer angle $\mathrm{\theta }$ was fitted to the function $P\left(\mathrm{\theta }\right)=I_{\text{max}}\text{co}{\mathrm{s}}^2(\mathrm{\theta }-\mathrm{\varphi })+I_{\text{min}}\text{si}{\mathrm{n}}^2(\mathrm{\theta }-\mathrm{\varphi })$ using a least-squares method, from which $I_{\text{min}}$, $I_{\text{max}}$, and $\mathrm{\varphi }$ were extracted.

Sample

BiTeI single crystals were provided by Muke-Nano Tech. They were grown by chemical vapor transport (CVT). First, polycrystalline BiTeI was synthesized by annealing stoichiometric amounts of Bi, Te, and I elements in an evacuated quartz ampoule. The resulting polycrystalline powder was then mixed with I₂ as a transport agent, sealed in an ampoule, and placed in a two-zone tube furnace with a controlled temperature gradient for CVT growth. After the growth process, high-quality layered BiTeI crystals were obtained through controlled cooling. For HHG experiments, single-crystal specimens with optically flat (111)-oriented surfaces and a size of ∼2 mm were prepared by mechanical exfoliation.

Simulation model

The theoretical framework and numerical simulations were founded on the SBEs within the Houston representation (52–56). Unless specified otherwise, atomic units are adopted uniformly throughout the calculations. The equation governs the time-dependent evolution of the electronic density matrix ${\mathrm{\text{ρ}}}_{\mathit{nm}}^{\mathbf{k}}\left(t\right)$ in the presence of a laser field, expressed as

$$\begin{array}{c}i{\mathrm{\partial }}_t{\mathrm{\rho }}_{\mathit{nm}}^{\mathbf{k}\left(t\right)}\left(t\right)=[{\mathcal{E}}_m^{\mathbf{k}\left(t\right)}-{\mathcal{E}}_n^{\mathbf{k}\left(t\right)}-\frac{i(1-{\mathrm{\delta }}_{\mathit{nm}})}{T_2}]{\mathrm{\rho }}_{\mathit{nm}}^{\mathbf{k}\left(t\right)}\left(t\right)\\ -\mathbf{E}\left(t\right)·{\sum }_l[{\mathbf{d}}_{\mathit{ln}}^{\mathbf{k}\left(t\right)}{\mathrm{\rho }}_{\mathit{lm}}^{\mathbf{k}\left(t\right)}\left(t\right)-{\mathbf{d}}_{\mathit{ml}}^{\mathbf{k}\left(t\right)}{\mathrm{\rho }}_{\mathit{nl}}^{\mathbf{k}\left(t\right)}\left(t\right)]\end{array}$$ (1)

where $\mathbf{k}\left(t\right)=\mathbf{K}+\mathbf{A}\left(t\right)$ denotes the quasimomentum of electrons under lasers, $\mathbf{K}$ is the crystal canonical momentum, $\mathbf{A}\left(t\right)$ is the vector potential of the laser field, ${\mathcal{E}}_n^{\mathbf{k}}$ is the energy of the nth band at k, and $T_2$ is the dephasing time of the system. We performed a robustness analysis of $T_2$ and selected $T_2$ = 3.17 fs (about one-quarter of the optical cycle period) as the reference value, which is commonly used in HHG simulations (54). Crucially, the anomalous ellipticity dependence of H6 persists across a wide range of $T_2$, confirming that this feature is intrinsic and not an artifact of dephasing assumptions. The dipole matrix element ${\mathbf{d}}_{\mathit{nm}}^{\mathbf{k}}=i⟨u_n^{\mathbf{k}}\mid {\mathbf{\nabla }}_{\mathbf{k}}\mid u_m^{\mathbf{k}}⟩$ is constructed from the periodic part of the Bloch wave functions $\mid u_{n,\mathbf{k}}⟩$. It can be decomposed into diagonal terms ($n=m$) and off-diagonal transition terms ($n\ne m$).

Our first-principles calculations were based on density functional theory and performed using the Vienna Ab initio Simulation Package (57). The structural relaxations and electronic self-consistency were calculated using the Perdew-Burke-Ernzerhof exchange-correlation functional and projector-augmented wave pseudopotentials, with the inclusion of SOC. The uniform Monkhorst-Pack k-point mesh of 8 × 8 × 8 was used for integration over the Brillouin zone (BZ). A plane-wave energy cutoff of 400 eV was used in all the calculations.

Using Wannier90 (58) and the Wannier tools package (59), we successfully constructed maximally localized Wannier functions (MLWFs). These MLWFs allow us to efficiently obtain the transition dipole matrix elements (60), which are essential for analyzing the electron dynamics in HHG. A high-precision grid sampling of 650 × 650 in the BZ was carried out to accurately capture the band structure and wavefunction of the Rashba splitting. We constructed a set of orthogonal and complete Bloch basis vectors using MLWFs with a well-defined Wannier gauge, which ensures smooth Bloch states between adjacent k-points in reciprocal space, enabling accurate construction of the dipole matrix and time evolution. The Bloch basis vectors are determined by the initial Hamiltonian, while the time-dependent Houston basis vectors evolve adiabatically from them in reciprocal space.

By adopting the fourth-order Runge-Kutta method to solve the SBEs, we obtained the time-dependent electron population in the adiabatic Houston representation. The time step is 0.16 arbitrary units (a.u.) in our time evolution simulation. The calculated total current can be decomposed into interband current and intraband current, defined respectively as follows

$${\mathbf{J}}_{\text{inter}}\left(t\right)=-{\sum }_{\mathbf{k}\in \text{BZ}}{\sum }_{n\ne m}{\mathrm{\rho }}_{\mathit{nm}}^{\mathbf{k}\left(t\right)}\left(t\right){\mathbf{p}}_{\mathit{nm}}^{\mathbf{k}\left(t\right)}$$ (2)

$${\mathbf{J}}_{\text{intra}}\left(t\right)=-{\sum }_{\mathbf{k}\in \text{BZ}}{\sum }_n{\mathrm{\rho }}_{\mathit{nn}}^{\mathbf{k}\left(t\right)}\left(t\right){\mathbf{p}}_{\mathit{nn}}^{\mathbf{k}\left(\mathrm{t}\right)}$$ (3)

Here, the momentum operator is given by $\hat{\mathbf{p}}\left(\mathbf{k}\right)={\mathrm{\partial }}_{\mathbf{k}}\hat{\mathbf{H}}\left(\mathbf{k}\right)$. In Houston representation, the momentum matrix element takes the form ${\mathbf{p}}_{\mathit{nm}}^{\mathbf{k}\left(t\right)}=⟨u_n^{\mathbf{k}\left(t\right)}\mid \hat{\mathbf{p}}\left(\mathbf{k}\right)\mid u_m^{\mathbf{k}\left(t\right)}⟩$, which can be calculated via

$${\mathbf{p}}_{\mathit{nm}}^{\mathbf{k}\left(t\right)}=i({\mathcal{E}}_n^{\mathbf{k}\left(t\right)}-{\mathcal{E}}_m^{\mathbf{k}\left(t\right)}){\mathbf{d}}_{\mathit{nm}}^{\mathbf{k}\left(t\right)},n\ne m$$ (4)

$${\mathbf{p}}_{\mathit{nn}}^{\mathbf{k}\left(t\right)}={\mathbf{\nabla }}_{\mathbf{k}}{\mathcal{E}}_n^{\mathbf{k}\left(t\right)}$$ (5)

The HHG spectra were obtained via Fourier transform of the current. These spectral features were quantitatively compared with experimental measurements to validate the authenticity of our theoretical model.

Supplementary Materials

This PDF file includes:

Sections S1 to S5

Figs. S1 to S18