Supercurrent effect in a charge density wave intertwined superconductor

Abstract

The energy-momentum (E-k) dispersion of quasiparticles constitutes a fundamental concept in condensed matter systems. The ability to modify the E-k dispersion, exemplified by supercurrent-induced Doppler shifts of Bogoliubov quasiparticle spectra in superconductors, enables manipulation of various emergent quantum properties. However, investigations into the supercurrent effect on superconductors intertwined with charge orders remain scarce. Here, we report that the Meissner current, generated by the diamagnetic response to an applied in-plane magnetic field, can tailor Bogoliubov quasiparticle excitations at the precursor charge density wave (CDW) vectors. Our scanning tunneling spectroscopic imaging reveals a field-driven symmetry breaking of CDW modulations, specifically a C_3v-to-C_s transition, in superconducting NbSe₂. Model calculations suggest that the observed anisotropy originates from a selective Doppler-shift-induced E-k dispersion reconstruction. Furthermore, altering the field direction enables on-demand tuning of anisotropic CDW modulations and visualization of their momentum-space distribution. These results highlight a novel mechanism for controlling emergent electronic phases through momentum-space engineering.

Scanning tunneling spectroscopic imaging reveals a field-driven symmetry breaking of CDW modulations in superconducting NbSe₂. Model calculations suggest that the observed anisotropy originates from a Doppler-shift-induced reconstruction of the dispersion of Bogoliubov quasiparticle excitations.

Introduction

Bogoliubov quasiparticles are the elementary excitations in superconductors and obey the energy-momentum dispersion:$E\left(\mathbf{k}\right)=\sqrt{{\left({ϵ}_{\mathbf{k}}-\mu \right)}^2+{\mathrm{\Delta }}^2}$, where ${ϵ}_{\mathbf{k}}$ denotes the normal-state dispersion, $\mu$ is the chemical potential, and $\mathrm{\Delta }$ is the superconducting gap. When a supercurrent flows through a superconductor, the dispersion of Bogoliubov quasiparticles acquires an additional term¹, i.e., the Doppler shift $E_{\mathrm{D}}={e\mathbf{v}}_{\mathbf{k}}\cdot \mathbf{A}$ (${\mathbf{v}}_{\mathbf{k}}$ is the Fermi velocity at momentum $\mathbf{k}$, $\mathbf{A}$ is the vector potential), which gives rise to many novel phenomena, including segmented Fermi surfaces², the Josephson diode effect³, and topological quantum phase transitions^4,5. Within engineered structures, supercurrents may also be harnessed to create and manipulate Majorana zero modes^6–9. Furthermore, the Doppler shift can tailor the optical and thermal properties of superconductors^10–12. Like superconductivity, charge density wave (CDW) represents another prototypical symmetry-broken phase, arising from Fermi surface instabilities¹³. CDW and superconductivity can either compete or coexist in quantum materials, and their interplay has captivated research interest for decades^14–19. In CDW intertwined superconductors, CDW order may superimpose a spatial modulation on the Bogoliubov quasiparticles in real space, thereby breaking the underlying crystal translational symmetry. However, the effect of supercurrents in such superconductors remains poorly understood. Here, we employ in-plane magnetic fields to generate shielding currents at the sample surface through the Meissner effect, and utilize dilution refrigerator–based scanning tunneling microscopy and spectroscopy (STM/STS) to reveal the supercurrent-induced symmetry breaking in momentum space and its impact on the orientation- and energy-dependent characteristics of Bogoliubov quasiparticle excitations at the precursor CDW vectors in 2H-NbSe₂.

2H-NbSe₂ is a layered transition-metal dichalcogenide comprises of hexagonally coordinated Se–Nb–Se trilayers stacked along the c-axis by weak van der Waals forces. It exhibits s-wave superconductivity with T_c ≈ 7.2 K and undergoes a 3×3 CDW phase transition at T_CDW ≈ 33 K. The coexistence of superconductivity and CDW order has established NbSe₂ as a canonical system in condensed matter physics^20–22, underpinning landmark studies—from real-space imaging of pair-density-wave states to explorations of electronic fluid flow^23–25.

Results

As illustrated in Fig. 1a, the triple-Q CDW modulation distinctly overlays the topmost hexagonal lattice of Se atoms on the NbSe₂ surface. The local commensurate domain reveals a CDW unit cell matching three times the lattice periodicity in real space (inset of Fig. 1a). Remarkably, C_3v point group symmetry is preserved in the CDW state, characterized by a modulation consisting of three symmetry-equivalent wavevectors oriented at intervals of 120°. Figure 1b shows the Fast Fourier Transform (FFT) of the STM image, where Bragg peaks and 3 × 3 CDW peaks simultaneously emerge. We denote the positions and intensities of the CDW peaks by vectors $\pm {\mathbf{d}}_i$ (i = 1, 2, 3) and scalars $P\left({\pm \mathbf{d}}_i\right)$, respectively (see Supplementary Fig. 1). Since the CDW is a real physical observable, its Fourier components exhibit complex conjugate symmetry, which guarantees that $P\left({\mathbf{d}}_i\right)=P\left({-\mathbf{d}}_i\right)$ (Supplementary Note 1). We therefore restrict our discussion to the set $\left\{{\mathbf{d}}_i\right\}$. In 2H-NbSe₂, $\left\{{\mathbf{d}}_1,{\mathbf{d}}_2,{\mathbf{d}}_3\right\}=\left\{\left(\left(\mathrm{0,\; 1}\right),\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right),\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)\right\}\right)4\sqrt{3}\mathrm{\pi }/\left(9a\right)$, where a is the lattice constant. The nearly equal intensities of all CDW peaks, i.e., $P\left({\mathbf{d}}_1\right)\approx P\left({\mathbf{d}}_2\right)\approx P\left({\mathbf{d}}_3\right)$, indicate a symmetric triple‑Q order. By incorporating a triple-Q CDW potential together with superconductivity into a tight-binding model simulation (Supplementary Note 1), we obtain the simulated real-space local density of states (LDOS) of NbSe₂ at E = 1.3Δ, which exhibits a clear 3 × 3 charge order (Fig. 1c). The corresponding FFT (Fig. 1d) reproduces the $P\left({\mathbf{d}}_1\right)=P\left({\mathbf{d}}_2\right)=P\left({\mathbf{d}}_3\right)$ pattern, demonstrating a triple-Q CDW order with C_3v symmetry (Supplementary Fig. 5).

Fig. 1. CDW order and Doppler-shifted superconducting spectra in 2H-NbSe₂.

a Atomically resolved topography showing 3 × 3 CDW superlattice superimposed on 1×1 atomic lattice (V = 3 mV, I = 100 pA). The inset shows a zoomed-in view, where the red rhombus marks the CDW unit cell. Scale bar, 5 nm. b Fast Fourier transformed (FFT) of the STM image shown in (a). The Bragg peaks are indicated by white triangles. The blue and orange circles and the green square mark the CDW peaks. The first Brillouin zone is outlined by a white hexagon. c Simulated real-space local density of states (LDOS) with CDW pattern at E = 1.3Δ. d FFT of the simulated CDW pattern shown in (c). The six CDW peaks exhibit equal intensities, indicating uniform modulation. e The superconducting gap spectra of NbSe₂ measured under in-plane magnetic fields. With increasing field magnitude, the coherence peaks gradually shift outwards along with quasiparticle filling at the Fermi energy, indicative of a Doppler shift (V = 3 mV, I = 500 pA, V_mod = 50 µV). f Spectra measured in a wide energy range remain essentially unchanged regardless of the field, confirming negligible alteration in normal state electronic structure (V = 100 mV, I = 800 pA, V_mod = 2 mV).

Applying a magnetic field parallel to a superconductor’s surface induces a diamagnetic surface current, i.e., the Meissner effect. To verify this effect in our 2H-NbSe₂ sample, we track the evolution of the superconducting gap under an in-plane magnetic field. In Fig. 1e, the averaged superconducting gaps expand as the applied field strength increases, accompanied by the emergence of a finite filling at the Fermi energy. The positions of the coherence peaks shift to higher energies with field up to 0.4 T (Supplementary Fig. 6). From the field dependence of the superconducting gap, we confirm that the screening currents induced by the in-plane field generate a Doppler shift. Expanding to a wider energy range, the spectra exhibit no discernible field‑induced changes in normal states (Fig. 1f), implying the applied field mainly affects Bogoliubov quasiparticles within the superconducting gap.

Next, we study the interplay between the CDW order and Bogoliubov quasiparticles. By performing spectroscopic imaging via energy-dependent dI/dV maps, we obtain real-space LDOS distributions, and the corresponding FFTs further reveal the CDW vectors in momentum space as a function of energy. At zero magnetic field, the superconducting gap of NbSe₂ shows pronounced coherence peaks at ±1.28 mV (Fig. 2a). Below 0.5 mV, the conductance drops to zero, indicating a fully gapped state. A clear kink appears near 1 mV, possibly arising from two-band pairing or an anisotropic gap structure^26–29. Figure 2b–g presents the FFTs of dI/dV maps at the indicated energies (see Supplementary Figs. 7 and 8 for the full dataset). Based on these reciprocal space patterns, one can find that the CDW modulations superimposed on the normal states (3 mV) and Bogoliubov quasiparticles (0.6-1.2 mV) both exhibit 3 × 3 orders with equal $P\left({\mathbf{d}}_i\right)$, indicative of preserved C_3v symmetry. Within the fully gapped region (≤ 0.5 mV) where the LDOS vanishes entirely, no CDW signal is detected.

Fig. 2. Supercurrent effect of CDW modulations on Bogoliubov quasiparticles.

a dI/dV spectra taken under zero magnetic field (V = 3 mV, I = 500 pA). b–g 3D representation of FFTed conductance maps (V = 3 mV, I = 500 pA) taken at indicated energies under zero field. Six CDW peaks (red circles) emerge above 0.5 mV with equal intensity. Bragg peaks are marked by white triangles. h dI/dV spectra (pink) measured under a 100 mT in-plane magnetic field along the Γ-M direction (V = 3 mV, I = 500 pA). The dashed gray curve is the zero-field spectrum shown in (a), overlaid for comparison. i–n 3D representation of FFTed conductance maps (V = 3 mV, I = 500 pA) taken at indicated energies under a magnetic field of 100 mT. CDW and Bragg peaks are marked by red circles and white triangles, respectively. Two CDW peaks oriented parallel to the field direction are enhanced at 0.2, 0.5, and 0.6 mV, consistently observed across the blue-shaded energy range (see Supplementary Fig. 10). The CDW peaks exhibit comparable intensities at higher energies (0.8, 1.2, and 3 mV) regardless of the magnetic field. Purple arrows denote the magnetic field directions.

Remarkably, Bogoliubov quasiparticle excitations at the precursor CDW vectors exhibit drastically distinct behavior under in-plane fields. Upon applying a 100 mT field along the Γ-M direction, the coherence peaks shift to ±1.4 mV, while the gap edge moves to 0.2 mV (Fig. 2h), consistent with a Doppler shift induced by the screening current. We subsequently carried out spectroscopic imaging at various energies. As shown in Fig. 2i–k, the FFTs of the dI/dV maps measured at E = 0.2, 0.5, and 0.6 meV exhibit two CDW peaks significantly enhanced along the field direction, while the other four are suppressed. We obtain the pattern $P\left({\mathbf{d}}_1\right)>P\left({\mathbf{d}}_2\right)=P\left({\mathbf{d}}_3\right)$ (here the vector parallel to the field direction is denoted as ${\mathbf{d}}_1$), which illustrates the symmetry breaking of the CDW from C_3v at zero field to C_s under the applied magnetic field. This enhancement remains robust throughout the 0.2–0.6 mV energy window (see Supplementary Figs. 9 and 10 for the full dataset). Outside this window, the CDW peaks retain an equal intensity pattern at E > 0.6 mV (Fig. 2l–n). Surface-vortex contributions are unlikely to account for the field-dependent CDW peak rearrangements, since the measurements were conducted in regions well separated from visible vortices (Supplementary Fig. 11). Given that the in-plane field predominantly modulates sub-gap Bogoliubov quasiparticles, we conclude that the field-induced enhancement of selected peaks is a direct consequence of the interplay between CDW order and Bogoliubov quasiparticles under the impact of supercurrent. In addition, the Bragg-peak intensities exhibit a field-direction dependence similar to that of the CDW (Fig. 2i–k), as expected from standard Bloch physics. The associated supercurrent-induced Doppler shift reshapes the constant-energy contours inside the superconducting gap, selectively modulates Bragg scattering between states at k and k + G, thereby providing fundamental momentum-space evidence of the Doppler shift acting on Bogoliubov quasiparticles.

To further investigate the response of the CDW to different field orientations, we conducted spectroscopic imaging under 400 mT vector fields aligned with six high-symmetry directions. Figure 3 presents real- and momentum-space images acquired at E = 0.6 mV, which reside in the energy window displaying CDW anisotropy. When the field is applied along the Γ-M directions, the real-space images display distinctly enhanced CDW modulations along the field directions, and the corresponding FFTs reveal marked intensity increase of two CDW peaks in those directions. Once the field is rotated to the Γ-K, the remaining four CDW peaks nearest the field direction exhibit pronounced intensity enhancements. In contrast, the CDW peaks in the FFTs of topographic images retain C_3v symmetry upon rotating the field (Supplementary Fig. 12), demonstrating that the field-direction dependence of the CDW order originates from Bogoliubov quasiparticles.

Fig. 3. Manipulation of the CDW modulations on Bogoliubov quasiparticles by in-plane magnetic fields.

a–c Real-space dI/dV maps and the corresponding 3D representation of FFT depicting the CDW modulations on Bogoliubov quasiparticles with magnetic field oriented along the Γ-M directions. Two CDW peaks aligned parallel to the field direction are selectively enhanced. d–f Replicating (a–c) with the magnetic field applied along the Γ-K directions, where four CDW peaks closest to the field direction display markedly enhanced intensity. All data were acquired within the same 22 × 22 nm² area, at a bias voltage of 0.6 mV and under an in-plane magnetic field of 400 mT (V = 2 mV, I = 1 nA, V_mod = 50 µV). Field orientations are indicated by arrows.

Discussion

The lowering of the overall spectral weight symmetry to C_s in momentum space is generically expected in superconductors due to the Doppler shift. The central question is the microscopic origin of the field-directional selectivity of the CDW response. To address this, we construct a real‑space tight‑binding model of NbSe₂, incorporating Ising-type spin-orbital coupling, 3 × 3 triple-Q CDW order, and superconductivity with a single s-wave pairing potential (see “Methods” for details). The supercurrent effect is depicted by considering Peierls substitution, $\mathbf{p}\to \mathbf{p}+e\mathbf{A}$, where $\mathbf{p}$ is the Fermi momentum of NbSe₂, and $\mathbf{A}$ is the magnetic vector potential and is assumed to be perpendicular to the magnetic field B according to the London gauge (Supplementary Fig. 13). Particularly, in superconductors, $e\mathbf{A}$ represents the momentum of Cooper pair (i.e., the supercurrent), which is able to modify the pristine Bogoliubov quasiparticle dispersion into $E_{\pm }\left(\mathbf{k}\right)=\sqrt{{\left({ϵ}_{\mathbf{k}}-\mu \right)}^2+{\mathrm{\Delta }}^2}\pm e{\mathbf{v}}_{\mathbf{k}}\cdot \mathbf{A}$. This framework allows us to track how supercurrents reshape low‑energy quasiparticle contours and induce symmetry breaking in a CDW intertwined superconductor.

Our calculated results in Fig. 4a, b demonstrate that when the field is applied along the Γ-M direction on the NbSe₂ surface, Doppler shift-induced modification of E-k dispersion emerges exclusively along the perpendicular Γ-K direction, and vice versa. When the Doppler shift satisfies $0<E_{\mathrm{D}}<\mathrm{\Delta }$, only half of the Bogoliubov quasiparticle branches are present on the constant-energy contours within the interval $E_{\mathrm{D}}<E<\mathrm{\Delta }$. In this scenario, the field-modified contour exhibits mirror symmetry, while the C_3v symmetry present at zero field is broken. Figure 4c, d displays the constant-energy cuts taken at E = 0.5Δ with indicated field directions. Beyond the confinement of spectral weight to one half of the Brillouin zone, two mutually perpendicular mirror planes M_x and M_y emerge, indicative of the resultant C_s point group symmetry. For $E\ge \mathrm{\Delta }$, the high density of states on the constant-energy contour renders Doppler-shift effect negligible, so the CDW orders retain C_3v symmetry and all six peaks exhibit equal intensity, as in the zero-field case (Supplementary Figs. 16 and 17).

Fig. 4. Numerical simulations of the supercurrent effect on CDW-mediated patterns.

a, b Calculated Bogoliubov quasiparticle dispersions with fields applied along Γ-M and Γ-K directions, respectively. The Doppler shift develops perpendicular to the magnetic field and parallel to the direction of screening supercurrent. Τhe blue and gray planes together represent constant-energy cuts at E = 0.5Δ, where the blue plane highlights regions of the constant-energy contour that become populated by Doppler-shifted quasiparticles, while the gray plane remains gapped. The black hexagon denotes the Brillouin zone, with high-symmetry points indicated. c, d Constant-energy contours at E = 0.5Δ extracted from (a, b). Under field-induced supercurrent, Bogoliubov quasiparticles persist only on one half of the contour, lowering the system’s symmetry down to C_s with a single mirror plane (e.g., M_x or M_y). e, f Simulated real-space CDW-modulated LDOS patterns based on the constant-energy contours in (c, d). The dominant CDW modulations are highlighted by red dashed lines. g, h Line cuts taken along the dashed-arrow directions in the corresponding FFTs, shown as insets and derived from (e, f). CDW peaks oriented parallel to the field direction exhibit pronounced enhancement, confirming the anisotropic CDW response to the supercurrent. All solid arrows indicate the in-plane field directions.

The anisotropic modification of the Bogoliubov quasiparticle dispersion gives rise to a symmetry transition in real-space electronic patterns (Fig. 4e, f). In contrast to the homogeneous LDOS map at zero field (Fig. 1c) showing CDW modulation with C_3v symmetry, our simulated real-space LDOS map under an in-plane field along Γ-M in Fig. 4e clearly displays a strip-shaped CDW pattern, consistent with C_s symmetry. Applying the field along Γ-K results in two overlapping strip-shaped modulations (Fig. 4f). The corresponding FFTs of the real-space maps (Fig. 4g, h) directly capture the supercurrent-induced modification of the CDW pattern superimposed on the Bogoliubov quasiparticles. Namely, from the FFT line cuts in Fig. 4g, h, $P\left({\mathbf{d}}_1\right)$ exceeds that of $P\left({\mathbf{d}}_2\right)$ when the field is applied along Γ-M, whereas the opposite ordering is observed for a field along Γ-K. Thus, a pattern with $P\left({\mathbf{d}}_1\right)>P\left({\mathbf{d}}_2\right)=P\left({\mathbf{d}}_3\right)\left[P\left({\mathbf{d}}_1\right)<P\left({\mathbf{d}}_2\right)=P\left({\mathbf{d}}_3\right)\right]$ is obtained when the magnetic field is parallel to Γ-M (Γ-K) direction. Our simulations in real and reciprocal spaces reproduce the experimental data, confirming that the manipulation of the CDW patterns is indeed driven by the supercurrent effect.

To elucidate the relationship between magnetic-field orientation and the selective enhancement of CDW peaks, we perform a momentum-space analysis. At zero field, the CDW peaks at ${\mathbf{d}}_1$, ${\mathbf{d}}_2$, and ${\mathbf{d}}_3$ are degenerate in intensity by virtue of the underlying C_3v symmetry. Under an applied magnetic field enforcing mirror symmetries M_x or M_y, the CDW peaks ${\mathbf{d}}_2$ and ${\mathbf{d}}_3$ are symmetry-locked (i.e., $P\left({\mathbf{d}}_2\right)=P\left({\mathbf{d}}_3\right)$), whereas ${\mathbf{d}}_1$ is unconstrained by these symmetries and departs from the original C_3v symmetry, enabling an independent amplitude (Supplementary Note 2). The corresponding CDW peak strengths, $P\left({\mathbf{d}}_1\right)$ and $P\left({\mathbf{d}}_2\right)$, are extracted through theoretical analysis (see Methods section for details):

$$P\left({\mathbf{d}}_i\right)\propto {\sum }_{\mathbf{k}}\delta {\left(E-E\left(\mathbf{k}\right)\right)\mathrm{\Delta }}_{\mathrm{CDW}}\left(\frac{1}{\mathrm{\Delta }E\left(\mathbf{k}+{\mathbf{d}}_i\right)}+\frac{1}{\mathrm{\Delta }E\left(\mathbf{k}-{\mathbf{d}}_i\right)}\right)$$ 1

where i = 1, 2, ${\mathrm{\Delta }}_{\mathrm{cdw}}$ is the CDW potential strength. ${\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)$ selects all $\mathbf{k}$ on the constant-energy contours, e.g., the red (blue) line in Fig. 4c (Fig. 4d). It’s worth noting that $P\left({\mathbf{d}}_i\right)$ is governed by the energy difference $\mathrm{\Delta }E\left(\mathbf{k}\pm {\mathbf{d}}_i\right)=E\left(\mathbf{k}\right)-E\left(\mathbf{k}\pm {\mathbf{d}}_i\right)$. Accounting for the Doppler-shift-induced spectral asymmetry of the constant-energy contours, we divide the Brillouin zone into K-dominated and M-dominated halves (Fig. 4c, d). Under a field along Γ-M, the Doppler shift increases $\mathrm{\Delta }E\left(\mathbf{k}\pm {\mathbf{d}}_2\right)$, which suppresses $P\left({\mathbf{d}}_2\right)$ while leaving $P\left({\mathbf{d}}_1\right)$ essentially unchanged (Fig. 4g). Conversely, with the field along Γ-K, $\mathrm{\Delta }E\left(\mathbf{k}{\mathit{\pm }\mathbf{d}}_1\right)$ is enhanced, so that $P\left({\mathbf{d}}_2\right)>P\left({\mathbf{d}}_1\right)$ (Fig. 4h). This contrast confirms that the CDW’s modulation of Bogoliubov quasiparticles at ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$, respectively, dominate the K- and M-dominated halves, underpinning the anisotropic CDW response to magnetic-field orientation.

Finally, it should be recognized that the applied magnetic fields in our experiment exceed H_c1 of NbSe₂, and vortices are therefore expected to be present in the sample. Although our STM measurements are performed in regions away from visible vortex cores at the surface, vortices are nevertheless present in the bulk. Under such conditions, where vortices are present and may overlap, the field distribution can become substantially more uniform than that expected from a Meissner-state screening profile, and the corresponding current distribution near the surface may therefore differ from that in a simple Meissner-screening picture. The pronounced symmetry-lowering observed in our data may therefore include additional effects beyond the most straightforward Doppler-shift scenario. One possible explanation is that vortices are repelled from the surface, for example, due to a strong surface-barrier effect, which would modify surface current distributions. Another possibility is that the Bogoliubov quasiparticles are intrinsically more sensitive to the in-plane magnetic field. Nevertheless, the Doppler shift is evidenced by the field evolution of the superconducting gap. The Doppler-shift framework therefore provides a natural and consistent explanation for the field-orientation-dependent redistribution of CDW peak intensities observed in our experiment.

In summary, our experiments and theoretical simulations uncover how supercurrent-induced Doppler shifts govern the interplay between superconductivity and CDW order in 2H-NbSe₂. We further demonstrate the capability to manipulate Bogoliubov quasiparticle excitations at the precursor CDW vectors, opening the possibility of in-plane magnetic field manipulation of the Cooper pair density waves in a broader class of CDW intertwined superconductors. These insights also provide a potential route toward controlling quantum phases through engineered supercurrents and Doppler effects in superconductors intertwined with other quantum orders.

Methods

STM measurement

The STM/STS measurements were performed in a scanning tunneling microscope operating under ultra-high vacuum and equipped with a dilution refrigerator and a vector magnetic field (H_x: 2 T, H_y: 2 T, H_z: 9 T). The 2H-NbSe₂ single crystals were cleaved in situ at room temperature and then transferred immediately into the STM head. All the measurements were carried out at a base temperature of 40 mK using tungsten tips treated with heating and silver decoration in situ. dI/dV signals were acquired by a lock-in amplifier with modulation of 100 µV at 991 Hz, unless otherwise noted.

Sample growth

2H-NbSe₂ single crystals were grown by the method of chemical vapor transport. High-purity Nb (powder, 99.99%) and Se (grain, 99.99%) were mixed in a molar ratio of 1:2 and sealed in an evacuated quartz tube with iodine as the transport agent. The quartz tube was subsequently placed in a dual-temperature zone tube furnace, with the raw material in the high-temperature zone and the crystal growth in the low-temperature zone. The furnace was initially heated to 573 K for 200 min and then maintained at this temperature for an additional 300 min. Following this, the high-temperature zone was further heated to 1065 K for 320 min, while the low-temperature zone was heated to 1025 K for 200 min. These respective temperatures were then sustained for a prolonged period of 15,000 min. Upon cooling to room temperature, the samples were successfully obtained on the low-temperature side of the tube.

Numerical simulation

The numerical simulations were performed using a real-space model electronic Hamiltonian $H_{\mathrm{e}}$ is

$$H_{\mathrm{e}}=H_{\mathrm{tb}}+H_{\mathrm{CDW}}$$ 2

which represent the tight-binding electronic structure and the CDW potential, respectively. Considering that STM primarily probes the surface of NbSe₂ and the diamagnetic current predominantly distributes on the material’s surface, we simplify the system simulation by constructing a two-dimensional tight-binding model Hamiltonian. According to first-principles calculations, the electronic states near the Γ-point and K-point are derived mainly from the Nb ${\mathrm{d}}_{z^2}$ and the ${\mathrm{d}}_{x^2-y^2}/{\mathrm{d}}_{xy}$ orbitals. As the CDW phenomena under consideration are not strongly influenced by valley physics, it is reasonable to concentrate on the Γ point in our analysis. Therefore, we consider only the ${\mathrm{d}}_{z^2}$ orbital on a triangular lattice. Accordingly, the tight-binding electronic Hamiltonian takes

$$H_{\mathrm{tb}}={\sum }_{{\mathbf{R}}_i,\sigma }\left({ϵ}_{\mathrm{d}}-\mu \right)c_{{\mathbf{R}}_i,\sigma }^{†}{c}_{{\mathbf{R}}_i,\sigma }+{\sum }_{<{\mathbf{R}}_i\sigma ,{\mathbf{R}}_j\sigma >}\left(t_1+\frac{1}{2i}\sigma \lambda v_{{\mathbf{R}}_i,{\mathbf{R}}_j}\right)c_{{\mathbf{R}}_i\sigma }^{†}{c}_{{\mathbf{R}}_j\sigma }+{\sum }_{\ll {\mathbf{R}}_i\sigma ,{\mathbf{R}}_j\sigma \gg }{t}_2{c}_{{\mathbf{R}}_i\sigma }^{†}{c}_{{\mathbf{R}}_j\sigma },$$ 3

where ${\mathbf{R}}_i=\left(x,y\right)$ is the site vector of the real space, ${ϵ}_d$ denotes the on-site energy of the orbital, $\mu$ is the chemical potential, $t_1$ and $t_2$ are spin-independent hopping amplitudes between the nearest and next-nearest neighboring sites respectively, $\lambda$ parameterizes the strength of the Ising spin-orbit coupling, $\sigma =\pm 1$ stands for spin-up and spin-down respectively, and $v_{{\mathbf{R}}_i,{\mathbf{R}}_j}$ $=\pm 1$ is a bond dependent factor. Compared to the first principles calculations³⁰, we estimate the parameters as:$t_1\approx 0.14\mathrm{eV},t_2\approx 0.11\mathrm{eV},\lambda \approx 0.03\mathrm{eV}.$ The lattice structure, as well as the corresponding band structure and Fermi surface, are presented in Supplementary Fig. 2.

To reproduce the experimentally observed 3 × 3 triple-Q CDW real-space pattern, the CDW-induced potential Hamiltonian takes $H_{\mathrm{CDW}}={\sum }_{{\mathbf{R}}_i\sigma }{\mathrm{\Delta }}_{\mathrm{CDW}}\left({\mathbf{R}}_i\right)c_{{\mathbf{R}}_i\mathrm{\sigma }}^{†}{c}_{{\mathbf{R}}_i\sigma }$, where

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\mathrm{CDW}}\left({\mathbf{R}}_i\right)={\sum }_{{\mathbf{d}}_1,{\mathbf{d}}_2,{\mathbf{d}}_3}2A_0\cos \left({\mathbf{d}}_i\cdot \mathbf{r}+{\varphi }_i\right)\hfill \\ \hfill =2A_0\cos \left(\frac{4\sqrt{3}\mathrm{\pi }}{9a}y-\frac{2\mathrm{\pi }}{3}\right)+2A_0\cos \left(\frac{2\mathrm{\pi }}{3a}x+\frac{2\sqrt{3}\mathrm{\pi }}{9a}y+\frac{2\mathrm{\pi }}{3}\right)+2A_0\cos \left(-\frac{2\mathrm{\pi }}{3a}x+\frac{2\sqrt{3}\mathrm{\pi }}{9a}y\right),\hfill \\ \hfill {\varphi }_1=-\frac{2\mathrm{\pi }}{3},{\varphi }_2=\frac{2\mathrm{\pi }}{3},{\varphi }_3=0.\hfill \end{array}$$ 4

When the system becomes superconducting, the corresponding Bogoliubov-de Gennes (BdG) Hamiltonian is

$$H_{\mathrm{BdG}}=\left(\begin{array}{cc}\hfill H_{\mathrm{e}}\hfill & \hfill H_{\mathrm{pairing}}\hfill \\ \hfill H_{\mathrm{pairing}}^{†}\hfill & \hfill -H_{\mathrm{e}}^{*}\hfill \end{array}\right),H_{pairing}={\mathrm{\Delta }}_{\mathrm{sc}}{\sum }_{{\mathbf{R}}_i}{c}_{{\mathbf{R}}_i\uparrow }^{†}{c}_{{\mathit{R}}_i\downarrow }^{†}$$ 5

where ${\mathrm{\Delta }}_{\mathrm{sc}}$ is the s-wave superconducting gap$.$

An applied magnetic field parallel to the x-y plane induces the diamagnetic current at the top and bottom surfaces due to the Meissner effect. For the slab geometry shown in Supplementary Fig. 13, the supercurrent J and the vector potential $\mathbf{A}$ are related by the London equation, $\begin{array}{c}\hfill \mathbf{J}=-\frac{n_{\mathrm{s}}{e}^2}{m^{*}}\hfill \end{array}\mathbf{A}$, within the London gauge ($\nabla \cdot \mathbf{A}=0$). Combined with Maxwell’s equations, the vector potential in a superconductor satisfies $\begin{array}{c}\hfill {\nabla }^2\mathbf{A}=\frac{\mathbf{A}}{{\lambda }_{\mathrm{L}}^2}\hfill \end{array}$, where ${\lambda }_{\mathrm{L}}$ is the London penetration length. When the magnetic field is applied along the Γ-M direction, based on the boundary conditions and choosing the Landau gauge $\begin{array}{c}\hfill {\partial }_z{A}_x\left(z=\frac{d}{2}\right)=B,{\partial }_z{A}_x\left(z=-\frac{d}{2}\right)=B\hfill \end{array}$, we can get the vector potential as $\mathbf{A}=\left(A_x,0,0\right)=\left(B{\lambda }_{\mathrm{L}}\frac{\sinh \left(\frac{z}{{\lambda }_{\mathrm{L}}}\right)}{\cosh \left(\frac{d}{2{\lambda }_{\mathrm{L}}}\right)},0,0\right)$. Based on the Peierls substitution, the presence of the vector potential modifies the hopping amplitude in $H_{tb}$ as

$$\begin{array}{c}\hfill {\tilde{t}}_{ij}\to t_{ij}{e}^{-i\frac{e}{\hslash }{\int }_{{\mathbf{R}}_i}^{{\mathbf{R}}_j}\mathbf{A}\mathit{\cdot }\mathbf{dl}}=t_{ij}{e}^{-i\frac{e}{\hslash }B{\lambda }_{\mathrm{L}}a\tanh \frac{d}{2{\lambda }_{\mathrm{L}}}\left(x_j-x_i\right)}\hfill \end{array}=t_{ij}{e}^{-i{\varphi }_{ij}}$$ 6

where ${\varphi }_{ij}=\frac{e}{\hslash }B{\lambda }_{\mathrm{L}}a\tanh \frac{d}{2{\lambda }_{\mathrm{L}}}\left(x_j-x_i\right)$ denotes the phase during the hopping process; therefore, in the tight-binding Hamiltonian $H_{\mathrm{tb}},$ the hopping terms are modified as

$${\sum }_{<{\mathbf{R}}_i\sigma ,{\mathbf{R}}_j\sigma >}\left(t_1+\frac{1}{2i}\sigma \lambda v_{{\mathbf{R}}_i,{\mathbf{R}}_j}\right)\begin{array}{c}\hfill c_{{\mathbf{R}}_i\sigma }^{†}{c}_{{\mathbf{R}}_j\sigma }\to {\sum }_{<{\mathbf{R}}_i\sigma ,{\mathbf{R}}_j\sigma >}\left(t_1+\frac{1}{2i}\sigma \lambda v_{{\mathbf{R}}_i,{\mathbf{R}}_j}\right)e^{-i{\varphi }_{ij}}{c}_{{\mathbf{R}}_i\sigma }^{†}{c}_{{\mathbf{R}}_j\sigma }\hfill \end{array}$$

$${\sum }_{\ll {\mathbf{R}}_i\sigma ,{\mathbf{R}}_j\sigma \gg }{t}_2\begin{array}{c}\hfill c_{{\mathbf{R}}_i\sigma }^{†}{c}_{{\mathbf{R}}_j\sigma }\to {\sum }_{\ll {\mathbf{R}}_i\sigma ,{\mathbf{R}}_j\sigma \gg }{t}_2{e}^{-i{\varphi }_{ij}}{c}_{{\mathbf{R}}_i\sigma }^{†}{c}_{{\mathbf{R}}_j\sigma }\hfill \end{array}$$ 7

The derivation follows the same procedure for a magnetic field applied along the Γ-K direction. To calculate the real-space LDOS for the large system sizes required, we employ the Kernel Polynomial Method³¹, an efficient linear-scaling algorithm. The real-space LDOS then takes

$${\rho }_E\left({\mathbf{R}}_i\right)={\sum }_n\delta \left(E-E_n\right)⟨{\mathbf{R}}_i\mid {\psi }_n⟩⟨{\psi }_n\mid {\mathbf{R}}_i⟩=\frac{1}{\mathrm{\pi }\sqrt{1-{\tilde{E}}^2}}\left(g_0+2{\sum }_{n=1}^N{g}_n⟨{\mathbf{R}}_i∣T_n\left(\tilde{H}\right)∣{\mathbf{R}}_i⟩T_n\left(\tilde{E}\right)\right).$$ 8

Here, $E_n$ represents the eigenenergy of the system, and ${\psi }_n$ denotes the corresponding eigenstate. $T_n$ are the Chebyshev polynomials of the first kind. $\tilde{H}$ and $\tilde{E}$ refer to the rescaled Hamiltonian and its corresponding eigen energies, respectively. The rescaling is given by $\tilde{H}=\frac{H-\frac{E_{\max }+E_{\min }}{2}}{\frac{E_{\max }-E_{\min }}{2}}$ which ensures that the energy spectrum lies within the interval $\left[-1,1\right]$, a necessary condition for performing the Chebyshev expansion. The kernel polynomial $g_n$ is used to dampen the Gibbs oscillations of the partial sum. In the following numerical calculations, the system consists of a 100 × 100 × 3 lattice. The kernel we choose is the Jackson kernel, and the number of Chebyshev polynomial moments is set to 2700. We use the random vector method to estimate the LDOS, with 2000 random vectors. The above numerical simulations are primarily carried out using the Kwant package³² in the Python programming language.

Symmetry analysis

To understand the observed CDW peaks, we perform a symmetry analysis by treating the CDW potential as a perturbation. Note that, even in the presence of the magnetic field, the two-dimensional Hamiltonian without CDW potential still preserves the translation symmetry, and its eigenstates are simply Bloch states denoted as $∣\mathbf{k}⟩.$ The CDW potential will break the translation symmetry and couples different $\mid \mathbf{k}⟩$ with $\mid \mathbf{k}\pm {\mathbf{d}}_1⟩$,$\mid \mathbf{k}\pm {\mathbf{d}}_2⟩$, and $\mid \mathbf{k}\pm {\mathbf{d}}_3⟩$ due to its specific form as shown in Eq. (4). According to first-order perturbation theory, the eigenstate with CDW becomes $∣\psi \left(\mathbf{k}\right)⟩=a_{0,\mathbf{k}}∣\mathbf{k}⟩+{\sum }_{l=1}^6{a}_{l,\mathbf{k}}\mid \mathbf{k}+{\mathbf{d}}_l⟩$ with $l\in \left\{\mathrm{1,\; 2,\; 3,\; 4,\; 5,\; 6}\right\}$ (The distribution of ${\mathbf{d}}_l$ can be seen in Supplementary Fig. 1, where they satisfy the relations ${\mathbf{d}}_4=-{\mathbf{d}}_1,{\mathbf{d}}_5=-{\mathbf{d}}_2,{\mathbf{d}}_6=-{\mathbf{d}}_3$) and $a_{0,\mathbf{k}}\gg \mid a_{l,\mathbf{k}}\mid$ the coefficients up to the first-order perturbation (Supplementary Note 1). Here, for the sake of convenience in the derivation, l is taken from 1 to 6, which differs from the definition in the main text where ${\mathbf{d}}_i$ is indexed with i = 1–3. The corresponding real-space LDOS can then be expressed as:

$${\rho }_E\left({\mathbf{R}}_i\right)={\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)⟨{\mathbf{R}}_i⟩\mid \psi \left(\mathbf{k}\right)⟩⟨\psi \left(\mathbf{k}\right)\mid {\mathbf{R}}_i⟩={\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)⟨{\mathbf{R}}_i\mid \left(a_{0,\mathbf{k}}\mid \mathbf{k}⟩+{\sum }_{l=1}^6{a}_{l,\mathbf{k}}\mid \mathbf{k}+{\mathbf{d}}_l⟩\right)\left(a_{0,\mathbf{k}}^{*}⟨\mathbf{k}\mid +{\sum }_{m=1}^6{a}_{m,\mathbf{k}}^{*}⟨\mathbf{k}+{\mathbf{d}}_m\mid \right)\mid {\mathbf{R}}_i⟩={\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)\frac{1}{N}\left(a_{0,\mathbf{k}}{e}^{i\mathbf{k}\cdot {\mathit{\cdot }\mathbf{R}}_i}+{\sum }_{l=1}^6{a}_{l,\mathbf{k}}{e}^{i\left(\mathbf{k}+{\mathbf{d}}_l\right)\mathit{\cdot }\cdot {\mathbf{R}}_i}\right)\left(a_{0,\mathbf{k}}^{*}{e}^{-i\mathbf{k}\cdot \mathit{\cdot }{\mathbf{R}}_i}+{\sum }_{m=1}^6{a}_{m,\mathbf{k}}^{*}{e}^{-i\left(\mathbf{k}+{\mathbf{d}}_m\right)\cdot {\mathit{\cdot }\mathbf{R}}_i}\right)\approx {\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)\frac{1}{N}\left\{a_{0,\mathbf{k}}^{\mathbf{2}}+a_{0,\mathbf{k}}\left[\left(a_{1,\mathbf{k}}+a_{4,\mathbf{k}}^{*}\right)e^{i{\mathbf{d}}_1\cdot {\mathit{\cdot }\mathbf{R}}_i}+\left(a_{2,\mathbf{k}}+a_{5,\mathbf{k}}^{*}\right)e^{i{\mathbf{d}}_2\cdot \mathit{\cdot }{\mathbf{R}}_i}+\left(a_{3,\mathbf{k}}+a_{6,\mathbf{k}}^{*}\right)e^{i{\mathbf{d}}_3\cdot \mathit{\cdot }{\mathbf{R}}_i}+c.c\right]\right\}.$$ 9

Here we can always take $a_{0,\mathbf{k}}$ to be real so that

$$a_{l,\mathbf{k}}=\frac{⟨\mathbf{k}+{\mathbf{d}}_l∣H_{\mathrm{CDW}}∣\mathbf{k}⟩}{E\left(\mathbf{k}\right)-E\left(\mathbf{k}+{\mathbf{d}}_l\right)}=\frac{{\mathrm{\Delta }}_{\mathrm{CDW}}{e}^{i{\varphi }_l}}{\mathrm{\Delta }E\left(\mathbf{k}+{\mathbf{d}}_l\right)},$$ 10

where ${\varphi }_4=-{\varphi }_1,{\varphi }_5=-{\varphi }_2,$ and ${\varphi }_6=-{\varphi }_3$. Notably, there are six terms in Eq. (9) which depend explicitly on the spatial position and arise from the coupling among different momentum states induced by the CDW potential.

To analyze the momentum-space structure of these patterns, we compute the Fourier transform of the LDOS with given Q as

$$P_E\left(\mathbf{Q}\right)={\sum }_{{\mathbf{R}}_i}{\rho }_E\left({\mathbf{R}}_i\right)e^{-i\mathbf{Q}\cdot \mathit{\cdot }{\mathbf{R}}_i}={\sum }_{{\mathbf{R}}_i}{e}^{-i\mathbf{Q}\cdot {\mathit{\cdot }\mathbf{R}}_i}{\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)\frac{1}{N}\left\{\mid a_{0,\mathbf{k}}{\mid }^2+\left[\left(a_{0,\mathbf{k}}{a}_{4,\mathbf{k}}^{*}+a_{1,\mathbf{k}}{a}_{0,\mathbf{k}}^{*}\right)e^{i{\mathbf{d}}_1\cdot \mathit{\cdot }{\mathbf{R}}_i}+\left(a_{0,\mathbf{k}}{a}_{5,\mathbf{k}}^{*}+a_{2,\mathbf{k}}{a}_{0,\mathbf{k}}^{*}\right)e^{i{\mathbf{d}}_2\cdot \mathit{\cdot }{\mathbf{R}}_{\mathit{i}}}+\left(a_{0,\mathbf{k}}{a}_{3,\mathbf{k}}^{*}+a_{6,\mathbf{k}}{a}_{0,\mathbf{k}}^{*}\right)e^{i{\mathbf{d}}_3\cdot {\mathit{\cdot }\mathbf{R}}_i}+c.c\right]\right\}={\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)\left\{a_{0,\mathbf{k}}^2{\delta }_{0,\mathbf{Q}}+a_{0,\mathbf{k}}\left[\left(a_{1,\mathbf{k}}+a_{4,\mathbf{k}}^{*}\right){\delta }_{{\mathbf{d}}_1,\mathbf{Q}}+\left(a_{2,\mathbf{k}}+a_{5,\mathbf{k}}^{*}\right){\delta }_{{\mathbf{d}}_2,\mathbf{Q}}+\left(a_{3,\mathbf{k}}+a_{6,\mathbf{k}}^{*}\right){\delta }_{{\mathbf{d}}_3,\mathbf{Q}}+c.c\right]\right\}.$$ 11

According to Eq. (10), the six position-dependent terms correspond to six peaks in momentum space located at ${\mathbf{d}}_l$ with the form

$$∣P_E\left({\mathbf{d}}_l\right)∣={\sum }_{\mathbf{k}}\delta \left(E-E\left(\mathbf{k}\right)\right)a_{0,\mathbf{k}}{\mathrm{\Delta }}_{\mathrm{CDW}}\left(\frac{1}{\mathrm{\Delta }E\left(\mathbf{k}+{\mathbf{d}}_l\right)}+\frac{1}{\mathrm{\Delta }E\left(\mathbf{k}-{\mathbf{d}}_l\right)}\right)$$ 12

Without applying the magnetic field, the system has C₃ rotational symmetry (an element of C_3v point group). Therefore, $∣P_E\left({\mathbf{d}}_l\right)∣$ for $l\in \left\{\mathrm{1,\; 2,\; 3,\; 4,\; 5,\; 6}\right\}$ possess equal values because ${\mathbf{d}}_l$ are transformed to each other under C₃ rotation (see details in Supplementary Note 1). When applying the magnetic field, the Meissner current breaks the C₃ rotational symmetry so that the $\mid P_E\left({\mathbf{d}}_l\right)\mid$ have not to take the same values (see details in Supplementary Note 2).

Author contributions

Z.Z., H.Z. and J.J. conceived the project; Z.Z. and D.L. conducted the STM measurement with the help of P. H., Q.Y., S.X., R.X., X.C., J.S., D.G., Y.L., S.W. and C.L.; W.C. and X.L. performed the numerical simulation and symmetry analysis; Y.Y. and Z.X. prepared the NbSe₂ samples; Z.Z., W.C., X.L., H.Z. and J.J. wrote the manuscript, with input from all authors. All authors discussed the results, interpretation, and conclusions.

Peer review

Peer review information

Nature Communications thanks Lun-Hui Hu, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

H.Z. discloses support for publication of this work from the National Natural Science Foundation of China [grant number 92365302], the Ministry of Science and Technology of China [grant number 2025YFE0201200], the Science and Technology Commission of Shanghai Municipality [grant number 24LZ1401000], and the Cultivation Project of Shanghai Research Center for Quantum Sciences [grant number LZPY2024-04]. J.J. discloses support for the research of this work from the National Natural Science Foundation of China [grant number 12488101], the Science and Technology Commission of Shanghai Municipality [grant numbers 2019SHZDZX01 and 20QA1405100], and the innovation program for Quantum Science and Technology [grant number 2021ZD0302500]. Z.Z. discloses support for the research of this work from the National Natural Science Foundation of China [grant number 12104292]. X.L. discloses support for the research of this work from the National Natural Science Foundation of China [grant number 92565305], the Science and Technology Commission of Shanghai Municipality [grant number 24LZ1400800], Cultivation Project of Shanghai Research Center for Quantum Sciences [grant number LZPY2024-05], and innovation program for Quantum Science and Technology [grant number 2021ZD0302700]. S.W. discloses support for the research of this work from the National Natural Science Foundation of China [grant numbers 22325203 and 92577203]. Y.L. discloses support for the research of this work from the National Natural Science Foundation of China [grant number 12474156]. W.C., D.L., P.H., Y.Y., Q.Y., S.X., R.X., X.C., J.S., D.G., C.L. and Z.X. declare no relevant funding.

Data availability

The main figure data generated in this study have been deposited in the Zenodo database at 10.5281/zenodo.20091865. All the raw data generated during this study and the datasets analyzed in this study are available from the corresponding authors upon request.

Code availability

The calculations were performed using the open-source packages Kwant³². The codes that support the findings of this study are available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-73778-7.