## Introduction

Imposing a new periodicity on a crystal leads to the reorganization of the electronic bands of the parent compound through their back-folding on the new Brillouin zone. New periodicities may be engineered in designer materials, for instance in artificial heterostructures with Moiré minigaps, or emerge due to a structural or electronic phase transition. The charge density wave state is an electronic order where the charge density develops a spatial modulation concomitantly to a periodic distortion of the atomic lattice and the opening of a gap in the quasi-particle spectrum. By reducing the electronic band energy, this gap compensates for the elastic and Coulomb energy costs associated with the formation of the CDW. It also lowers the degeneracy of the electronic states at the crossings of the folded bands. These are the points in the band structure of the parent compound that are connected by the wavevector of the new periodicity. Although a gap should open at all the crossings of the folded bands, previous studies only focused on the primary CDW gap around the Fermi-level, which leads to the highest energy gain of the reconstructed system. The existence of secondary gaps and associated charge modulations (CMs) remains largely unexplored.

In many cases, only a tiny fraction of the electrons are involved in the CDW formation. Therefore, the CDW gap manifests only as a slight reduction of the density of states (DOS)—which can depend on momentum—rather than a full depletion of the DOS. This makes it challenging to measure the CDW gap using spectroscopic probes such as angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling spectroscopy. This is particularly true for 2H-NbSe21,2,3,4,5 (hereinafter simply NbSe2). However, the effect of the redistributed electrons can be readily detected in topographic STM images, even for minute changes brought upon by the opening of the CDW gap as demonstrated in the following.

NbSe2 is an iconic material of correlated electron physics. It hosts a nearly commensurate charge density wave below TCDW = 33.5 K and a superconducting (SC) order below TSC = 7.2 K6,7,8,9,10. NbSe2 is a layered material with a three-fold symmetric crystal structure around the direction perpendicular to the layers (Fig. 1a). Each unit cell is composed of two slabs of Se–Nb–Se trilayers, where the Se lattices are 60° rotated, while the Nb atoms are aligned on top of each other in a trigonal prismatic coordination with the Se atoms.

The Fermi surface (FS) of NbSe2 is mainly determined by the bonding and antibonding combinations of the Nb-4d orbitals11,12 leading to double-walled barrel-shaped pockets around the K and Γ points of the hexagonal Brillouin-zone2,3,13,14 (Fig. 1b). The charge-ordered state consists of three CDWs, which form along the three equivalent ΓM directions with wavevectors $$(1-\delta )\frac{2}{3}| {{\Gamma }}M|$$, where δ ≈ 0.02 and depends on temperature10. In real space, this yields a locally commensurate 3a0 × 3a0 superstructure delimited by discommensurations15,16, where a0 is the atomic periodicity.

The 3a0 × 3a0 reconstruction is readily accessible to topographic STM imaging. Its bias-dependent contrast has been the focus of previous studies, with particular emphasis on the contrast inversion expected in a classic Peierls scenario between images acquired above and below the CDW gap17,18, and on the role of defects in stabilizing the CDW5. Sacks et al.18 calculate the bias dependence of the CDW phase in a perturbative approach, considering a single band normal state description of NbSe2. They find that the phase-shift of the CDW component of the local DOS can be very different from the 180° expected in a one-dimensional (1D) case (Supplementary Note VII) when changing the imaging bias across the Fermi level (EF). However, their model does not reproduce the full bias dependence of the CDW amplitude and phase that we find.

## Results and discussion

### Bias dependent STM topography, CDW phase and amplitude

In Fig. 2a–c, we present a selection from numerous topographic STM images of the same region on a cleaved NbSe2 surface at different sample biases (Vb) between −0.5 V and 0.5 V. They show a triangular atomic lattice with a superimposed 3a0 × 3a0 CDW modulation (see also Supplementary Fig. 1a), consistent with previous STM studies of unstrained bulk NbSe24,5,16,17,19,20,21,22. A defect-free region with a well resolved CDW outlined in red is magnified in Fig. 2d, f, h for each Vb. In order to identify the origin of the bias dependence of the topographic contrast in these images, we separate the atomic lattice and CDW contributions using Fourier filtering (Supplementary Note I). This analysis shows that the bias-dependent STM contrast is due to the changing CDW signal (Fig. 2e, g, i) since the corresponding atomic lattice contrast remains unchanged (see Supplementary Fig. 1).

The observed CDW pattern can be modelled as the sum of three plane waves as described in ref 16. While each plane wave has its own phase φi(r), which depends on a selected reference point, the dephasing parameter Θ(r) = φ1(r) + φ2(r) + φ3(r) mod 360 is uniquely defined for each particular CDW pattern, independent of any reference point. Θ(r) represents the internal CDW structure, quantifying the local relative position of the wavefronts of the three CDWs. In Fig. 2j–l, we show Θ(r) corresponding to the STM images in Fig. 2a–c, respectively. They were obtained by fitting the CDW contrast following the method described in ref. 16.

Each bias voltage is characterized by a dominant dephasing parameter (Fig. 2j–l), except in the vicinity of defects discussed later. This visual assessment is confirmed by the peaked histograms of Θ(r) (Supplementary Fig. 2). Fitting a Gaussian to these histograms allows to extract a well-defined dephasing parameter Θ0(Vb) for each imaging bias (Supplementary Fig. 3). For a quantitative analysis of the bias dependence of Θ0, we note that a given local CDW structure is represented by any arbitrary combination of φi(r) summing up to the same dephasing parameter, in particular the one where all three phases are equal. Moreover, the threefold symmetry of the system implies there is no privileged plane wave among the three used to describe the CDW. These observations allow us to map the problem to a one-dimensional (1D) description with an apparent CDW phase φ0(Vb) = Θ0(Vb)/3 (Supplementary Note III), and model φ0(Vb) to understand the bias dependent CDW pattern.

Plotting φ0(Vb) in Fig. 3a reveals a striking non-monotonic bias dependence, with an inflexion point around −0.15 V and a minimum slightly above the Fermi-level (EF = 0 V). This dependence is robust as long as φ0(Vb) is extracted from topographic STM images away from defects (Supplementary Fig. 4). Close to defects, the dephasing parameter Θ0(Vb) is different and tends to depend much less on imaging bias (Supplementary Fig. 5). This is consistent with earlier findings that defects (and impurities) can act as strong pinning centres23,24 locking the local phase of the CDW or driving the formation of CDW domains25,26.

The CDW amplitude can be extracted in a similar way to the phase, by fitting the histogram of the local amplitudes ai(r) of each plane wave measured over the entire field of view with a Gaussian, and extracting the peak value ai(Vb). The bias dependence and magnitude of ai(Vb) is nearly the same for all three CDWs (Supplementary Fig. 6b). For the analysis, we consider the average of these three amplitudes at each bias A0(Vb) = (a1(Vb) + a2(Vb) + a3(Vb))/3 plotted in Fig. 3b.

### Modelling and calculations

To understand the bias dependence of the CDW amplitude and phase in Fig. 3, we simulate topographic STM traces using a 1D model system (Supplementary Note VIII). In the simplest configuration corresponding to the Peierls reconstruction, we consider the contribution to the tunneling current of a single CM and its associated gap centred on EF (Fig. 4a–c). In this case, traces at the same polarity are always in phase, whereas traces at opposite polarities always show contrast inversion (or a 180° phase shift in the present harmonic model). The latter, often considered as an identifying hallmark of the CDW state27, clearly does not reproduce the data in Fig. 3a.

A single CM can only produce two sets of STM traces differing by contrast inversion in the vicinity of the gap. To generate a more complex bias dependence of the phase, we consider the possibility of a second CM whose associated gap opens in another band and away from EF (Fig. 4d–f). If these two harmonic CMs are in phase (Fig. 4d), the resulting STM traces are either in-phase or 180° out of phase (Fig. 4e), unable to reproduce the data in Fig. 3a. To generate more structures in the bias dependence of the phase, we need to introduce a phase shift between the two CMs (Figs. 4g–i). This leads to a phase that is no longer bi-modal, limited to two values differing by 180° as in Fig. 4c, f. It takes many different values (Fig. 4i), where the precise bias dependence is defined by the magnitude of the two gaps, their position relative to EF and by the relative phase shift between the two CMs. The simulated STM topographic traces in Figs. 4b, e and h also reveal a pronounced bias-dependent imaging amplitude with distinct line-shapes in the three model cases discussed above (Supplementary Fig. 7).

The broad parameter space of our 1D model makes it challenging to run a self-converging fit to the data. Visually optimizing the size and position of the two gaps in Fig. 4i, we find a range of parameters (Supplementary Note X) simultaneously reproducing the experimental bias dependent CDW phase and amplitude data remarkably well (Fig. 3). As for the relative phase between the two CMs, it is chosen to minimize the Coulomb interaction of the CMs and to conform with the strong commensuration energy that locally locks them to the lattice. Reducing the Coulomb energy is obtained by introducing a phase shift between the two CMs, which can only be ±120° (Fig. 4g) to satisfy the lock-in criterion with the lattice given the 3a0 periodicity of the CDW.

In the following, we turn to theoretically modelling multiple CDW gaps on different bands in NbSe2. We deploy self-consistent calculations to include the CDW gap within the random phase approximation on the two-dimensional two-band tight-binding fit to the NbSe2 band structure constrained by ARPES3. The corresponding FS shown in Fig. 1b consists of inner (red) and outer (blue) bands originating from symmetric and antisymmetric combinations of the Nb d$${}_{3{z}^{2}-{r}^{2}}$$ orbitals. The model (see “Methods”) was previously shown to accurately reproduce the full range of experimental measurements on the charge-ordered state12,28. The resulting DOS for the gapped and ungapped cases in each band are shown in Fig. 5a. To emphasize the DOS reduction accompanying the CDW phase transition, we plot the difference between the gapped and ungapped DOS for each band in Fig. 5b.

Our theoretical modelling shows a clear gap on the inner band at EF, consistent with the gap measured by ARPES around the K-point3. Interestingly, Fig. 5b reveals further DOS reductions, for example near 100 meV on the inner band and −50 meV on the outer band. These features are indicative of CDW gaps opening away from EF in addition to the (primary) gap at EF, supporting the simple model we propose to understand the bias dependence of the CDW appearance in STM images of NbSe2. According to Fig. 5b, there could even be more than two gaps. Consequently, we have included up to n = 8 gaps to our 1D model. However, the agreement with the data is similar for n = 2 and n = 3 (Supplementary Note XI), and we see no improvements adding more gaps.

In summary, the remarkable match between the bias dependence of the CDW contrast in STM topography and the simple 1D model proposed here provides compelling evidence that the CDW in NbSe2 is composed of at least two out-of-phase CMs on the inner and outer bands. While a 180° phase shift between these two CMs would minimize the Coulomb energy, the complex bias dependence of the CDW amplitude and phase observed by STM can only be reproduced when considering also the commensuration energy. This highlights the importance of the coupling of charge order to the lattice, which manifests in the formation of discommensurations15,16 and ultimately enables the observation of the multiband CMs uncovered here. The present study further highlights the power of topographic imaging to gain unique insight into detailed features of the CDW too faint to be detected accurately by tunneling spectroscopy. The formation of multiple modulations in response to new periodicities of a primary transition directly observed here, is extremely general and should in principle be present in all charge (and spin) density wave materials, and suggests new directions to explore in the physics of spatially modulated electronic orders.

## Methods

### Crystal growth and STM measurements

Single crystals of NbSe2 were grown via iodine-assisted chemical vapour transport and cleaved in-situ at room temperature. STM experiments were done in UHV (base pressure below 2  10−10 mbar) using tips mechanically cut from a PtIr wire and conditioned in-situ on a clean Ag(111) single crystal. The bias voltage was applied to the sample. STM images were recorded in constant current mode with at least 64 pixel/nm resolution. Details of the CDW amplitude and phase fitting procedure can be found in ref 16.

### Self-consistent calculations

We carried out a 22-orbital Slater–Koster tight-binding fit29 to the NbSe2 band structure, constrained by ARPES measurements2,3 and local density approximation numerical calculations11. This provided not only the band structure, but the orbital composition of the bands. We found that the two bands crossing the Fermi level are predominantly composed of the niobium $${d}_{3{z}^{2}-{r}^{2}}$$ orbitals (>60% across the Brillouin zone). A third, small, pancake-shaped pocket centred on Γ derives primarily from the selenium p-orbitals and is therefore not expected to mix significantly with the other bands, in-keeping with experimental observations that it plays no role in the charge ordering. We then re-fit the two bands of interest using only the two relevant orbitals; the fit was indistinguishable from the phenomenological fit to ARPES data provided in ref. 3.

The Coulomb interaction can be neglected in NbSe2, as the large DOSs at the Fermi level leads to strong screening. This remains true down to the lowest temperatures (above the SC transition at 7.2 K) since the CDW gap only opens on small regions of the FS. The relevant interaction is the electron-phonon coupling, for which we constructed an analytic expression following Ref. 30. It has long been suggested that the CDW in NbSe2 originates not from FS nesting, but from the dependence of the electron–phonon coupling on the momentum transfer in the phonon-mediated electron–electron scattering11. Our calculation includes the dependence of the coupling on the ingoing and outgoing electron momenta, as well as the orbital composition of the electronic bands scattered between. Only by taking all of these factors into account were we able to find a consistent explanation of the full range of experimental observations, including ARPES2,3, scanning tunneling spectroscopy/microscopy4, and inelastic X-ray scattering31. Our model has only one free parameter, the overall magnitude of the electron–phonon coupling, which we fixed using TCDW = 33.5 K.

We modelled the effect of the CDW on the electronic band structure using the random phase approximation. We employed the Nambu–Gor’kov method to work within the gapped phase; this method consists of promoting the electronic Green’s function to a 9 × 9 matrix, representing the tripling of the real-space unit cell in both lattice directions induced by the CDW formation. The CDW gap appears in off-diagonal elements, and diagonalisation then results in a gapped electronic band structure. We solved for the CDW gap self-consistently at high-symmetry points in the Brillouin zone, and used the results to constrain a tight-binding fit for the gap structure. We assumed the gap to be independent of energy. The CDW gap serves as an order parameter, and so our model naturally accounts for long-range CDW correlations. Further details of the method are given in refs. 12,28.

We found that a CDW gap opens at the Fermi level on the outer pocket centred on the K-point, along the MK line, in agreement with ARPES2,3. However, since the order parameter is non-zero at all points in the Brillouin zone, and at all energies, gaps also open wherever band crossings are introduced. This is the origin of the multiband CDW, evidenced by the suppression of DOS at energies below EF seen in Fig. 5. We calculated the DOS at different energies, with and without the CDW gap, by summing over the Brillouin zone the spectral function found from the electronic Green’s function.