Introduction

Charge-density-wave (CDW) order, a periodic modulation of the charge carrier density, often coexists or competes with superconductivity, e.g., in copper-oxide superconductors1,2,3,4 and transition-metal-based kagome metals5,6,7,8,9,10. Strong electronic correlations originating in the electronic band structures featuring flat bands, Dirac cones and van Hove singularities are expected to be responsible for the reported phase competition. On the other hand, classic CDW materials such as the layered transition-metal dichalcogenides (TMD) continue to provide a rich playground for emergent physics including excitonic insulators11,12,13, dimensionality-dependent correlated electronic phases14,15 as well as intriguing magnetic properties16 with potential applications in spintronic devices17. CDW order is widespread in metallic TMDs18,19. In the seminal model by Peierls20 an electronic instability in the presence of finite electron–phonon coupling (EPC) stabilizes the CDW ground state. While this scenario applies to several known quasi-1D materials21,22,23,24, recent work for layered TMDs25,26,27,28 showed that the full momentum dependence of both the electronic band structure and the EPC have to be taken into account to explain CDW formation and the existence of closely related superconducting phases29,30,31,32,33,34.

A hallmark of CDW order is a periodic lattice distortion and the softening of a symmetry-related phonon to zero energy at the phase transition temperature TCDW. Thus, investigation of this soft phonon mode can yield important insights about the underlying mechanism governing the phase transition and has been indispensable to understanding the physics of many CDW materials24,25,27,28. However, a full softening is not always observed questioning the displacive nature of the CDW phase transition. For instance, quasi-1D materials NbSe335 and (TaSe4)2I36 as well as cuprates2,3,37 and kagome metals8 feature no or only little softening at TCDW.

2H-TaSe2 is a prototypical CDW compound featuring a large periodic lattice distortion38 and a momentum-dependent energy gap in the electronic band structure in its low-temperature state39,40. It is a layered material (see inset in Fig. 1a) for which CDW order with a transition temperature TCDW = 122.3 K was reported in the 1970s41. On cooling through TCDW, 2H-TaSe2 first enters a CDW phase with an incommensurate ordering wave vector qCDW = (0.323,0,0) (all wave vectors are given in reciprocal lattice units; see the “Methods” section), which gradually evolves on cooling and reaches the commensurate value qCDW = (1/3,0,0) at TCDW-C ≈ 90 K42. The superconducting transition temperature of 2H-TaSe2 is Tsc = 0.1 K. However, Tsc increases up to 8.2 K under pressure and a close connection between fluctuations of the pressure-suppressed CDW order and superconductivity is anticipated31. In 2H-TaSe2, the CDW soft phonon mode corresponds to the longitudinal acoustic (LA) phonon propagating along the [100] direction and has been investigated by inelastic neutron scattering (INS) experiments in the 1970s38,41. Surprisingly, the observed phonon softening was not complete, i.e., the energy of the LA mode at qCDW softened from above 7 meV at room temperature only to 4.5 meV at TCDW. However, a resolution-limited, static central peak at zero energy transfer developed already below 150 K, well above TCDW where phonon softening was reported to be strongest. Similar observations, i.e., the rise of a central elastic peak before the corresponding phonon fully softens, had been made earlier in ferroelectric SrTiO343,44,45 and more recently in quasi-1D CDW compounds22 as well as in cuprates2. The origin of the central peak is not fully understood45 though it was ascribed to ordered fluctuations in some CDW-related materials2,22. The report of such an elastic peak rising above the phase transition temperature in 2H-TaSe2 is one of the earliest documented, although the authors of the original INS study38 pointed out that the coarse momentum resolution in their experiments may have obscured a better view of the lattice dynamics. Hence, the lattice dynamical behavior in 2H-TaSe2 remains unclear and is highly relevant not only for its intertwined superconducting and charge-ordered phases but also for our understanding of other materials featuring CDW order such as quasi-1D (TaSe4)2I36, NbSe335, ZrTe322, and underdoped cuprates2,3.

Fig. 1: Probing of charge-density-wave order in 2H-TaSe2.
figure 1

a Schematic dispersion (black solid lines) of an acoustic phonon with a soft mode at qCDW ≈ (1/3,0,0) for energy-loss (positive energies) and energy-gain (negative energies). The red dot indicates the position of the corresponding superlattice peak in the ordered phase. Thick vertical (green) and horizontal (blue) arrows illustrate the scans done on the HERIX spectrometer to investigate the phonon softening and superlattice peak formation in 2H-TaSe2. Labels b–d refer to the following panels showing typical results for the corresponding scans in more detail. Insets in (a) show the crystal structure of 2H-TaSe2 (\(P{6}_{3}/{mmc}\), a = b = 3.44 Å, c = 12.7 Å, #194) and the layout of the Brillouin zone with high symmetry points labeled. b Energy scan at q ≠ qCDW. For a realistic picture we convoluted a damped harmonic oscillator function with the pseudo-voigt-like experimental resolution (ΔEFWHM = 1.5 meV). c Energy scan at q = qCDW. The signal from the superlattice peak is approximated by the resolution function whereas the damped phonon is represented by a damped harmonic oscillator function convoluted with the resolution function. The scattering contributions from the superlattice peak [thick (red) solid line] are easily distinguished from the phonon contribution [thick (blue) dashed line]. d Momentum scan at zero energy transfer, E = 0, across the CDW superlattice peak (red dot in (a)). Because of the finite energy resolution and the broad phonon lineshapes, it is not clear how strongly soft phonon mode and superlattice peak contribute to the scattering at zero energy transfer.

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Here, we employ inelastic x-ray scattering (IXS) with meV resolution to study the CDW soft phonon mode in 2H-TaSe2. The much better momentum resolution of IXS compared to INS enables the detailed study of the soft-mode properties as a function of wave vectors as well as temperature. Combined with the energy resolution of 1.5 meV, IXS enables an unambiguous separation between the static superlattice peak (resolution limited in energy) and in- or quasi-elastic (not resolution limited) scattering from the soft phonon mode. We observe a full phonon softening of the LA phonon mode at qCDW = (0.323,0,0) and that the static CDW superlattice peak intensity rises strongly only on cooling to temperatures below that of the full phonon softening. This is in contrast to previous results38 and rules out the central-peak scenario for 2H-TaSe2. Yet, the most surprising finding is that the phonon softening occurs at a temperature T* = 128.7 K, i.e., well above TCDW = 121.3 K. Analysis of high momentum resolution scans at E = 0 reveals that the phase at TCDW ≤ T ≤ T* is characterized by lattice fluctuations, observed as the overdamped soft phonon mode, coexisting with static but only medium-range CDW order. Furthermore, we performed ab initio lattice dynamical calculations to investigate the origin of CDW order as well as the role of the CDW soft phonon mode with regard to the increasing superconducting temperatures at high pressure. The results show that both electron–phonon coupling, as well as the Fermi surface topology, have to be taken into account to understand the origin of CDW order in 2H-TaSe2. It also seems to be an ideal candidate to investigate the interplay of a CDW quantum critical point and emergent superconductivity relevant to a number of other systems of interest, including cuprates and 122 Ni-based superconductors.

Results

The strategy of our IXS experiment is outlined in Fig. 1: The LA mode at q = (h,0,0) shows a pronounced softening close to the temperature of the phase transition at qCDW (black lines in Fig. 1a) where eventually the CDW superlattice peak (red dot at zero energy transfer) emerges. We performed energy scans at constant momentum transfer [vertical (green) arrows] and momentum scans at zero energy transfer [horizontal (blue) arrows]. Typical results for three different scenarios are sketched in Fig. 1b–d. At q ≠ qCDW, the LA mode can already have a low energy, e.g., Ephon ≈ 4 meV. Taking into account the energy resolution and damping due to electron–phonon coupling, the resulting scan shows broad phonons peaked at ±Ephon and with finite scattering intensity even at zero energy transfer (Fig. 1b). Constant momentum scans at q = qCDW and T ≈ TCDW typically show a resolution-limited signal corresponding to the rising static CDW superlattice peak (solid (red) line in Fig. 1c) easily distinguishable from the broad scattering from the damped soft phonon mode [dashed (blue) line]. Due to the small phonon scattering intensities, constant momentum scans employ a relatively broad momentum resolution, Δq = 0.09 Å−1, not suited to investigate the correlation length of the rising CDW superlattice peak. Therefore, we performed additional momentum scans at zero energy transfer across q = qCDW with high resolution, Δq = 0.018 Å−1. Generally, it is not clear which part of the scattering intensity at zero energy transfer originates from the static elastic peak and which part is due to the (nearly) completely damped LA phonon (Fig. 1d). Yet, the combined analysis of both types of scans allows the unambiguous determination of the temperature dependence of the elastic CDW superlattice peak concomitant with that of the soft phonon mode.

Figure 2 illustrates raw data along the dispersion of the LA mode (Fig. 2a–f) and highlights its strong temperature dependence at q = (0.325,0,0) [≈qCDW = (0.323,0,0)] (Fig. 2b and e). The temperature-dependent energy of the LA mode at qCDW (Fig. 3a) shows a clear soft-mode behavior. Surprisingly, the softening is, within our experimental resolution, complete already just below 130 K, well above the reported CDW transition temperature38 of about 122 K (vertical blue dashed line in Fig. 3a). Fitting a power law of the form |T−T*|δ (red solid line in Fig. 3a) to the temperature dependence of the soft mode for T ≥ 130 K yields δ = 0.32(2) and T* = 128.7(3) K (vertical red dashed line in Fig. 3a). Below T*, the energy of the soft mode stays close to zero. The observed value of δ is lower than the mean-field one, δMF = 0.5, observed e.g. for the CDW transition in the iso-structural 2H-NbSe225. Since mean-field theory neglects fluctuations of the order parameter, δ = 0.32(2) indicates the presence of strong order parameter fluctuations in 2H-TaSe2. We expect reduced fluctuations far away from the transition and, indeed, the soft-mode energies for T ≥ 175 K follow mean-field behavior, which means that the square of the phonon energies depends linearly on temperature (see Fig. 3b). A linear fit for T ≥ 175 K extrapolates zero energy for the soft mode at T ≈ 85 K, which is close to TC-IC ≈ 88 K41,42, the temperature at which 2H-TaSe2 acquires a commensurate CDW.

Fig. 2: Energy scans of the soft phonon mode in 2H-TaSe2.
figure 2

Energy scans taken at q || [100] at a–c T = 250 K and d–f 130 K. Dashed (red) lines are fits consisting of DHO functions convoluted with the experimental resolution (blue solid lines), estimated background (straight dotted line) and a resolution limited pseudo-Voigt function for the elastic line (black dash-dotted line). Error bars represent standard deviations. Error bars represent s.d.

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Fig. 3: Temperature and momentum dependence of the soft phonon mode in 2H-TaSe2.
figure 3

a Temperature dependence of the energies of the LA phonon modes at q = (0.325,0,0) (=qCDW, spheres) and q = (0.28,0,0) (open circles). The line is a power law fit to the data for q = qCDW and T ≥ 130 K of the form |T−T*|δ yielding T* = 128.7 K (vertical red dashed line) and δ = 0.32 ± 0.02 (data in gray-shaded box are shown in more detail in Fig. 4d). The vertical blue dashed line indicates TCDW = 121.3 K. b Square of the phonon energy at q = qCDW as function of temperature. The solid line is a linear fit corresponding to δ = 0.5 in (a) for T ≥ 175 K. TIC-C ≈ 88 K indicates the transition temperature into the commensurate CDW phase (see inset in Fig. 5c). c Dispersion and d damping ratio \(\varGamma /\omega\) of the LA mode along q = (h,0,0) for 250 K ≥ T ≥ 125 K. Lines are guides to the eye. Error bars represent s.d.

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Before we discuss the temperature range T ≈ T* in more detail, we present the dispersion and the damping of the LA soft mode in Fig. 3c, d, respectively. We find, within our experimental resolution, a V-like dispersion and a sharply peaked damping centered at qCDW at T = 130 K (black spheres in Fig. 3c, d) again in contrast to 2H-NbSe2, where the LA soft mode displays a much broader, U-shaped dispersion at the respective TCDW25,46. Phonon energies of the LA mode rise much faster going away from qCDW in 2H-TaSe2 than in 2H-NbSe2. A V-like dispersion (Fig. 3c) is expected when the EPC of the soft mode is dominated by a Fermi surface (FS) nesting. In that case, the strong increase of electronic decay channels at the nesting wave vector leads to sharply localized phonon anomalies. The dispersion of 2H-TaSe2 is sharper than in 2H-NbSe2, in which FS nesting was shown to be completely absent47, but not as sharp as in some quasi-1D compounds, such as KCP21 and ZrTe322. This is in line with results from angle-resolved photoemission spectroscopy (ARPES) revealing a moderately strong FS nesting in 2H-TaSe248. The range of wave vectors over which the LA mode shows significant renormalization on cooling to T* is 0.032(10) r.l.u. along the [100] direction (half-width of phonon anomaly, Fig. 3c) indicating a short coherence length of the CDW order parameter of \({\xi }_{0}=17\pm 4\) Å, in reasonable agreement with estimates from specific heat \({\xi }_{0}\approx 14\) Å49.

Figure 4 shows energy scans at qCDW taken for temperatures TCDW ≤ T ≤ T* along with an analysis of the temperature dependence of the observed static CDW superlattice peak. The results show that the CDW superlattice-peak intensity (dash-dotted red lines in Fig. 4a–c) increases in this temperature range by a factor of 10. Yet, the spectrum at T = 122.5 K still shows similar spectral weights of the CDW superlattice peak (dash-dotted red line) and the soft phonon mode (solid blue line in Fig. 4c). Cooling further by only ΔT = 1 K, the scattering at zero energy jumps by a factor of 20 (inset of Fig. 4c) and we will see further below that this jump marks the onset of long-range CDW order. However, the strong increase of the superlattice peak prohibits further analysis of in- or quasi-elastic scattering at lower temperatures with the given energy resolution, because the much weaker quasi-elastic scattering is hidden under the tail of the elastic peak. The temperature dependences of the soft mode’s energy, its intensity, and that of the CDW superlattice peak are summarized in Fig. 4d, e, respectively. The intensity of the soft phonon mode dominates the spectra except for close to and, of course, below TCDW. Nevertheless, our results clearly show that a small static CDW superlattice peak is already present at T ≈ 130 K. The increase of the CDW superlattice peak intensity accelerates sharply below T* (vertical red dashed line in Fig. 4e) before it jumps by a factor of 20 crossing TCDW (see also inset in Fig. 4c).

Fig. 4: Evolution of elastic and inelastic scattering at TCDW ≤ T ≤ 150 K.
figure 4

a–c Energy scans taken at Q = τ + q with τ = (3,0,1) and q = qCDW [=(0.325,0,0)] at various temperatures 122.5 K ≤ T ≤ 127.5 K, i.e, TCDW < T < T*. Thin solid (black) lines are fits consisting of DHO functions convoluted with the experimental resolution (blue solid lines), constant background, and a resolution-limited pseudo-Voigt function for the rising CDW superlattice peak (red dash-dotted line). The inset in c shows data taken at T = 121.5 K (≈TCDW). Here, no analysis of inelastic scattering is possible because of the increase of the CDW superlattice peak by more than a factor of 20. d Temperature-dependent energy of the CDW soft phonon mode (blue spheres). The red solid line represents the same power-law fit as shown in Fig. 3a. e Temperature-dependent integrated intensities of the CDW superlattice peak (red spheres, see also red dash-dotted lines in (a–c)) and the soft phonon mode (blue spheres, see also blue solid lines in (a–c)). Small letters, a–c, reference panels showing the corresponding IXS raw data. Vertical blue and red dashed lines (in d and e) denote TCDW = 121.3 K and T* = 128.7 K, respectively. Error bars represent s.d.

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Our results identify a precursor region at TCDW ≤ T ≤ T* characterized by the simultaneous presence of soft-mode lattice fluctuations and static CDW order. A critically damped phonon indicates that there is quasi-zero energy needed for small lattice distortions that are symmetry-related to the phonon oscillation pattern. On the other hand, the intensity of a superlattice peak is proportional to the square of the order parameter, i.e., the atomic displacement from the high-temperature equilibrium position. Thus, the presence of a weak CDW superlattice peak indicates small but finite displacements. A possible interpretation is that there exist small domains of static CDW order with a finite correlation length ξcorr, in a matrix of material with critical lattice fluctuations. On cooling, one would expect that ξcorr increases and finally diverges at the temperature at which the sample reaches long-range CDW order, i.e., at TCDW.

However, the momentum resolution employed for energy scans (Δq = 0.09 Å−1) is too coarse for a detailed analysis of the CDW correlation length ξcorr. Therefore, we performed momentum scans at constant energy transfer of E = 0 (indicated by the horizontal blue arrow in Fig. 1a and illustrated in Fig. 1d) with an improved momentum resolution of Δq = 0.018 Å−1 (see the “Methods” section and Supplemental Note 2). Raw data at selected temperatures are shown in Fig. 5a. Data at T = 122 K (red spheres in Fig. 5a) show the rise of scattering intensity for a broad momentum range q = (0.3–0.35,0,0) on cooling. Just 1 K below, at T = 121 K, we find a more than twenty-fold increase of the scattering intensity centered at qCDW = (0.323,0,0). On further cooling the intensity increases further and the peak position moves towards the commensurate wave vector qCDW,C = (1/3,0,0) illustrated by the data for T = 80 K (orange spheres in Fig. 5a). From a peak fit we obtain the detailed temperature dependences of the integrated intensity (Fig. 5b), the peak position qCDW (Fig. 5c) and its line width Γexp,FWHM (Fig. 5d). The integrated peak intensity follows a power law of the form |T−TCDW|2β for T ≤ 121 K with TCDW = 121.3(2) K and β = 0.21(1) (red line in the inset of Fig. 5b). On further cooling, the intensity continues to increase and qCDW shifts to the commensurate value qCDW,C = (1/3,0,0) at temperatures close to TC-IC ≈ 88 K42 (inset in Fig. 5c). Γexp,FWHM(T) shows that peaks are resolution limited for data taken at T ≤ 121 K. A power law fit of the form (T−TCDW)δ to Γexp,FWHM(T) for T ≥ 122 K corroborates the transition temperature to long-range CDW order TCDW = 121.3(2) K in good agreement with the literature38,41,42,49. A more detailed analysis which takes into account the experimental resolution and estimates the phononic background (see Fig. S2 and Supplemental Note 2) shows that the correlation length ξcorr of the static CDW domains in the precursor region increases along [100] on cooling from 100 to 200 Å just above TCDW (black spheres in Fig. 5e). Values along [010] are essentially the same while the correlation length along [001] is 4–5 times smaller. Thus, the precursor region in 2H-TaSe2 is characterized by medium-range-sized CDW domains, which only form a long-range ordered state at T ≤ TCDW.

Fig. 5: High-resolution momentum scans at E = 0 and temperatures T ≤ 140 K.
figure 5

a Momentum scans at zero energy transfer along the [100] direction, i.e., across qCDW for temperatures above and below the CDW transition. Data were taken in a high-momentum resolution setup (see Supplemental Note 2) to resolve the evolution of qCDW on cooling below TCDW. b–d Temperature-dependent b integrated intensity, c position qCDW and d line width Γexp,FWHM of peak fits to the momentum scans at zero energy transfer. Color-coded dots denote temperatures with data shown in (a). The inset in b shows the integrated intensity on a linear scale along with a power law fit (red line) of the form |T−TCDW|δ yielding TCDW = 121.3(2) K. The inset in c focuses on temperatures around TC-IC ≈ 88 K42. Lines are linear fits to the range below and above TC-IC. The solid red line in d is another power law fit to the corresponding data for T ≥ 122 K and yields TCDW = 121.3(2) K, indicated by the vertical blue dashed line. The dashed horizontal line in d denotes the experimental momentum resolution. e The temperature-dependent correlation length \({\xi }_{{{\rm {corr}}}}\) (spheres) of the static scattering was obtained by subtracting the data at T = 129 K (≈T*) as background and analyzing the linewidth of the static scattering rising below T* (see text, Supplemental Note 2 and Fig. S2). The solid line is a guide to the eye. The vertical blue and red dashed lines denote TCDW = 121.3 K and T* = 128.7 K, respectively. Green squares denote the temperature-dependent pseudo-gap deduced from ARPES measurements (see text, Supplemental Note 3 and Fig. S3). Dashed lines are linear fits to the data for T < TCDW and T > TCDW. Error bars represent s.d.

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The precursor region has not been observed previously by other techniques. INS experiments38,41 did not have sufficient momentum resolution while XRD42 was energy integrated and so could not distinguish between static CDW and soft-mode intensity contributions. We performed synchrotron-based ARPES to cross-check results on our own samples with previous reports40. In particular, we looked for quantities previously used as a measure of the onset of CDW order such as a pseudo-gap40 and the evolution across T*. Measurements were done with an incident photon energy of 80 eV (resolution <10 meV) over a temperature range 132 K ≥ T ≥ 113 K, i.e., cooling across T* and TCDW. Technical details and detailed results of our analysis are given in Supplemental Note 3. Here, we summarize, that—in agreement with previous reports—the electronic band structure exhibits a pseudo-gap (as defined in ref. 40, see Supplemental Note 3 and Fig. S3 for more details), which increases slowly on cooling in the upper part of the temperature range (open squares in Fig. 5e). The increase of the pseudo-gap on cooling accelerates by a factor 4 at lower temperatures and linear fits within the two temperature regions (dashed lines in Fig. 5e) intersect very close to TCDW = 121.3 K. However, no particular response is detectable on crossing T*. The presence of the pseudo-gap in 2H-TaSe2 has been reported up to nearly room temperature48 as well as in the high-temperature phase of 2H-NbSe250. The latter report found that the pseudo-gap phase in 2H-NbSe2 at T > TCDW is related to incoherent CDW fluctuations. Long-range CDW order only sets in once phase coherence is established at T < TCDW. On the other hand, electrons adjust quasi-instantaneously to lattice motions. It is argued in the case of 2H-NbSe250 that fluctuations always yield a pseudo-gap irrespective of the time-scale of the fluctuations. Therefore, we argue that electronic degrees of freedom do not distinguish between lattice fluctuations with a finite lifetime, i.e., the overdamped soft mode, and static correlations but follow the evolution of the energy-integrated signal as probed by standard x-ray diffraction42. Consequently, the pseudo-gap is rather insensitive to T* but responds to the large increase of scattering intensity on cooling below TCDW.

We performed calculations based on density-functional perturbation theory (DFPT, for details see the “Methods” section and Supplemental Note 4) to investigate the microscopic origin of CDW order and the impact of the CDW soft mode regarding the strongly increasing superconducting transition temperature at high pressure31. Results based on ambient-pressure lattice constants38 correctly assess the CDW structural instability as an anomaly of the longitudinal acoustic phonon propagating along the Γ−M direction (thick solid line in Fig. 6a). The soft mode features imaginary phonon frequencies where the leading instability around q ≈ (0.35,0,0) is close to the experimentally observed qCDW. We further calculated the electronic contribution to the phonon linewidth

$${\gamma }_{{{\rm {EPC}}},{{\rm {DFT}}}}^{{{{{{\bf{q}}}}}}}={\pi \omega }_{{{{{{\bf{q}}}}}}\lambda }\mathop{\sum}\limits_{{{{{{\bf{k}}}}}}}{\left|{g}_{{{{{{\bf{k}}}}}}+{{{{{\bf{q}}}}}},{{{{{\bf{k}}}}}}}^{{{{{{\bf{q}}}}}}\lambda }\right|}^{2}\delta \left({\epsilon }_{{{{{{\bf{k}}}}}}}-{\epsilon }_{F}\right)\delta \big({\epsilon }_{{{{{{\bf{k}}}}}}+{{{{{\bf{q}}}}}}}-{\epsilon }_{F}\big)$$
(1)

where \({g}_{{{{{{\bf{k}}}}}}+{{{{{\bf{q}}}}}},{{{{{\bf{k}}}}}}}^{{{{{{\bf{q}}}}}}\lambda }\) is the EPC matrix element for a phonon mode \(\lambda\) at a wave vector \({{{{{\bf{q}}}}}}\), \({\omega }_{{{{{{\bf{q}}}}}}\lambda }\) is the mode’s energy and \(\mathop{\sum}\limits_{{{{{{\bf{k}}}}}}}\delta ({\epsilon }_{{{{{{\bf{k}}}}}}}-{\epsilon }_{F})\delta ({\epsilon }_{{{{{{\bf{k}}}}}}+{{{{{\bf{q}}}}}}}-{\epsilon }_{F})\) gives the number of electronic states at the Fermi level connected by the phonon wave vector q, i.e., the electronic joint density-of-states (eJDOS).

Fig. 6: Ab-initio lattice dynamical calculations.
figure 6

a Calculated dispersion for longitudinal phonons along the Γ-M direction in 2H-TaSe2—performed in the hexagonal unit cell, \(P{6}_{3}/{mmc}\). Imaginary phonon energies denote an unstable structure and are shown as negative values. b Calculated electronic contribution to the phonon linewidth \({\gamma }_{{{\rm {EPC}}},{{\rm {DFT}}}}\) (FWHM) of the LA soft phonon mode (thick solid and dashed lines in (a)). Symbols represent the linewidth (FWHM) of the DHO function approximated to the IXS data (right-hand scale, error bars represent s.d.). c Calculated electronic joint density-of-states (eJDOS) for the band transition mediating EPC of the soft mode (see Figs. S4–S6 and Supplemental Note 4). Solid lines denote calculations for ambient-pressure lattice constants (a = b = 3.436 Å, c = 12.7 Å). Dashed lines represent calculations based on the high-pressure lattice constants (a = b = 3.242 Å, c = 11.09 Å, see text). d Calculations for the high-pressure lattice constants showing the phonon density-of-states (PDOS), Eliashberg function \({\alpha }^{2}F\left(\omega \right)\), and electron–phonon coupling constant \({\lambda }_{{\rm {{EPC}}}}\) as function of phonon energy.

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The momentum dependence of \({\gamma }_{{{\rm {EPC}}},{{\rm {DFT}}}}^{{{{{{\bf{q}}}}}}}\) of the CDW soft phonon mode is in good qualitative agreement with our experimental observations (solid line and symbols in Fig. 6b). The fact that the experimental values are significantly larger than the calculated ones has been already observed in several other IXS measurements in CDW systems25,51,52. While anharmonic effects close to the structural phase transition which are not included in our quasi-harmonic calculations might play a role, the finite momentum resolution in concert with steeply dispersing phonon modes will also lead to increased linewidths near the phase transition temperature (see Supplemental Note 2 for more details). For 2H-TaSe2, the eJDOS for electronic band transitions mediating strong coupling to the CDW soft phonon mode shows a clear maximum near qCDW (solid line in Fig. 6c, for more details see Figs. S4–S6 and Supplemental Note 4). The peak in the eJDOS is not very sharp in agreement with the reported moderately strong FS nesting in 2H-TaSe248. A more detailed analysis (see Figs. S4–S6 and Supplemental Note 4) explains that a previously reported19, and by us reproduced, nesting feature at \({{{{{{\bf{q}}}}}}}_{{{\rm {nest}}}}=(1/{{{{\mathrm{3,1}}}}}/{{{{\mathrm{3,0}}}}})\) is not related to any significant EPC and, thus, irrelevant for the LA soft phonon mode. Thus, 2H-TaSe2 is an excellent example where the CDW properties are linked to an interplay of momentum-selective EPC matrix elements and FS topology. Regarding the superconducting properties at ambient pressure, it is expected that the EPC of the low-energy phonons is mostly gapped out by the opening of the CDW energy gap in the electronic band structure39, and, thus, cannot mediate superconductivity at T \(\ll\) TCDW. On the other hand, the coupling of the optic branches with energies above 20 meV is relatively small. Estimates for a corresponding superconducting transition temperature are zero or very small (see Supplemental Note 4) consistent with the experimentally observed Tsc ≈ 0.1 K at ambient pressure31.

Interestingly, calculations based on lattice constants obtained for p = 23 GPa53 [Tc,max (p = 23 GPa) = 8.2 K]31 predict a stable phonon spectrum for the hexagonal structure of 2H-TaSe2 although the anomaly around qCDW is still clearly visible (thick dashed line in Fig. 6a). The stabilization of the LA mode at high pressure is accompanied by a strong reduction of its linewidth \({\gamma }_{{{\rm {EPC}}},{{\rm {DFT}}}}\) (dashed line in Fig. 6b) and eJDOS (dashed line in Fig. 6c). In particular the peak in the eJDOS near qCDW is flattened suggesting that the pressure-induced changes of the electronic band structure play an important role in the suppression of charge order at high pressure in 2H-TaSe2. Thus, we can analyze the EPC of 2H-TaSe2 when the CDW ground state is suppressed at high pressure (Fig. 6d). We find that more than 70% of the overall EPC constant λEPC = 1.03 originates from phonons below 15 meV (red solid line in Fig. 6d). The estimated value of Tc,DFPT = 10.6 K is in reasonable agreement with the experimentally observed Tc,exp = 8.2 K at p ≈ 23 GPa31. We conclude that the EPC of the low-energy phonons in 2H-TaSe2 including the CDW soft phonon mode can drive the large increase of the superconducting transition temperature at high pressure.

Discussion

The presented IXS investigation puts the CDW transition in 2H-TaSe2 in a new light in that there is a full phonon softening. Our analysis shows that the tail of the scattering above TCDW (Fig. 5b), observed in synchrotron XRD42 and previous neutron scattering work38,41, is dominated by scattering from the soft LA mode (see Fig. 4e). Thus, we can rule out the central-peak scenario for 2H-TaSe2 as it was observed, e.g., in SrTiO343,44,45, and other CDW compounds2,3,22,35,36.

A new striking feature of the CDW transition in 2H-TaSe2 is the precursor region for TCDW = 121.3 K ≤ T ≤ T* = 128.7 K defined by the presence of an overdamped phonon mode and the slow increase of the static elastic line. Both features appear at qCDW. Thus, the characteristic features of the precursor region are not competitive but rather precursor-like with regard to the CDW phase at T ≤ T*. The presence of an overdamped phonon over a finite temperature range bears similarities to a liquid-like scenario for the soft phonon mode54,55. For instance, Ag-based argyrodites, which are promising candidates for thermoelectric applications, feature a structural phase transition temperature Ts at which a phonon softens on heating towards Ts. This soft optical phonon mode becomes and stays over-damped at temperatures T ≥ Ts and is linked to the diffusion of Ag ions in these materials54,56. Normally, the transition from a full soft phonon mode to a long-range CDW order occurs at the same temperature, e.g., in 2H-NbSe225, 1T-VSe228 or the rare-earth tritellurides51,57. A possible reason for the anomalously wide temperature range, ΔT ≈ 7.4 K, in which this evolution happens might be strong fluctuations that prohibit a direct transition to a CDW long-range ordered phase. This transition only occurs at the lower end of ΔT, i.e., TCDWaccompanied by a large increase of the intensity of the, then resolution limited, superlattice peak. It is worth noting that the 20-fold increase of the elastic scattering within ΔT = 1 K from above to below TCDW (see Fig. 4c and inset) is much more abrupt than in similar measurements in 2H-NbSe2. In the latter compound, the rise is rather gradual and the intensity is not even doubled on cooling ΔT = 1 K at any temperature close to or below TCDW,NbSe225. Similarly, specific heat data for 2H-TaSe2 shows a sharp peak at TCDW with a width of <1 K49, whereas equivalent data for 2H-NbSe2 feature a width of more than 5 K58. It requires additional theoretical modeling which is able to include the effects of fluctuations to understand the details of the CDW transition in 2H-TaSe2 beyond the scope of the work reported here.

Our results regarding the elastic superlattice peak compared to the weak pseudo-gap in the electronic band structure demonstrate that structural order precedes full electronic order in 2H-TaSe2. Early specific heat measurements in 2H-TaSe249 as well as theoretical considerations59 already suggested a dominant lattice, i.e., phonon contribution to the phase transition entropy on entering the incommensurate CDW phase at TCDW. More recently, CDW order in quasi-1D ZrTe3 has been shown to feature a structural transition at the typically reported TCDW = 63 K while charge modulations only become long-range ordered below TLO = 56 K60. The authors argued that Friedel oscillations seeded by impurities produce the short-range charge modulations at TLO ≤ T ≤ TCDW, whereas only the long-range ordered modulations below TLO bear the true CDW nature. A similar scenario in 2H-TaSe2 might explain the weak electronic response to the transition at TCDW (see Fig. 5e, 8 and Supplemental Note 3 and refs. 39,40) and a strong one at the onset of commensurate order at TC-IC40,48. Yet, lattice dynamics in ZrTe322 are qualitatively different from those in 2H-TaSe2 and a more detailed comparison is necessary to understand these common characteristics. Overall, 2H-TaSe2 seems to be between mean-field-like 2H-NbSe2 and quasi-1D ZrTe3 in terms of the strength of FS nesting and the impact of fluctuations. Thus, the study of equally important electronic and lattice degrees of freedom in 2H-TaSe2 presents an interesting topic for future research.

Discussions on the interplay of CDW order and superconductivity have been revitalized by the discovery of CDW states in cuprates2,3,61. More recently, a sixfold increase of the superconducting transition temperature Tsc was reported for Ba1-xSrxNi2As2, a non-magnetic sister compound of the 122 iron-based superconductors, featuring a CDW state suppressed near the maximum Tsc value62. Inelastic x-ray scattering measurements of the parent compound, BaNi2As2, found a full phonon softening at the incommensurate CDW ordering wave vector well above TCDW61,63,64 similar to the precursor region observed by us in 2H-TaSe2. Interestingly, one report interpreted these findings for BaNi2As2 as a gradual slowing down on cooling of the already fully softened phonon mode before they coalesce into the static long-range CDW order64. This is not comparable with our observations for 2H-TaSe2 where we can always clearly distinguish between the resolution-limited elastic line and the soft phonon mode (see Fig. 4a–c). Furthermore, it is argued for BaNi2As264 that the intense lattice fluctuations above TCDW can be responsible for the strong increase of Tsc when static CDW order is suppressed. However, Ba1−xSrxNi2As2 features a 1st order triclinic phase transition just below the onset of CDW order65,66 and both structural transitions seem to be closely related and are suppressed near the same critical doping62. Thus, it will be difficult to establish the presence of a quantum critical point, let alone investigate CDW fluctuations and their impact on superconductivity. Here, we argue that 2H-TaSe2 under pressure is likely a good candidate to study the soft phonon mode near a CDW quantum critical point. No other structural transition is reported67 and Tsc increases by more than a factor of eighty31. Furthermore, the here-reported full phonon softening at ambient pressure necessitates a revision of discussions on the nature of the CDW transition at ambient and finite pressure31. Combined with our calculations for high-pressure lattice constants (Fig. 6) and estimated Tsc of 10.6 K, our observations suggest that the high-pressure CDW phase might remain 2nd order and can feature a CDW quantum critical point. This would be a rare opportunity since the relation in other CDW compounds often turned out to be less clear (e.g., 2H-NbSe227,68, 1T-TiSe212,29,69,70, ZrTe371,72,73). Intercalated 2H-PdxTaSe2 also seems to be an interesting candidate material74. Here, x-ray diffraction measurements are necessary to unambiguously determine the CDW phase diagram because the reports on 1T-TiSe2 showed that x-ray diffraction is more accurate than resistivity in determining the CDW phase diagram when CDW order is suppressed by pressure69,75 or intercalation29,30.

Methods

Inelastic x-ray scattering experiments were carried out at the 30-ID beamline, HERIX spectrometer76, at the Advanced Photon Source, Argonne National Laboratory. The sample was a single crystal grown at the University of Kiel and resistivity measurements on a sample from the same growth batch indicate a CDW transition of TCDW,resist. = (121.7 ± 0.5) K (see Fig. S1 and Supplemental Note 1). All x-ray measurements presented in this work (Figs. 2–5, S2) were done in the same experiment on the HERIX spectrometer. Phonon excitations measured in constant-momentum scans were approximated by damped harmonic oscillator (DHO) functions77 convoluted with a pseudo-Voigt resolution function [full-width at half-maximum (FWHM) = 1.5 meV, Lorentz ratio = 0.4]. The resolution function was further used to approximate resolution-limited elastic scattering at zero energy transfer. Measurements were done at scattering wave vectors Q = τ−q, where τ is a reciprocal lattice vector and q is the reduced wave vector in the first Brillouin zone. Wave vectors are expressed in reciprocal lattice units (r.l.u.) \((2\pi /a,2\pi /b,2\pi /c)\) with the lattice constants a = b = 3.44 Å and c = 12.7 Å of the high-temperature hexagonal unit cell (#194). All measurements were done in the Brillouin zone adjacent to \({{{{{\boldsymbol{\tau }}}}}}=(3,0,1)\). Throughout the text, results are presented in reduced wave vectors q = τ−Q. Measurements were done at constant energy or constant momentum transfer. For the former, we employed the best momentum resolution possible on the HERIX spectrometer (Δq = 0.018 Å−1) by decreasing the effective size of the backscattering analyzers to a circular diameter of 18 mm (FWHM) compared to 95 mm opening in energy scans at constant momentum transfer (Δq = 0.09 Å−1). For more details on the HERIX setup and data analysis see Supplemental Note 2.

Angle-resolved photoelectron spectroscopy measurements were performed at the Bloch endstation of the R1 synchrotron at the MAXIV institute in Lund, Sweden. More experimental details and results are given in Supplemental Note 3.

Ab-initio lattice dynamical calculations based on density-functional-perturbation-theory (DFPT) were performed in the framework of the mixed basis pseudopotential method42. The exchange-correlation functional was treated in the local-density approximation (LDA). Spin–orbit interaction was taken into account consistently. More details are given in Supplemental Note 4.