Precursor region with full phonon softening above the charge-density-wave phase transition in 2H-TaSe2

Research on charge-density-wave (CDW) ordered transition-metal dichalcogenides continues to unravel new states of quantum matter correlated to the intertwined lattice and electronic degrees of freedom. Here, we report an inelastic x-ray scattering investigation of the lattice dynamics of the canonical CDW compound 2H-TaSe2 complemented by angle-resolved photoemission spectroscopy and density functional perturbation theory. Our results rule out the formation of a central-peak without full phonon softening for the CDW transition in 2H-TaSe2 and provide evidence for a novel precursor region above the CDW transition temperature TCDW, which is characterized by an overdamped phonon mode and not detectable in our photoemission experiments. Thus, 2H-TaSe2 exhibits structural before electronic static order and emphasizes the important lattice contribution to CDW transitions. Our ab-initio calculations explain the interplay of electron-phonon coupling and Fermi surface topology triggering the CDW phase transition and predict that the CDW soft phonon mode promotes emergent superconductivity near the pressure-driven CDW quantum critical point.


Introduction
Layered transition-metal dichalcogenides (TMD) continue to provide a rich playground for emergent physics including excitonic insulators [1][2][3] , dimensionality-dependent correlated electronic phases 4,5 as well as intriguing magnetic properties 6 with potential applications in spintronic devices 7 .Charge-density-wave (CDW) order, a periodic modulation of the charge carrier density, is widespread in metallic TMDs 8,9 .In the seminal model by Peierls 10 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 materials [11][12][13][14] , recent work for layered TMDs 15- 18 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 phases [19][20][21][22][23][24] .
2H-TaSe2 is a prototypical CDW compound featuring a large periodic lattice distortion 25 and a momentumdependent energy gap in the electronic band structure in its low-temperature state 26,27 .It is a layered material [see inset in Fig. 1] for which CDW order with a transition temperature TCDW = 122.3K was reported in the 1970s 28 .On cooling through TCDW, 2H-TaSe2 first enters a CDW phase with an incommensurate ordering wave vector qCDW = (0.323,0,0), 1 which gradually evolves on cooling and reaches the commensurate value qCDW = (1/3,0,0) at TCDW-C ≈ 90 K. 29 As a continuous structural phase transition, a CDW transition is typically associated with a phonon mode softening to zero energy at qCDW on cooling towards TCDW.Thus, the investigation of this soft phonon mode can yield important insights about the underlying mechanism governing the phase transition and has been indispensable to understand the physics in many CDW materials 14,15,17,18 .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 1970s 25,28 .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.Similar observations had been made earlier in ferroelectric SrTiO3 [30][31][32] and more recently in quasi-1D CDW compounds 12 as well as in cuprates 33 .The origin of the central peak is not fully understood 32 though it is often ascribed to order fluctuations. 12,33Currently, 2H-TaSe2 is considered as one of the earliest materials for which a central-peak was observed, although the authors of the original INS study 25 pointed out that the coarse momentum resolution in their experiments may have obscured a better view of the lattice dynamics.Hence, the origin of the central-peak in 2H-TaSe2 remains unclear but is highly relevant, e.g., for our understanding of other materials featuring CDW order such as quasi-1D (TaSe4)2I, 34 NbSe3, 35 ZrTe3, 12 and underdoped cuprates 33,36 .
Here, we employ inelastic x-ray scattering (IXS) with meV resolution to study the CDW soft phonon mode in 2H-TaSe2.The much higher momentum resolution of IXS compared to INS enables the detailed study of the softmode properties as function of wave vector 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 results 25 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 = 121.3K. 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.

Methods
The IXS experiments were carried out at the 30-ID beamline, HERIX spectrometer 37 , at the Advanced Photon Source, Argonne National Laboratory.Phonon excitations measured in constant-momentum scans were approximated by damped harmonic oscillator (DHO) functions 38 convoluted with a pseudo-voigt resolution function (full-width at half-maximum (FWHM) = 1.5 meV).The resolution function was further used to approximate resolution limited elastic scattering at zero energy transfer.Measurements were done at scattering wave vectors  =  − , where τ is a reciprocal lattice vector and q is the reduced wave vector in the 1 st Brillouin zone.Wave vectors are expressed in reciprocal lattice units (r.l.u.) (2/, 2/, 2/) with the lattice constants  =  = 3.44 Å and  = 12.7 Å of the high-temperature hexagonal unit cell (#194).All measurements were done in the Brillouin zone adjacent to  = (3,0,1) .In the following, results are presented in reduced wave vectors  =  −  .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 the Supplemental Material (SM).Complementary ARPES 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 SM.

Results
The strategy of our IXS experiment is outlined in Figure 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. 1(a)] 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) arrow].Typical results for three different scenarios [vertical and horizontal arrows in Fig. 1(a)] are sketched in Figures 1(b)-(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.1(b)].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. 1(c)] 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 central peak and which part is due to the (nearly) completely damped LA phonon [Fig.1(d)].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.3(a)] shows a clear soft-mode behavior [solid red line in Fig. 3(a)] as a function of temperature.Surprisingly, the softening is complete already just below 130 K, well above the reported CDW transition temperature 25 of about 122 K [vertical blue dashed line in Fig. 3(a)].Data obtained at a slightly offset wave vector [open circles in Fig. 3(a)] allow measurements above and below the CDW transition and clearly reveal the hardening of the LA mode on cooling into the CDW phase.Fitting a power law of the form |T-T*| δ [red solid line in Fig. 3(a)] 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. 3(a)].Below T*, the energy of the soft mode stays close to zero.The observed value of δ is lower than the meanfield one, δMF = 0.5, observed e.g. for the CDW transition in the iso-structural 2H-NbSe2. 15Since 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 depend linearly on temperature [see Fig. 3(b)].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 K, 28,29 the temperature at which 2H-TaSe2 acquires a commensurate CDW.
Before we discuss the temperature range T ≈ T* in more detail, we present the dispersion and the damping of the LA soft mode in Figures 3(c) and (d), respectively.We find a V-shaped dispersion and a sharply peaked damping centered at qCDW at T = 130 K [black spheres in Fig. 3(c)] again in contrast to 2H-NbSe2,where the LA soft mode displays a much broader, U-shaped dispersion at the respective TCDW. 15,39Phonon energies of the LA mode rise much faster going away from qCDW in 2H-TaSe2 than in 2H-NbSe2.A classic V-shaped dispersion [Fig.3(c)] 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 absent 40 , but not as sharp as in some quasi-1D compounds, such as KCP 11 and ZrTe3. 12This is in line with results from angle-resolved photoemission spectroscopy (ARPES) revealing a moderately strong FS nesting in 2H-TaSe2 41 .The range of wave vectors over which the LA mode shows significant renormalization is about 0.08 -0.1 r.l.u.along the [100] direction, in reasonable agreement with the coherence length of 14 Å estimated from specific heat measurements. 42gure 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 [dash-dotted red lines in Figs.4(a)-(c)] increases in this temperature range by another factor of 10.Yet, the spectrum at T = 122.5K still shows similar spectral weights of the CDW superlattice peak (dash-dotted red line) and the soft phonon mode [solid blue line in Fig. 4(c)].Cooling further by only ΔT = 1 K, the scattering at zero energy jumps by a factor of 20 [inset of Fig. 4(c)] 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 quasielastic 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 Figures 4(d) and (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 present already at T ≈ 130 K.The increase of the CDW superlattice peak intensity accelerates sharply below T* [vertical red dashed line in Fig. 4(e), note the break of the vertical scale] before it jumps by another factor of 20 crossing TCDW [see inset in Fig.

4(c)].
Our results identify a precursor phase 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 which are symmetrically 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 longrange 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. 1(a) and illustrated in Fig. 1(d)] with an improved momentum resolution of Δq = 0.018 Å -1 (see methods).Raw data at selected temperatures are shown in Figure 5(a).Data at T = 122 K [red spheres in Fig. 5(a)] show the rise of scattering intensity for a broad momentum range q = (0.3 -0.35,0,0) on cooling.Just one degree 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. 5(a)].From a peak fit we obtain the detailed temperature dependences of the integrated intensity [Fig.5(b)], the peak position qCDW [Fig.5(c)] and its line width Γexp,FWHM [Fig.5(d)].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 inset of Fig. 5(b)].At T < TCDW, 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 K 29 [inset in Fig. 5(c)].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 literature. 25,28,29,42A more detailed analysis which takes into account the experimental resolution and estimates the phononic background (see Fig. S4 and SM) shows that the correlation length ξcorr of the static CDW domains in the precursor phase increases along the [100] on cooling from 100 Å to 200 Å just above TCDW.Values along [010] are essentially the same while the correlation length along [001] is 4-5 times smaller.Thus, the precursor phase in 2H-TaSe2 is characterized by medium-range-sized CDW domains, which only form a long-range CDW ordered state at T ≤ TCDW.
The precursor phase has not been observed previously by other techniques.INS experiments 25,28 did not have sufficient momentum resolution while XRD 29 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 reports 27 .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 SM.Here, we summarize, that -in agreement with previous reports -the electronic band structure exhibits a pseudo-gap (as defined in Ref. 27 , see SM and Fig. 7 for more details), which increases slowly on cooling in the upper part of the temperature range [open squares in Fig. 4(e)].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. 4(e)] intersect very close to TCDW = 121.3K.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 temperature 41 as well as in the high-temperature phase of 2H-NbSe2. 43The 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 quasiinstantaneously to lattice motions.It is argued for the case of 2H-NbSe2 43 that fluctuations always yield a pseudogap 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 life time, i.e., the overdamped soft mode, and static correlations but follow the evolution of the energy-integrated signal as probed by standard x-ray diffraction 29 .Consequently, the pseudo-gap is rather insensitive to T* but strongly responds to the large increase of scattering intensity on cooling below TCDW.

Discussion
The presented IXS investigation puts the CDW transition in 2H-TaSe2 in a new light in that there is a full phonon softening.As already suggested in the original publications 25,28 , the coarse momentum resolution of INS in concert with the V-shaped soft mode dispersion prevented the observation of the full softening.Moreover, our analysis shows that the tail of the scattering at T ≥ TCDW [Fig.4(b)], also observed in synchrotron XRD 29 is dominated by scattering from the soft LA mode for T ≥ T* [see Fig. 4(e)].The central peak at T > T* is a factor of twenty smaller than the soft mode intensity [see Fig. 2(a)].4][35][36] One can ask the question whether other central-peak compounds might show similar surprises as 2H-TaSe2.In general, we believe that applying IXS to study the structural instabilities in materials such as SrTiO3, KMnF3 or Nb3Sn, which have been investigated at a time when only INS was available, will provide new insights because of the good momentum resolution as well as the ability to investigate very small samples having less defects/vacancies.Reports of central-peaks in ZrTe3 and other quasi-1D CDW materials are already based on IXS.Quasi-1D compounds show inherently stronger fluctuations and this could be the reason that a central peak is present while it is not in 2H-TaSe2.
2H-TaSe2 seems to be one of the few examples among layered TMDs where FS nesting defines the periodicity of the charge modulation.For comparison, FS nesting is completely absent in iso-structural 2H-NbSe2 (TCDW,NbSe2 = 33 K) 40 and its impact in 1T-VSe2 (TCDW,VSe2 = 110 K) 44 was recently questioned by a study combining IXS and anharmonic ab-initio calculations 18 emphasizing the 3D character of the CDW transition in 1T-VSe2.Another difference is that the phonon softening in both 2H-NbSe2 15 and 1T-VSe2 18 feature critical exponents of δ ≈ 0.5, i.e., the mean-field value.In a detailed study on 2H-NbSe2, some of us did not see any indication of a central peak above TCDW. 15δ = 0.32 in 2H-TaSe2 [see Fig. 3(d)] indicates the presence of fluctuations.Both, the decisive role of FS nesting and the presence of fluctuations are in line with previous studies in 2H-TaSe2 using ARPES 41 and high-momentum resolution x-ray diffraction 29 , which assigned temperature dependent nesting features to competing CDW fluctuations reflected in the gradual evolution of qCDW on cooling from an incommensurate to a commensurate value for T ≤ TCDW [see Fig. S1(b)].Similarly, our analysis of the phonon softening indicates deviations from mean-field behavior for T ≤ 150 K and estimates the full phonon softening in absence of fluctuations close to TC-IC = 88 K [see inset in Fig. 2(c)].
However, the most striking feature of the CDW transition in 2H-TaSe2 is the precursor phase for TCDW = 121.3K ≤ T ≤ T* = 128.7 K characterized by a fully damped soft phonon mode and a gradually rising static CDW superlattice peak (see Fig. 3).Though the central-peak scenario does not apply, 2H-TaSe2 acquires (at least partially) a distorted lattice without concomitant electronic signatures evidenced by previous 26,27 and our own ARPES data (see SM). Furthermore, structural probes including our own data reveal the strongest phase-transition-like signatures on cooling below TCDW (see Fig. 4) while changes in the electronic band structure are much more pronounced at the incommensurate-to-commensurate transition at lower temperatures 27,41 .A dominant lattice, i.e., phonon contribution to the phase transition entropy on entering the incommensurate CDW phase at TCDW was also suggested based on specific heat measurements 42 and theoretical considerations. 45Thus, structural order accompanied by a (weak) pseudogap (see SM and 27 ) precedes full electronic order (featuring a well-defined gap on the FS) in 2H-TaSe2 in the presence of competing CDW fluctuations.
Recently, CDW order in quasi-1D ZrTe3 has been shown to feature a structural transition at the typically reported TCDW = 63 K while electronic long-range order only sets in at TLO = 56 K 46 .The soft phonon mode shows a temperature dependence best described by a power law with δ ≈ 1/8 assigned to fluctuations 12 in the presence of FS nesting 13 .Yet, lattice dynamics in ZrTe3 are qualitatively different to those in 2H-TaSe2 in several aspects: the softening is complete only at TCDW, a central elastic peak rises already above TCDW 12 and CDW order remains always incommensurate 46 .
2H-TaSe2 seems to be between mean-field-like 2H-NbSe2 and quasi-1D ZrTe3 in terms of strength of FS nesting, critical exponent δ of the phonon softening and impact of fluctuations.The CDW transition in 2H-NbSe2 is a show case for momentum-dependent EPC [15][16][17] and it has been argued that the latter must be considered for a quantitative understanding of the CDW transition also in ZrTe3 as well. 47As 2H-TaSe2 seems to be between these two examples, the study of equally important electronic and lattice degrees of freedom presents an interesting topic for future research.Finally, the full phonon softening and medium-range CDW order above TCDW, which, to our knowledge, is unique to 2H-TaSe2 among CDW compounds, is likely a manifestation of intense orderparameter fluctuations, which are already duped responsible for the strongly temperature dependent CDW properties at TCDW,IC ≤ T ≤ TCDW.Indeed, the analysis of the phonon softening shows deviations from mean field behavior and, thus, indicates the pronounced impact of fluctuations in the temperature range 90 K ≤ T ≤ 150 K [see inset in Fig. 2(c)].We expect that momentum dependent EPC and the FS topology have to be taken into account for a full understanding of CDW order in 2H-TaSe2 which is beyond the scope of the present work.
We note that IXS spectra generally contain a small, resolution-limited peak at zero energy transfer due to crystal imperfections.In our sample of 2H-TaSe2 this defect scattering peak is tiny with amplitudes < 1 count/mon×10 5 observed at temperatures far above TCDW [Figs.2(a)-(c)], confirming its high quality.Below T < 150 K , the zero energy transfer intensity at qCDW starts to slowly rise due to the onset of the static CDW superlattice peak.For T = 130 K, the data exhibit a resolution-limited peak at E = 0 [dash-dotted (black) line in Fig. 2(e)] with an amplitude that is ten times the value of the elastic line due to crystal imperfection observed at temperatures T > 150K.Yet, this intensity is still small compared to that of the soft phonon mode [solid (blue) line in Fig. 2(e)].The static superlattice peak becomes similarly strong as the overdamped phonon in scattering intensity only below T* = 128.7 K.
IXS is not able to detect phonon intensities when there is strong elastic scattering, e.g., at a reciprocal lattice point or at qCDW (for T < TCDW).Therefore, the CDW soft phonon mode in 2H-TaSe2 cannot be investigated anymore for T ≤ TCDW due to the rise of the superlattice peak at qCDW.However, neighboring wave vectors are not affected by the superlattice peak.The phonon softening is detectable at q = (0.28,0,0)where the LA phonon energy softens by about 2 meV between T = 250 K and 130 K [Fig.3(a)].Cooling below TCDW, we observe a hardening and narrowing of the phonon mode in the CDW phase.The temperature dependence of phonon energy levels off around TC-IC ≈ 90 K in reasonable agreement with results from Raman spectroscopy of the CDW soft phonon for T < TCDW. 51n the following and Figure 6, we explain in detail the analysis of the momentum scans from which we deduce the correlation length of the CDW domains for TCDW < T < T* [see Fig. 5(e)].Momentum scans at zero energy transfer at q = (0.3 -0.37,0,0) were performed with the best momentum resolution possible on the HERIX spectrometer, Δq = 0.018 Å -1 .Here, we decreased the effective size of the backscattering analyzers by closing a circular slit to a diameter of 18 mm (FWHM) compared to 95 mm opening in regular inelastic scans.From a fit we can determine the line width Γexp,FWHM(T) [see Fig. 5(d)].However, we cannot simply take the analysis of Γexp,FWHM(T) at face value to analyze the CDW correlation length because our energy scans demonstrate the significant two-component nature of the scattering at zero energy transfer for T > TCDW [see Fig. 4(a)-(c)].On the other hand, the energy scans [see Fig. 4] also show that the scattering from the soft phonon mode (1) dominates for T ≥ T* and ( 2) is essentially constant for T ≤ T*.We conclude that scattering observed in momentum scans at zero energy transfer is mostly due to the soft phonon mode for T ≥ T*.Therefore, data taken at T = 129 K [blue circles in Fig. 6(a)] represent an estimate of the phononic background in our momentum scans and a fit [solid blue line in Fig. 6(a)] was subtracted from data taken at lower temperatures.The resulting scans indicate the evolution of the static CDW superlattice peak only [Fig.6(b)].Thereby, we can investigate the correlation length ξcorr of the static CDW superlattice peak at T ≤ T* by approximating the resulting phonon-corrected momentum scans with a Voigt function.We fixed the Gaussian width of the Voigt function to the experimental resolution [see Fig. 5(d)].The temperature dependences of the Lorentzian linewidth ΓLor,HWHM(T) and the corresponding   = /(2 × Γ , ) are shown as black and red spheres in Figure 5(e), respectively.ξcorr increases below T* to about 200 Å just above TCDW.Thus, the precursor phase in 2H-TaSe2 is characterized by medium-range-sized CDW domains, which only form a long-range CDW ordered state at T ≤ TCDW.

Appendix B: Angle-resolved photoemission spectroscopy
Angle-resolved photoemission spectroscopy (ARPES) is one of the most powerful experimental techniques to study the electronic band structure of solids.In CDW materials, ARPES has been indispensable to investigate the gap in the electronic excitation spectrum upon entering the ordered phase.Previous ARPES measurements in 2H-TaSe2 reported some anomalous behaviour: The band gap observed in energy distribution curves (EDCs) opens only below the onset of commensurate CDW order at TC-IC ≈ 90 K. 26 Subsequent studies 27,52 corroborated this observation but reported the opening of a pseudo-gap already on cooling below TCDW.The pseudo-gap opens in the bands of the K barrel and its size can be determined from the shift of the leading edge of the EDC.More details are given below in the analysis of our own data set.
In our experiment, we wanted to check the temperature evolution of the FS in 2H-TaSe2 in a narrow temperature range focusing on TCDW and T*.ARPES measurements at temperatures 132 K ≥ T ≥ 113 K were performed at the Bloch endstation of the R1 synchrotron at the MAX IV laboratory in Lund, Sweden, using linearly polarized light with 80 eV photon energy with a total resolution of < 10 meV in energy and < 0.2° in angle.The spot size was about 15 µm × 10 µm.The sample was cleaved using tape at 10 -8 mbar and measurements were performed at < 10 -10 mbar.Photoelectrons were recorded using a ScientaOmicron DA30-L hemispherical analyser.Sample orientation and cooling were achieved with a six axis "Carving" manipulator from SPECS GmbH cooled by a closed-loop liquid helium cryostat.Sample temperature was determined from previous calibration measurements.After each change in temperature setpoint, the sample temperature was allowed to stabilise for 20 minutes before the next measurement was performed.The sample used for ARPES measurements was a different single crystal from the same growth batch as that used for IXS measurements.
In agreement with previous studies 26,27 , we observe hole-like circular FS sections centered on the Γ � and K � points and electron-like "dogbones" around the M � point [Fig.7(a)].Here, Γ � , K � , and M � denote the positions of highsymmetry points projected to the basal plane, i.e. kz = 0.
EDCs were taken from ARPES spectra centred around M along the K � -M � direction [see blue dots #1 and #2 in Fig. 7(a)] as determined from Fermi surface maps taken at every temperature step.Each EDC is integrated over a 1° range.Each EDC was approximated with a single Gaussian peak modified by a Fermi-Dirac distribution with the Fermi level (EF) kept constant across all temperatures in order to determine the K � barrel peak positionand and the position of the leading edge for K � barrel and M � dogbone [k corresponding to blue dot #1 in Fig. 7(a)].We observe good agreement with the data down to at least -0.1 eV binding energy for all spectra.The pseudo-gap size is defined as the difference of the energies at half-height in EDCs taken on the K � barrel [blue dot #1 in Fig. 7(a)] and the M � dogbone [blue dot #2 in Fig. 7(a)].Examples for T = 132 K and 113 K [Figs.7(b) and (c)] are normalised for ease of comparison and the horizontal arrows denote the position at which the pseudo-gap size was defined.The obtained temperature dependences are summarized in Fig. 7(d).The shown values of the K � barrel peak position (red dots, right-hand scale) and pseudogap size (black open squares, left hand scale) represent the average of the values obtained for the two pairs of K � and M � bands at positive and negative momentum relative to the M � point visible in Fig. 7(a).The individual values show qualitatively the same behaviour across the studied temperature range as their average shown here.For both the K � band peak position and the pseudo-gap size, we observe different slopes in their temperature dependence at low and high temperatures in our data set.Linear fits in each region [solid/dashed lines in Fig. 7(d)] cross close to TCDW = 121.3K, i.e., the transition temperature deduced from x-ray momentum scans at zero energy transfer [see Fig. 4].The observed kinks in the temperature dependence of the pseudogap and K � -band peak at TCDW agree also with previous results 27 .Finally, our analysis of the ARPES measurements reveals no particular electronic changes at T*.

Figure 2
Figure 2 illustrates the dispersion of LA mode [Figs.2(a)-(c) and 2(d)-(f)] and highlights its strong temperature dependence at q = (0.325,0,0) [≈ qCDW = (0.323,0,0)] [Figs.2(b)(e)].The energy of the LA mode [Fig.3(a)] shows a clear soft-mode behavior [solid red line in Fig.3(a)] as a function of temperature.Surprisingly, the softening is complete already just below 130 K, well above the reported CDW transition temperature25 of about 122 K [vertical blue dashed line in Fig.3(a)].Data obtained at a slightly offset wave vector [open circles in Fig.3(a)] allow measurements above and below the CDW transition and clearly reveal the hardening of the LA mode on cooling into the CDW phase.Fitting a power law of the form |T-T*| δ [red solid line in Fig.3(a)] 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.3(a)].Below T*, the energy of the soft mode stays close to zero.The observed value of δ is lower than the meanfield one, δMF = 0.5, observed e.g. for the CDW transition in the iso-structural 2H-NbSe2.15Since 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 depend linearly on temperature [see Fig.3(b)].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 K,28,29 the temperature at which 2H-TaSe2 acquires a commensurate CDW.

52FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 . 7 KFIG. 5 .
FIG. 1. Charge-density-wave order in 2H-TaSe2 (a) Schematic dispersion (black solid lines) of an acoustic phonon with a soft mode at qCDW ≈ (1/3,0,0) for energyloss (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),(c) and (d) refer to following panels showing typical results for the corresponding scans in more detail.Insets in (a) show the crystal structure of 2H-TaSe2 (6 3 /,  =  = 3.44 Å,  = 12.7 Å, #194) and the layout of the Brillouin zone with high symmetry points labelled.(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.

Fig. 6 .
Fig. 6.(a) High resolution momentum scans at zero energy transfer for T = 123 K (black spheres) and 125 K (green open squares) and T =129 K (≈ T*, blue circles).The solid blue line denotes a fit to the data at T = 129 K and was subtracted from the data taken at lower temperatures as phonon dominated background (see text).(b) Background subtracted high resolution momentum scans at zero energy transfer for T = 123 K (black spheres) and 125 K (green open squares).Solid lines are approximated Voigt functions where the Gaussian widths were fixed to the experimental resolution (FWHM indicated by the horizontal bars).(c) Temperaturedependent line width of the Lorentzian part of the Voigt function, ΓLor,HWHM (black spheres, left-hand scale).The corresponding correlation length   = /(2 × Γ , ) is shown in red (right-hand scale).Solid lines are guides to the eye.The vertical blue and red dashed lines denote TCDW = 121.3K and T* = 128.7 K, respectively.

FIG. 7 .
FIG. 7. (a) Fermi surface map measured at T = 126 K. White characters denote high-symmetry points of the Brillouin zone.Blue points 1 and 2 indicate the momentum position of the EDCs shown in red and black, respectively, in panels (b) and (c).(b)(c) EDCs at (b) T = 132 K and (c) 113 K obtained at the FS pockets around the  � point [red solid line, blue point #1 in (a)] and the  � point [black solid line, blue point #2 in (a)].The horizontal and vertical arrows indicate the deduced sizes of the pseudo-gap and  � -band peak position, respectively.(d) Temperature dependence of the pseudo-gap (open squares) and  � -band peak position (dots).Lines are linear fits to the data for T < TCDW and T > TCDW.The vertical blue dashed line denotes TCDW = 121.3K deduced from elastic x-ray scattering.