Enhanced superconductivity near a pressure-induced quantum critical point of strongly coupled charge density wave order in 2H-Pd0.05TaSe2

It is widely documented that superconductivity (SC) emerges near the quantum critical point (QCP) of various symmetry-broken phases^1,2,3,4. Defined as a continuous phase transition occurring at absolute zero temperature (T = 0), the QCP and its corresponding fluctuations are currently believed to promote superconductivity in various quantum materials. For example, QCPs of antiferromagnetic orders have been found near the superconducting phase in various cuprates¹ and heavy fermion superconductors^2,3. Similarly, it is well known that fluctuations arising from nematic QCPs are correlated to the superconductivity of iron pnictides and chalcogenides⁴. A charge density wave state, which forms periodic spatial modulations of charge carrier densities, has also been found alongside superconductivity in many systems so far, ranging from cuprates¹ to the recently investigated topological kagome systems⁵. Therefore, similar to the other neighboring orders, one can naturally question whether low-energy fluctuation of CDW order might also be closely associated with the formation of the Cooper pair.

However, contrary to the other orders, the exact role of CDW fluctuation on the emergence of SC is far from complete understanding. While competing behavior between CDW and SC has been well established since the early 1970s⁶, the mechanism for stabilizing superconductivity or the origin of a dome shape in the superconducting properties is still poorly understood^7,8. One of the major bottlenecks is the scarcity of the material family that exhibits a QCP of CDW order. While materials that exhibit CDW are not rare, most CDW systems have been found to exhibit a 1^st order quenching of the CDW order by external tuning parameters, resulting in merely an extended region of superconductivity after the suppression of CDW order without showing an optimization of SC^9,10,11. Therefore, it is highly desirable to search for systems in which the CDW order exhibits a continuous phase transition at zero temperature and the optimization of superconducting order parameters.

One promising approach of inducing such a QCP of the CDW order (CDW-QCP) is to tune a temperature dependent 2^nd order CDW transition to 0 K, which has been successful in provoking CDW QCPs with chemical doping/substitution as in e.g., Lu(Pt_1−xPd_x)₂In¹² and (Ca_xSr_1−x)₃Rh₄Sn₁₃¹³. In this regard, 2H-TaSe₂, a van der Waals material with a 2^nd order CDW transition at a CDW transition temperature T_CDW = 122 K¹⁴ and SC with a superconducting transition temperature T_c = 0.14 K¹⁵, can be another candidate for observing the QCP by applying an external tuning parameter. However, the CDW-QCP has not been induced by intercalation or doping of other chemical element yet as T_CDW remains robust; for instance, T_CDW is reduced by only ~16 K with Pd intercalation between the TaSe₂ layers¹⁶, and by ~ 42 K by S doping in the Se site¹⁷ before the CDW order is smeared by disorder effects or destabilized in a first order manner, respectively.

Closely related to this robustness of T_CDW, recent studies have shown that the CDW order in 2H-TaSe₂ exhibits various physical properties that can’t be described by a weak-coupling Peierls-type CDW instability^{18,19,20,21,22,23,24}. For example, a large gap of 2Δ = ~6.4 k_BT_CDW = ~66 meV has been found at ~20 K below T_CDW by the angle resolved photoemission spectroscopy (ARPES), and the gap feature with 2Δ = ~20 meV persists even at 290 K, far above T_CDW¹⁸. This behavior suggests the presence of short-ranged CDW or local lattice distortions even far above T_CDW, which have been independently found by X-ray diffraction¹⁹ and nuclear magnetic resonance²⁰ studies. Consistent with these observations, an inelastic X-ray scattering study has found evidence of the Kohn anomaly, namely, substantial softening of an acoustic phonon mode near the CDW q-vector, even at 300 K^14,21. This characteristic Kohn anomaly has resulted in the persistent, strong intensity of the two-phonon Raman scattering spectra even at room temperature^22,23. All these observations consistently support the strongly coupled CDW state in 2H-TaSe₂^{18,19,20,21,22,23,24,25}, indicating that q-dependent electron-phonon coupling linked to the formation of periodic local lattice distortions are the main driving force of the CDW formation, rather than the conventional Fermi surface nesting picture²⁶.

In this regard, application of hydrostatic pressure (P) is expected to effectively tune the crystal lattice, which may in turn suppress the local lattice distortions and the corresponding CDW energy scales continuously. Moreover, it should be emphasized that pressure can control the system without introducing substantial disorders to the lattice, contrary to other chemical methods e.g. doping or intercalations. To better understand the interplay between strong coupling CDW and SC, we therefore employ electrical transport and Raman scattering measurements at high-pressure conditions to uncover clear evidence of pressure-induced CDW quantum critical point in a Pd 5% intercalated 2H-Pd_0.05TaSe₂ single crystal.

Single crystal growth and X-ray diffraction measurements

Single crystals of 2H-Pd_0.05TaSe₂ were grown by the chemical vapor transport method using SeCl₄ as a transport agent as described in our previous report¹⁶. It has been confirmed that the Pd intercalation does not alter the 2H structure of the pristine compound, as shown in the X-Ray diffraction (XRD) pattern of the 2H-Pd_0.05TaSe₂ in Supplementary Figure 1. For the XRD measurements, XRD θ-2θ scans of grinded powders of single crystalline 2H-Pd_0.05TaSe₂ were performed using a high-resolution x-ray diffractometer (Empyrean^TM, PANalytical).

In-plane resistivity and Hall effect measurements

In-plane resistivity and Hall effect measurements were performed in the PPMS^TM (Quantum Design) using a standard van der Pauw configuration. For the ambient pressure measurements, the electrical contacts were attached by silver paint (Dupont 4929 N). The high-pressure measurements were performed inside a homemade NiCrAl diamond anvil cell using a pair of diamonds with a culet size of 300 μm. Silicon oil was used as a pressure transmitting medium. For the high-pressure measurements, the electrical contacts were attached by using silver epoxy (Circuitworks, CW2500). It is noted that the silver epoxy is known for providing reliable electrical contacts under high pressure conditions and has been thus widely used in the high pressure measurements involving liquid media^27,28. The applied pressure was estimated from the fluorescence line of ruby particles inserted inside the gasket near the sample.

Raman spectroscopy measurements

For the Raman measurements, two pieces were cut out from a large piece of sample #3 (S3) and were used for acquiring the low temperature and room temperature spectra independently. The samples were mechanically cleaved to obtain fresh surfaces before the measurements. The Raman spectra at ambient pressure were measured by a commercial Raman spectrometer (Nanobase, XperRam200) equipped with a Nd:Yag laser with a 532 nm wavelength. For high-pressure Raman measurements performed at room temperature, a diamond anvil cell was employed using 300 μm culet diamonds. Fine-grounded NaCl was used as a pressure-transmitting medium and the pressure was determined from the shift of the ruby fluorescence line. The Raman scattering spectra were obtained by using a 532 nm laser beam. For high-pressure Raman measurements performed at low temperatures, Neon gas was used as a pressure-transmitting medium and the Raman scattering spectra were obtained by using a 488 nm laser beam. Pressure was determined from the shift of the ruby fluorescence line.

Figure 1a shows the crystal structure of 2H-Pd_0.05TaSe₂, which consists of two hexagonal 1H-TaSe₂ layers stacked along c-axis with in-plane orientation of one layer rotated by an angle of 180° to another layer. In each 1H-TaSe₂ layer, one Ta ion is located in the center of a trigonal prism created by six adjacent Se ions to form in-plane covalent bonding between each other. The intercalated Pd ions are located between the van der Waals layers, indicated by a red shaded area. To measure various physical properties, we have taken three samples S1, S2 and S3 from the same batch. Figure 1b displays the temperature (T)-dependent in-plane resistivity ρ_ab(T) of 2H-Pd_0.05TaSe₂ of S1, which exhibits a variety of electronic phase transitions upon the temperature being lowered. Notably, ρ_ab(T) exhibits quite linear T-dependence at high temperatures below 300 K down to T_CDW, indicating an unconventional metallic state associated with the periodic lattice distortions and the pseudogap behavior²⁹. As the temperature is further lowered, a hump near T_CDW = 115 K appears with the formation of a CDW phase, followed by a superconducting (SC) transition near T_c = 2.6 K. It is noted in the 2H-Pd_xTaSe₂ system that the intercalation of Pd ions effectively results in electron doping³⁰, increased lattice constants a and c¹⁶, and a reduced c/a ratio³¹. As a result, Pd intercalation moderately suppresses the CDW phase in 2H-Pd_0.05TaSe₂ (T_CDW = 115 K) while it sharply enhances the SC phase (T_c = 2.6 K), as compared to those of pristine 2H-TaSe₂ (T_CDW = 122 K and T_c = 0.15 K)^14,15.

a The 2H crystal structure of Pd_0.05TaSe₂ composed of two layers of 1H-TaSe₂. The red shaded area represents the positions of the intercalated Pd ions. b Temperature dependence of in-plane resistivity, ρ_ab, of the sample S1 of 2H-Pd_0.05TaSe₂ at 0 ≤ P ≤ 34.7 GPa. The black arrow marks the CDW transition temperature T_CDW. c Temperature dependence of dρ_ab/dT of S1 for 0 ≤ P ≤ 21.5 GPa. Plots are shifted downwards by a constant value for clarity. The colored arrows indicate dips in the dρ_ab/dT data realized at T_CDW. d Temperature dependence of the Hall coefficient R_H measured in the sample S2 at 0 ≤ P ≤ 31.8 GPa. R_H is obtained from a linear fit to the Hall resistivity measured at magnetic fields of -4 T ≤ μ₀H ≤ 4 T applied along the c-axis. The red colored arrow denotes the T_CDW estimated from the R_H curve; the temperature where a negative drop in the R_H curve starts in is determined by linear extrapolations of high and low temperature traces.

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To investigate the effect of pressure on the CDW order, we have measured ρ_ab(T) at various pressures. It can be seen that the resistivity at 300 K decreases by almost 50% from 0.18 mΩ cm at ambient pressure to 0.10 mΩ cm at P = 34.7 GPa, indicating that pressure plays a role of increasing electrical conductivity. Furthermore, with increase of pressure, the hump feature in ρ_ab(T) curves becomes weak and T_CDW decreases monotonically to lower temperatures, suggesting the gradual suppression of the CDW order. We can identify T_CDW as a dip from the dρ_ab/dT curve, which shifts to lower temperatures down to ~45 K at 20.4 GPa (Fig. 1c). Figure 2a presents the traces of T_CDW as determined from the dip position (solid circles, sample S1). Another independent measurement on S2 provides a similar trace. This dip fades out near a critical pressure P_c = ~21.5 GPa, implying the disappearance of the CDW order at the quantum phase transition realized at P_c.

a Electronic phase diagram of 2H-Pd_0.05TaSe₂ with pressure. The filled blue circles represent T_CDW as obtained from the ρ_ab(T) measurements of S1 while the unfilled purple triangles represent the T_CDW estimated from the R_H measurements in S2. A dashed black curve represent a guide to the eye. The filled green triangles represent the T_c as obtained from ρ_ab measurements from S1. T_c is plotted after being multiplied by a factor of 3 for clarity. b Pressure-dependence of the Hall coefficient R_H (purple diamonds, left axis) at 10 K shown in Fig. 1d. Pressure dependence of the integrated intensity of the two-phonon Raman scattering at 300 K (right axis) presented in Fig. 5e. c Pressure dependence of (({left|{boldsymbol{d}}{{boldsymbol{H}}}_{{boldsymbol{c}}{boldsymbol{2}}}^{{boldsymbol{c}}}/{boldsymbol{dT}}right|}_{{{boldsymbol{T}}}_{{bf{c}}}}/{{boldsymbol{T}}}_{{boldsymbol{c}}}))^1/2 obtained from the analysis of Fig. 3. A vertical dotted line indicates the critical pressure P_c ~ 21.5 GPa.

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The temperature-dependent Hall coefficient R_H measured at various pressures between 0 ≤ P ≤ 31.8 GPa provides further insight into the electronic structure evolution with the pressure. At ambient pressure, R_H is positive and exhibits a weak T-dependence above T_CDW (Fig. 1d). Near the onset of the CDW transition as determined by the dρ_ab/dT, R_H steeply decreases toward lower values to become negative, which is attributed to the opening of the CDW gap near the K-barrel (a hole pocket) of the Fermi surface according to the ARPES results^18,32. It is noticed that the abrupt change in R_H found below ~T_CDW shifts toward lower temperatures and becomes broader with the pressure increase. To estimate the T_CDW from the R_H curve, the temperature below which a negative drop in the R_H curve starts was determined by linear extrapolations at both high and low temperature traces of the R_H curves. The inverted triangles in Fig. 2a summarize T_CDW estimated from the drop of R_H curves, which shows nearly consistent behavior with T_CDW estimated from the ρ_ab data.

The R_H values measured at 10 K (Fig. 2b) also provides the insight into the pressure driven evolution of the CDW states. R_H values above T_CDW at ambient pressure are positive and nearly temperature-independent, indicating the transport dominated by hole carriers above T_CDW. On the other hand, below T_CDW, R_H starts to develop a downward shift, indicating that electron pockets contribute to the transport more significantly. The negative contribution increases as temperature is lowered; at 10 K, R_H reaches nearly a saturating value of −4.7 × 10⁻¹⁰ m³/C. With increase of pressure, R_H around 10 K increases systematically to positive values until the P reaches 22.1 GPa, and at the pressure above 22.1 GPa, it stays nearly constant, indicating that the suppression of CDW state leads to the disappearance of the negative contribution. The almost P-independent R_H from 22.1 to 31.8 GPa confirm complete suppression of the CDW order near 22.1 GPa. The linear extrapolation of the R_H data at a low-pressure regime provides an estimated critical pressure of 22.8 ± 1.4 GPa, which is close to the P_c = 21.5 GPa of the CDW quantum phase transition estimated from the ρ_ab(T) measurements.

While the CDW state is continuously suppressed by pressure application, T_c is found to be maximized near P_c. This behavior is illustrated in the low temperature ρ_ab(T) as shown in Fig. 3a. At ambient pressure, a clear drop in ρ_ab(T) to zero value is found, indicating T_c = 2.6 K as defined by the 50% resistivity criterion. With the application of pressure, T_c increases from 2.6 K at ambient pressure to attain a broad maximum value of ~8.5 K around P_c = 21.5 GPa (~8.2 K at 20.4 GPa and ~8.4 K at 22.8 GPa) and then decreases to ~7.2 K at 34.7 GPa. The T_c vs. P curves summarized in the phase diagram of Fig. 2a indeed demonstrate that the T_c trace forms a dome shape centered around P_c = 21.5 GPa, showing optimized superconductivity in the vicinity of P_c. The overall phase diagram in Fig. 2a also suggests that at ambient- and low-pressure regimes, the CDW and SC phases in 2H-Pd_0.05TaSe₂ might coexist each other. As pressure increases toward P_c, T_c is systematically enhanced with the simultaneous suppression of T_CDW, thus exhibiting competing nature of the CDW and superconducting orders.

a Low temperature behavior of ρ_ab in 2H-Pd_0.05TaSe₂ (S1) for 0 ≤ P ≤ 34.7 GPa. b Temperature dependence of upper critical fields μ₀({{boldsymbol{H}}}_{{boldsymbol{c}}{boldsymbol{2}}}^{{boldsymbol{c}}}) at the magnetic field applied along the c-axis for 0 ≤ P ≤ 34.7 GPa. Also plotted are the best fit results of the single-band Werthamer–Helfand–Hohenberg (WHH) model. Pressure dependence of (c) the upper critical field at zero temperature μ₀({{boldsymbol{H}}}_{{boldsymbol{c}}{boldsymbol{2}}}^{{boldsymbol{c}}})(0) and d the in-plane superconducting coherence length ξ_ab,0 obtained from the WHH fit results. A vertical dotted line indicates the critical pressure P_c =~ 21.5 GPa.

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To further characterize the optimized superconductivity near P_c, we have measured the P,T-dependent upper critical fields under magnetic fields H applied along the c-axis, μ₀({H}_{c2}^{c})(T), (Fig. 3b). In order to extract the pressure evolution of the μ₀({H}_{c2}^{c})(T = 0), we have fitted the μ₀({H}_{c2}^{c})(T) curves in each pressure by employing the Werthamer–Helfand–Hohenberg (WHH) equation³³,

Here, t is the reduced temperature T/T_c, (bar{h}) = (4/π²)(H_c2(T)/|dH_c2/dT | _Tc), λ_so is the spin-orbit scattering constant, and α is the Maki parameter. The best fit results for a minimal model of λ_so = α = 0, which neglects the spin-orbit coupling and assumes a pure orbital limiting case, are shown in dashed lines in Fig. 3b. The pressure dependence of μ₀({H}_{c2}^{c})(T = 0) is clearly maximized from 0.87 T at 2.2 GPa to a 7-fold increased value of 6.3 T at 21.5 GPa, and then decreases back to 2.8 T at 34.7 GPa. As a result, P-induced evolution of μ₀({H}_{c2}^{c})(0) plotted in Fig. 3c shows that they are sharply enhanced near P_c = 21.5 GPa. The P dependence of the superconducting coherence lengths ξ_ab(0)’s, estimated by the Ginzburg-Landau expression³⁴ μ₀({H}_{c2}^{c})(0) = Φ₀/2πξ_ab(0)², consistently shows that the minimum ξ_ab(0) of 7.23 nm is realized at P_c (Fig. 3d). Therefore, along with the dome-shape in T_c centered at P_c, μ₀({H}_{c2}^{c})(0) (ξ_ab(0)) values exhibit a sharp maximum (minimum) at P_c.

Figure 2c summarizes the pressure dependence of the (({|d{H}_{c2}^{c}/{dT}|}_{{{boldsymbol{T}}}_{{bf{c}}}}/{T}_{c}))^1/2 obtained from the analysis of Fig. 3, where ({|d{H}_{c2}^{c}/{dT}|}_{{{boldsymbol{T}}}_{{bf{c}}}}) represents the (d{H}_{c2}^{c})/dT values obtained near T_c. It has been shown by the Ginzburg-Landau theory that if the Fermi wavevector of Bogoliubov quasiparticles is smoothly varying over pressure variation, the relationship ({|d{H}_{c2}^{c}/{dT}|}_{{{boldsymbol{T}}}_{{bf{c}}}}/{T}_{c}) (propto) m^*2 holds^35,36,37. Here, m^* is the average effective mass of the quasiparticles in the ab-plane. Therefore, the sudden peak behavior in ({|d{H}_{c2}^{c}/{dT}|}_{{{boldsymbol{T}}}_{{bf{c}}}}/{T}_{c}) indicates that the effective mass of the quasiparticles in the superconducting state is peaked at P_c. This supports that the CDW-QCP and associated critical fluctuation can increase the effective mass near the critical pressure.

All these evolutions of the T_CDW and T_c with pressure clearly show that the optimization of the superconductivity occurs at or in close vicinity to P_c, suggesting that the suppression and fluctuation of the CDW order expected at the CDW-QCP could be helpful in enhancing the superconducting pairing strength. This is a rare case of observing a CDW-QCP linked to the optimization of superconductivity among the class of 2D van der Waals materials. Note that most of other 2D transition-metal chalcogenides with CDW instability often exhibit a first-order transition of the CDW state that mostly competes with SC under variation of external turning parameters^9,10 or mere coexistence of SC with the CDW state without showing any evidence of a link between the optimized superconducting properties and a putative CDW-QCP^38,39.

In order to check whether the fluctuation of CDW order expected at the CDW-QCP can affect the electrical transport in the normal state before forming the Cooper pair, we have applied external magnetic field along the c-axis (μ₀({H}^{c})) of 9 T to investigate ρ_ab(T) behavior above ~1.8 K at each pressure (Fig. 4a). Note that T_c becomes always less than 3 K at μ₀({H}^{c},)= 9 T, allowing us to investigate more easily transport of the normal state at lower temperature regions. At ambient pressure, ρ_ab at μ₀({H}^{c},)= 9 T can be successfully described by a quadratic power law fit ρ_ab(T) = ρ₀ + AT² (black solid line), at least up to ~10 K, defined here as an effective Fermi liquid temperature T_FL. Above the T_FL, the experimental data steeply increase upward, deviating from the Fermi liquid behavior. This suggests that the electron-electron scattering inherent to the Fermi liquids dominate the transport for 2H-Pd_0.05TaSe₂ at ambient pressure below T_FL while the extra scattering, i.e., electron-phonon scattering starts to contribute more conspicuously above T_FL.

a The ρ_ab curves of S1 measured at a magnetic field of μ₀({{boldsymbol{H}}}^{{boldsymbol{c}}}) = 9 T applied along the c-axis. The data in the normal state are fitted by a quadratic power law of ρ = ρ₀ + AT ² (black lines). The plots of (ρ_ab – ρ₀) vs. T² for (b) 0 ≤ P ≤ 21.5 GPa and c 22.8 ≤ P ≤ 34.7 GPa. The black lines are linear guidelines. Black arrows mark the deviating temperature from the linear behavior for each pressure, T_FL, below which the Fermi liquid behavior starts. d The pressure dependence of T_FL. The pressure-evolution of fitting parameters e ρ₀ and f A^1/2 obtained from the best fit to ρ = ρ₀ + AT ² curves below T_FL.

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It turns out that such a quadratic power law behavior is persistently found at all the pressure ranges investigated in the low temperature region of the ρ_ab(T) curves at μ₀({H}^{c},)= 9 T. Figure 4b, c shows the (ρ_ab – ρ₀) vs. T² plots of 2H-Pd_0.05TaSe₂, confirming that (ρ_ab – ρ₀) exhibit linear behavior to T² below T_FL. With the application of pressure, it is found that T_FL does not change up to 19.5 GPa, indicating negligible P-dependence of the Fermi liquid properties. However, from 20.4 GPa, T_FL starts to exhibit an increase to ~13 K and exhibit a maximum T_FL ~ 15 K near P_c = 21.5 GPa, indicating an enhanced Fermi-liquid energy scale just below and at P_c. Above P_c = 21.5 GPa and up to ~25.6 GPa, the T_FL is slowly decreased down to ~13 K and becomes nearly saturated up to 34.7 GPa (see, Fig. 4d). Moreover, ρ₀ determined by extrapolation of ρ_ab(T) to 0 K exhibits negligible change with pressure up to 19.5 GPa and subsequently a systematic drop at 19.5 GPa ≤ P ≤ 22.8 GPa, roughly consistent with the pressure window of exhibiting the T_FL increase (Fig. 4e). Moreover, at the regime at 22.8 GPa ≤ P, ρ₀ maintains nearly constant values initially up to 25.6 GPa but slowly decreases above. This reduced residual resistivity and the increased T_FL far above P_c as compared with those far below P_c could be related to the expected recovery of the Fermi surface with the suppression of the CDW gap in a mean field level. However, there exist a pressure window of exhibiting the extra increase of T_FL beyond the expected behavior of a constant increase across the collapse of the CDW state, which is roughly between 19.5 and 25.6 GPa. This indirectly indicates that the fluctuation of the CDW order expected at the CDW-QCP may affect the low-T transport behavior near P_c.

It is also surprising to find that the square root of the quadratic power law coefficient A^1/2, which is 3.3 ×10⁻⁵ Ω^1/2cm^1/2/K at ambient pressure, gradually increases to 6.3 ×10⁻⁵ Ω^1/2cm^1/2/K at 19.5 GPa and is quickly enhanced to 7.1 ×10⁻⁵ Ω^1/2cm^1/2/K in a pressure window between ~19.5 GPa and ~25.6 GPa near P_c (Fig. 4f). Namely, A^1/2 exhibits a sharp peak at P_c = 21.5 GPa but is enhanced in an extended P region of ~6 GPa near P_c. According to the Kadowaki-Woods relation⁴⁰ of A = α_KW γ₀² and the expression of the Sommerfeld coefficient⁴¹ γ₀ = 1/3π²k_B²(1+λ_ep+λ_ee)N(ε_F), which is valid for the Fermi liquid states, A^1/2 should be proportional to N(ε_F)(1+λ_ep+λ_ee). Here, α_KW is the Kadowaki-Woods ratio, k_B is the Boltzmann constant, N(ε_F) is the electronic density of states, and λ_ep (λ_ee) is the electron-phonon (electron-electron) coupling constant. Note that while electron-phonon coupling might be important for pairing mechanism for the superconductivity within the BCS mechanism, many-body effects resulting in the enhancement of the Sommerfeld coefficient might also include both electron-phonon and electron-electron interactions. As a result, 2H-Pd_0.05TaSe₂ exhibits the enhancement of N(ε_F) (1+ λ_ep + λ_ee) by a factor 2.15 near P_c. All these observations suggest that the CDW-QCP and associated critical fluctuation can increase the effective mass (Sommerfeld coefficient) of the normal state carriers in an extended P region near P_c at low temperatures.

Note that our observation of the Fermi liquid behavior near the CDW-QCP is in contrast to the magnetic QCP case wherein the non-Fermi liquid behavior is often observed, for example, ρ_ab(T) = ρ₀ + A_nT ⁿ with n ~ 1. In previous studies investigating transport behavior near the CDW-QCP with the fitting scheme of ρ_ab(T) = ρ₀ + A_nT^n12,35,42, it was found that n exhibits a minimum value between ~1.8 and 2.1 near the CDW-QCP while it is systematically increased up to toward higher values ~ 2.8–3.0 as the system reaches the phase regions far away from the CDW-QCP. When we apply a similar fitting scheme of ρ_ab(T) = ρ₀ + A_nTⁿ to the data between the temperature window of ~1.8 and 20 K shown in Fig. 4a, we also find out a similar behavior in the evolution of n with pressure (see, Supplementary Note 2 for further details).

To find spectroscopic signatures on the evolution of CDW order with temperature and pressure, Raman scattering measurements have been performed in a single crystal of 2H-Pd_0.05TaSe₂. Figure 5a presents the temperature-dependent Raman shift in a frequency window of 10–100 cm⁻¹ at ambient pressure. At T = 20 K, three CDW related modes can be identified; following similar modes at 82, 64, and 49 cm⁻¹ as reported in a pristine 2H-TaSe₂²², they can be assigned as an A_1g amplitudon mode ({{boldsymbol{A}}}_{{boldsymbol{1}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) (yellow triangles) at 80 cm⁻¹, an A_1g phason mode ({{boldsymbol{A}}}_{{boldsymbol{1}}{boldsymbol{g}}}^{{boldsymbol{pha}}}) (gray arrow) at 57 cm⁻¹, and an E_2g amplitudon mode ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) at ~45 cm⁻¹, respectively. On the other hand, the E_2g phason mode, found at 41 cm⁻¹ in 2H-TaSe₂²², can’t be discerned in this study, possibly because it may be too close to the ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) mode or lower than our frequency limit of ~10 cm⁻¹. Upon raising the temperature close to T_CDW = 115 K, the frequency of ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) mode particularly shifts to lower frequencies close to zero, thereby proving the suppression of the CDW excitation energy nearly to zero with T approaching T_CDW. By fitting the peak frequency of the ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) mode to the T-dependent power law, T_CDW = ~ 115 K (green arrow) could be extracted, being consistent with the T_CDW = 115 K from the ρ_ab(T) in Fig. 1. (see, Supplementary Figure 3 for the detailed fitting results). Note that although the ({{boldsymbol{A}}}_{{boldsymbol{1}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) mode frequency also shows a steep decrease with T increase, the mode intensity gets too small at T > ~ 70 K, making the ({{boldsymbol{A}}}_{{boldsymbol{1}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) mode untraceable at higher temperatures. Moreover, the A_1g phason mode shows a slight decrease upon T being increased to T_CDW before disappearing. As a result, the frequency of the ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) mode seems to be most sensitively reflecting the systematic suppression of the CDW excitation energy with T evolution.

a–d Color contour plots of the temperature-dependent Raman shift at 0 ≤ P ≤ 12 GPa. The yellow triangles and white diamonds represent the frequencies of the CDW amplitudon modes A_1g^amp and E_2g^amp, respectively. The A_1g phason mode, A_1g^pha, is only detected at ambient pressure, possibly due to the higher signal-to-noise ratio. The green arrows mark T_CDW as obtained by the in-plane resistivity measurements shown in Fig. 1. e Raman spectra of 2H-Pd_0.05TaSe₂ at pressures between 0 ≤ P ≤ 35.2 GPa at room temperature. The curves have been vertically shifted for clarity. The orange shaded areas indicate that the broad two-phonon peak systematically decreases its intensity with application of P up to 20.7 GPa and disappears above 21.8 GPa.

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To investigate how the CDW excitation energy will evolve with pressure close to P_c, we have performed low-temperature Raman scattering under high-pressure conditions. Figure 5b–d presents the Raman shifts vs. temperature summarized at 5, 10, and 12 GPa, respectively. The ({{boldsymbol{A}}}_{{boldsymbol{1}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) and ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) modes can be clearly identified at 3 K for all three pressures studied. Notably, the frequencies of both the amplitudon modes are found to decrease with applied pressure; the ({{boldsymbol{A}}}_{{boldsymbol{1}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) from 80 cm⁻¹ at ambient pressure to 61 cm⁻¹ at 12 GPa, and ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) from 45 cm⁻¹ at ambient pressure to 35 cm⁻¹ at 12 GPa. Upon raising the temperature, the frequency of ({{boldsymbol{E}}}_{{boldsymbol{2}}{boldsymbol{g}}}^{{boldsymbol{amp}}}) modes for each pressure, shifts to lower frequencies close to zero, indicated by the green arrows corresponding to the T_CDW obtained by the resistivity measurements in Fig. 1. However, the CDW amplitudon modes could not be traced above 12 GPa, as the mode intensity becomes weak and its shape gets broadened with increase in pressure (see, Supplementary Figure 4 for the raw data). As a result, from the temperature-dependent Raman measurements, both the amplitudon peak energies and T_CDW are confirmed to be suppressed with pressure at least up to 12 GPa.

In order to trace a continuous change in the lattice dynamics up to P_c ~ 21.5 GPa by the suppression of the CDW order, we have further investigated the pressure evolution of the two-phonon scattering resulting from the Kohn anomaly in the room-temperature Raman spectra (Fig. 5e). The three peaks observed in the ambient-pressure spectrum can be assigned as a two-phonon mode at 141.5 cm⁻¹, an in-plane E_2g¹ vibrational mode at 206.0 cm⁻¹, and an out-of-plane A_1g mode at 232.3 cm⁻¹, based on the comparison with the data in a pristine sample²². Here, we highlight the evolution of the two-phonon mode with pressure, which can effectively show the presence of the Kohn anomaly coming from the CDW correlation at room temperature^14,23. It has recently been shown by an inelastic X-ray scattering study that the Kohn anomaly in 2H-TaSe₂, i.e., the anomalous softening of the longitudinal acoustic phonon dispersion at q_CDW = (0.3, 0, 0) in the momentum space, is persistently found at T far above T_CDW and survives even at room temperature. The Kohn anomaly shortens the bonding between the three Ta atoms, thereby increasing the intralayer overlap of Ta-5({{boldsymbol{d}}}_{{boldsymbol{xy}}}) and Ta-5({{boldsymbol{d}}}_{{{boldsymbol{x}}}^{{boldsymbol{2}}}{boldsymbol{-}}{{boldsymbol{y}}}^{{boldsymbol{2}}}}) orbitals²⁴. The prevailing two-phonon Raman scattering at high temperatures directly support the presence of the short-range CDW correlation with local lattice distorting involving three Ta sites, a characteristic of a CDW state in a strong coupling regime.

As indicated by the orange-shaded area in Fig. 5e, the intensity of the two-phonon Raman mode exhibits a systematic decrease with application of pressure. With increase of pressure from ambient pressure up to 20.7 GPa, the intensity progressively decreases and disappears above 21.8 GPa. As a result, when the intensity of the two-phonon mode is integrated with a liner background subtracted, the integrated intensity summarized in Fig. 2b (right axis) exhibits a nearly linear decrease. Upon a linear extrapolation, the critical pressure of reaching zero integrated intensity is estimated as 21.7 ± 0.6 GPa, which is quite close to P_c = 21.5 GPa, the critical pressure where the QCP of the CDW state is realized at low temperatures. Namely, the two-phonon Raman scattering at room temperature, i.e. the Kohn anomaly due to the short-range CDW correlation, truly disappears at the pressure close to P_c. Our observations support that the local lattice distortion as a main characteristic of short-range CDW correlation also controls the stability of the CDW state near the QCP. Therefore, it is concluded that both short-range and long-range CDW state can be continuously suppressed by squeezing lattice isotopically by the pressure, leading to the nearly same critical pressure of P_c. In this regard, the pressure seems to be an effective tuning parameter for suppressing continuously the local three Ta site distortions, the building block of the strong-coupling CDW state realized in 2H-TaSe₂ at both room and low temperature regions.

A noteworthy aspect in the temperature-pressure phase diagram of 2H-Pd_0.05TaSe₂ (Fig. 2) is that the P_c = 21.5 GPa showing optimal superconductivity in 2H-Pd_0.05TaSe₂ is lower than the P_c = ~23–28 GPa showing optimal superconductivity in a pristine 2H-TaSe₂⁴³. As our results indicate that superconductivity optimatized at the CDW-QCP in 2H-Pd_0.05TaSe₂, this strongly suggests that a CDW-QCP should exist in the pristine 2H-TaSe₂ as well, presumably near the critical pressure of ~23–28 GPa. While the full suppression of the T_CDW had not been traced in a previous work in 2H-TaSe₂⁴³, it is found that the T_CDW = 122 K of 2H-TaSe₂, which is higher than the T_CDW = 115 K of 2H-Pd_0.05TaSe₂ at ambient pressure, evolves to T_CDW = ~ 90 K at 20 GPa. This T_CDW of 2H-TaSe₂ at the high pressure is indeed higher compared to the T_CDW = 52 K (45 K) of 2H-Pd_0.05TaSe₂ at 19.5 GPa (20.4 GPa), supporting the scenario that a CDW-QCP should exist at higher pressures in the pristine 2H-TaSe₂. Therefore, these observations strongly suggest that the pressure for optimized superconductivity should be linked to the critical pressure for the CDW-QCP in both 2H-TaSe₂ and 2H-Pd_0.05TaSe₂.

A generic origin for the sharply enhanced N(ε_F)(1+λ_ep+λ_ee) near P_c in pressurized 2H-Pd_0.05TaSe₂ needs further considerations. One natural outcome of the continuous suppression of CDW state is the increase of effective carriers across P_c to result in the increase of N(ε_F) above P_c. However, with the fluctuating CDW amplitude expected near P_c, the effective bandwidth of the system can be reduced sharply near P_c to possibly induce a sharp enhancement in N(ε_F) in a mean field level near P_c, thereby resulting a maximized T_c at P_c. However, the enhancement via the many-body effects should be also considered. Although a sharp enhancement of λ_ep near P_c can’t be ruled out, it is usually expected that the electron-phonon interaction is more or less fixed in one crystal structure. If so, it is more likely that fluctuating CDW amplitude might cause an additional effective increase of λ_ee near P_c⁴⁴. Whichever scenarios are suitable for the microscopic mechanism, our observations unambiguously point out that the CDW fluctuations expected at the CDW-QCP can be helpful in optimizing superconductivity in the 2H-TaSe₂ as well as Pd intercalated 2H-TaSe₂.

In conclusion, the pressure-dependent electrical transport and Raman scattering spectra have been investigated in 2H-Pd_0.05TaSe₂. Through comprehensive analysis of the resistivity and Hall effect, we have demonstrated the continuous suppression of CDW order with pressure-increase, resulting in T_CDW approaching to zero Kelvin at a pressure-induced QCP at P_c ~ 21.5 GPa. At this QCP, both the superconducting transition temperature T_c and the upper critical field H_c2 reach their maximum values. The simultaneous disappearance of the CDW amplitudon mode and the suppression of two-phonon scattering under pressure further corroborate the existence of a CDW-QCP. These observations provide direct evidence for the formation of a CDW-QCP at P_c, indicating that charge and lattice fluctuations associated with the QCP of strongly coupled CDW order can enhance SC in pressurized 2H-Pd_0.05TaSe₂.