Incommensurate two-dimensional checkerboard charge density wave in the low dimensional superconductor Ta4Pd3Te16
Zhenzhong Shi S. J. Kuhn F. Flicker T. Helm J. Lee W. M. Steinhardt S. E. Dissanayake D. Graf J. C. Ruff G. Fabbris D. Haskel Sara Haravifard
cond-mat.str-el





Department of Physics, Duke University, Durham, NC 27708, USA

Department of Physics, Duke University, Durham, NC 27708, USA

Rudolph Peierls Centre for Theoretical Physics, University of Oxford, Department of Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK

Max Planck Institute for Chemical Physics of Solids, Dresden, Germany

Cornell High Energy Synchrotron Source, Cornell University, Ithaca, NY 14853, USA

Department of Physics, Duke University, Durham, NC 27708, USA

Department of Physics, Duke University, Durham, NC 27708, USA

National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA

Cornell High Energy Synchrotron Source, Cornell University, Ithaca, NY 14853, USA

Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA

Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA

Department of Physics, Duke University, Durham, NC 27708, USA
Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA





We report the observation of a two-dimensional (2D) checkerboard charge density wave (CDW) in the low-dimensional superconductor . By determining its CDW properties across the temperature-pressure (T-P) phase diagram and comparing with prototypical CDW materials, we conclude that  features: a) an incommensurate CDW with a mixed character of dimensions (Q1D considering its needle-like shape along the b-axis, Q2D as the CDW has checkerboard wavevectors, and 3D because of CDW projections along all three axes); and b) one of the weakest CDWs compared to its superconductivity (SC), i.e. enhanced SC with respect to CDW, suggesting an interesting interplay of the two orders. 






Incommensurate two-dimensional checkerboard charge density wave in the low dimensional superconductor 
    S. Haravifard
    December 30, 2023
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Charge density waves (CDWs), electronic instabilities originally considered in one dimension (1D) <cit.>, have now been found in various forms and exist not only in quasi-one dimensional (Q1D) materials but also in quasi-two dimensional (Q2D) and three dimensional (3D) systems <cit.>. Q1D systems such as the transition metal trichalcogenide compounds are generally expected to be unstable to CDW formation via the Peierls instability, i.e. Fermi surface (FS) nesting (FSN) <cit.>. However, in systems with higher dimensions, such as the transition metal dichalcogenides (Q2D) and cubic system R_3T_4X_13 (3D) <cit.>, FSN is not the natural explanation for CDW formation. A single wavevector will always nest the two points of the FS onto one another in 1D; in 2D and 3D a special shape of FS would be required to match large portions of the FS onto itself. Instead, CDWs in higher dimensional materials such as NbSe_2, often have a more complex origin involving the momentum and orbital dependence of the electron-phonon coupling matrix <cit.>. 

The electron-phonon coupling required for the CDW order may also facilitate superconductivity (SC) based on the standard Bardeen Cooper Schrieffer (BCS) mechanism <cit.>, and the two orders are often found to be proximal or to coexist <cit.>. However, the interplay of the CDW and SC may extend well beyond the BCS picture, where the pairing mechanism may not be phonon mediated and the SC becomes unconventional. In cuprate, for example, the relative strength of the SC with respect to the charge order is much stronger than that in conventional superconductors <cit.> and the interplay of charge order and SC is still highly debated <cit.>. It is possible that SC in some unconventional superconductors could be enhanced by quantum fluctuations accompanied by the suppression of the CDW to a quantum critical point (QCP) <cit.>. In reality, however, it is extremely difficult to discern the different yet possibly intertwined mechanisms contributing to the interplay of the CDW and SC, including dimensionality and critical fluctuations. Here we report a CDW system that may serve as an ideal platform for such studies.

Featuring three chains of PdTe_2, TaTe_3, and Ta_2Te_4 along its b-axis, aligned into planes perpendicular to c, the monoclinic crystalline structure of  with lattice parameters a = 17.687 Å, b = 3.735 Å, c = 19.510 Å, and β = 110.42^∘ <cit.> is reminiscent of those of trichalcogenide compounds such as ZrSe_3, TaSe_3, NbSe_3, and TaS_3. However, transport measurements on  indicate resistivity ratios along the a^*:b:c directions of 4:1:13 at room temperature (T), evolving to 10:1:20 just above the superconducting transition T_c ∼ 4.6 K <cit.>. The ratios suggest a rather weak anisotropy, and the material itself is better thought of as having a character of mixed dimensionality: Q1D (along b), Q2D (cleavage plane perpendicular to c), and 3D (relatively weak anisotropy), which, as we will show below, underlies an interesting 2D checkerboard CDW in this system with wavevectors projected along all three axes.

Previous studies of  have revealed an anisotropic SC ground state <cit.>. Under pressure (P), a dome-shaped SC phase with maximum T_c ∼ 6 K at 0.2 ∼ 0.3 GPa was discovered <cit.>. Some evidence suggests that the SC order parameter in  might have a d-wave symmetry and thus an unconventional origin <cit.>. Others have argued that the experimental findings may be explained by the more conventional s-wave SC order parameter and the multi-band nature of the compound <cit.>. While phase-sensitive measurements are needed to eventually reveal the pairing symmetry, the SC in  is also interesting because of hints of its proximity to a CDW phase <cit.>. NMR/NQR <cit.> and thermodynamic measurements <cit.> suggest a CDW transition temperature (T_CDW) of ∼ 20 K, while Raman scattering measurements support the emergence of CDW fluctuations below 140 K ∼ 200 K <cit.>. Additionally, STM measurements have shown period-4 commensurate CDW-like stripes along the b-axis that coexist with SC <cit.>. However, the suggested wavevector is seemingly at odds with transport measurements, which reveal a resistivity anomaly primarily along the a^*-direction while being much weaker along the b and c directions <cit.>. Therefore, a direct measurement of the bulk CDW in  is critical for reconciling these experimental results and understanding the interplay of SC and CDW in this material. 
 
In this paper, we report the direct observation of a delicate CDW in  using synchrotron X-ray diffraction (See Supplemental Material <cit.>). We establish the T-P phase diagram, which reveals a low T_CDW ∼ 16 K at ambient P, quickly suppressed with increasing P, becoming undetectable where T_c reaches its maximum. Our de Haas-van Alphen (dHvA) oscillation measurements (See Supplemental Material <cit.>) do not show any sign of FS reconstruction through the CDW transition, consistent with an earlier report <cit.>. The observed CDW has a checkerboard pattern in real space, which resembles more closely the checkerboard CDW in quasi-2D systems such as NbSe_2 <cit.> rather than those in typical Q1D systems such as NbSe_3, where multiple CDWs, if they exist, are typically unidirectional and independent <cit.>. However, unlike the CDWs seen in most Q2D systems, the CDW in  does not lie within the Q2D planes. These peculiar dimensional properties may explain its weak CDW, as we discuss in detail below. 






We obtained the most direct evidence of a checkerboard CDW in  using ambient-P X-ray diffraction <cit.>, which provides an unambiguous identification of the bulk CDW and its wavevectors. Using a high-dynamic-range area detector, we scanned wide areas across multiple Brillouin zones in many directions, including along the wavevector [0 0.25 0] suggested by a previous STM study <cit.>. However, our X-ray scans did not reveal any CDW peak along this direction. Instead, we identified two peaks at wavevectors Q1 = [-0.2 0.21 -0.3] and Q2 = [0.2 0.21 0.3] in all the Brillouin zones within our detection range [Fig. <ref>(a)]. The integrated intensity of the CDW peak is ∼ 10^-3 the strength of that at the nearby Bragg peak. The Q1 and Q2 wavevectors are related by a 180^∘ rotation about the b-axis. Together they constitute a commensurate pattern of the superlattice along the H and L (or a^* and c^*) directions. 

We repeated the measurements at different T from 3.5 K up to 30 K. Figure <ref>(b) shows the line-cuts and their Lorentzian fits along the K direction for the Q1 CDW near the (-1 4 1) Bragg peak for a few selected T. The entire set of data consisting of the line-cuts at all measured T is summarized in a color contour plot in Fig. <ref>(c). It is clear that, with T increasing from 3.5 K, the Q1 CDW peak intensity first increases and plateaus at 5 K ∼ 9 K, then diminishes and becomes undetectable at T_CDW ∼ 16 K, and this is accompanied by a slight shift in the peak position. Our results are consistent with the T_CDW ∼ 20 K inferred from NMR/NQR <cit.> and transport measurements <cit.>. We do not see any evidence of a static CDW above ∼ 16 K, though our measurements cannot rule out the possibility of fluctuating CDW order at higher T. The signatures of phonon anomalies at higher T <cit.> therefore might be attributed to a pseudogap phase, similar to that in NbSe_2 <cit.>. A comparison between T_CDW (∼16 K) and the energy gap (Δ = 20 ∼ 30 meV) observed by STM <cit.> suggests that the CDW in  is in the strong-coupling limit (3.52k_BT_c≪ 2Δ), again similar to the case in NbSe_2 <cit.> and cuprates <cit.>. It is interesting to point out that our observation is consistent with an early band structure study on rare-earth tellurides which suggests that a checkerboard state would occur with sufficiently low T_CDW, while a unidirectional (stripe) CDW is favored with higher T_CDW <cit.>.








To confirm our observations at ambient P, and to further explore the T-P phase diagram, we also conducted X-ray diffraction under pressure at a different synchrotron source <cit.>. With the lowest pressure achieved in our measurement (P = 0.2 GPa <cit.>), we again observed the Q1 and Q2 CDWs near the (3 -1 0) Bragg peak, at T = 4.3 K, as shown in Fig. <ref>(a). The ratio of the integrated intensity of the CDW peak and the nearby Bragg peak is again ∼ 10^-3, in good agreement with our ambient-P results. The intensities of the two CDWs are comparable within error, consistent with the picture of one checkerboard CDW. The T-dependence of the q_k line cuts are measured for the Q2 CDW at P = 0.2 GPa. The results as seen in Fig. <ref>(b) clearly show that the peak intensity is quickly suppressed with increasing T, and becomes undetectable at 12 K. Moreover, as shown in Fig. S3, our experiments do not show any sign of CDW <cit.> at higher P (0.35 GPa), where the peak position of the reported SC dome resides <cit.>. Therefore, we find that the CDW becomes weaker, characterized by a smaller T_CDW, with increased P, and disappear at higher P when the SC is strongest (maximum T_c). This seems to suggest a competitive nature of the interplay between CDW and SC, and is consistent with the fact that the ambient-P CDW peak intensity plateaus at 9 K ∼ 5 K, below which the SC emerges and CDW peak intensity is weakened [see Fig. <ref>(c) and Fig. <ref>(a)]. However, as discussed below, we do not rule out the effect of quantum fluctuations near a CDW QCP on the SC. In cuprates, for example, while a reduced CDW intensity below T_c was reported as evidence for a CDW competing with SC <cit.>, a recent study also revealed the importance of a charge order quantum critical point <cit.>.

The T-dependence of the CDW peak position along q_k, as shown in Fig.<ref>(c) and Fig.<ref>(b), shows some unexpected behaviors. At ambient P the CDW peak center along q_k shifts towards a commensurate value 0.2 (a real-space periodicity of 5) with decreasing T, but saturates before locking into that value. The peak also shifts further away from q_k = 0.2 under pressure. Therefore, it appears that the CDW remains incommensurate along q_k down to the lowest T. Typically, one expects the CDW wavevector to evolve continuously with T until abruptly jumping to a commensurate value, in a first-order lock-in phase transition <cit.>. Examples include Q1D TTF-TCNQ and Q2D TaSe_2. However, many other materials show an evolution of the CDW wavevector without an eventual lock-in <cit.>, such as Q1D NbSe_3 and TbTe_3 and Q2D NbSe_2. Our observations indicate that  falls into this latter category.

The CDW wavevectors, Q1 = [-0.2 0.21 -0.3] and Q2 = [0.2 0.21 0.3], featuring modulations along all three axes, appear at first to be at odds with the CDW reported in the previous STM study, which found a period-4 commensurate charge modulation only along the b-axis <cit.>. This discrepancy underlies the importance of the synchrotron X-ray measurement in directly observing the bulk CDW, in contrast to the STM, which is sensitive only to the projection of the charge modulation onto the sample surface [i.e. the (-1 0 3) cleavage plane]. A careful analysis of the Q1 and Q2 wavevectors and their projections on different planes (see Fig. S2 and the associated discussions in Supplemental Material <cit.>) confirms that the Q1 and Q2 wavevectors revealed in our X-ray data actually reconciles the apparently conflicting results from STM <cit.> and transport measurements <cit.>.

The CDW correlation length can be extracted from the width of the CDW peak, which is measured by the full width at half maximum (FWHM) of the Lorentzian fits [Fig. 1(b) and Fig. 2, also see Supplemental Material <cit.> for more details]. The T-dependent FWHM at P = 0 and 0.2 GPa are plotted in Fig. <ref>(c). At low T, FWHM ∼ 0.01 r.l.u., suggesting a CDW correlation length ξ_CDW ∼ 120 Å along q_k <cit.>. A slight increase of FWHM to 0.015 r.l.u. is seen below T_c, suggesting a possible competitive interplay of CDW with SC. Near T_CDW the FWHM increases drastically, and the inferred ξ_CDW reduces to ∼ 30 Å, which is close to the CDW wavelength (i.e. CDW modulation period) along the b-axis λ_b ∼ 5b ∼ 20 Å, signaling the melting of the static CDW at higher T.









In Fig. <ref>(d) we establish the T-P phase diagram using T_CDW(P) from our X-ray diffraction measurements and T_c from Ref. <cit.>. The SC dome peaks at the pressure where the CDW vanishes. Although such observations may generally be expected within standard BCS theory (i.e. T_c increases as the density of states released upon CDW gap closure becomes available for the formation of Cooper pairs), we note that other scenarios including SC related to a CDW QCP <cit.> may not be ruled out: (a) the energy scales of the CDW and SC are comparable, while the former is typically much larger than the latter in conventional superconductors; and (b) the CDW transition in  appears to be a second order phase transition, as hinted by the very weak kink-like feature in resistivity <cit.> and the smooth evolution of the CDW order parameter seen in our X-ray data, although we do not have evidence regarding whether the transition remains second order or sufficiently weak first order at T=0.

Our results reveal a delicate checkerboard CDW in  (see the Supplemental Material <cit.>). It is found that  features (a) a character of mixed dimensionality, (b) two CDWs related by a lattice symmetry developing at the same T, (c) CDW wavevectors that are commensurate along two lattice directions but incommensurate along the third even at low T, (d) no clear signatures of FS reconstruction as the CDW develops (see our dHvA oscillation results), and (e) an interesting interplay of the CDW and SC. The uniqueness of the CDW in  is further demonstrated when comparing to other CDW materials.

On the one hand,  is often considered Q1D, structurally similar to the transition metal trichalcogenides such as NbSe_3, NbS_3, ZrSe_3, and TaSe_3, each of which has a monoclinic phase formed from chains within planes, and also the rare-earth tri-tellurides RTe_3 (R=Sm, Gd, Tb, Dy, Ho, Er, Tm) with weakly orthorhombic structures. All these materials host multiple CDWs <cit.>, which, unlike , are not related by a lattice symmetry, and develop at different T (also see Supplemental Material <cit.>). On the other hand, the resistivity ratios of  suggest that the material is better thought of not as Q1D but as a dimensional hybrid with Q1D chains within Q2D planes within a 3D material. DFT calculations confirm the existence of FS pockets of each dimensionality <cit.>. Thus a fairer comparison might be found with Q2D CDW materials such as the transition metal dichalcogenides (TMDC) 2H-NbSe_2, 2H-TaSe_2, and 1T-TaS_2 <cit.>, which, however, feature CDW entirely within the Q2D planes. 






It has previously been argued that CDWs never truly develop by a Q1D Peierls-type mechanism <cit.>. From this point of view, all compounds with a purported low-dimensional character might more reasonably be thought of as dimensional hybrids, with  simply being a particularly clear example. A consequence can be seen in the behavior of the CDW under pressure, and its interaction with SC. As noted in Ref. <cit.>, increasing P has the effect of increasing interchain (or interplane) coupling, which might have one of two opposite effects on T_CDW depending on how Q1D the system really is. In extreme Q1D systems, T_CDW is heavily suppressed by fluctuations to well below the value predicted by mean-field theory <cit.>. The reduction in interchain spacing (increased interchain coupling) decreases the one-dimensional character of the system, suppressing these fluctuations, and increasing T_CDW. On the other hand, if T_CDW is not dictated by Q1D fluctuations, the primary effect of the increased interchain coupling is to decrease the density of states at the Fermi level, suppressing T_CDW. In the standard BCS picture, the phonon mode formerly suppressed by the CDW could become available to mediate SC at high P and an SC phase could develop, although the resulting SC phase does not necessarily form a dome centered around the vanishing of the CDW. As shown in Fig. <ref> and Fig. S4, such a picture seems to apply to many Q2D and 3D materials (e.g. 2H-NbSe_2 <cit.>, 2H-TaS_2 <cit.>, 1T-TaS_2 <cit.>, 1T-TaSe_2 <cit.>, 1T-TiSe_2 <cit.>, (Ca,Sr)_3Ir_4Sn_13 <cit.>) (see Supplemental Material <cit.>), as well as to  to a certain extent. However, as explained above, we do not rule out the enhancement of T_c due to possible QCP-related fluctuations in  and the possibility of an incommensurate density wave quantum criticality in 2D metals has recently been discussed <cit.>. 

In summary, we report a direct observation of a bulk CDW with its checkerboard wavevectors in . It shows several interesting characteristics: (a) a unique mixed character of dimensionality and (b) the lowest value of the T_CDW/T_c among prototypical CDW materials. Future studies with probes sensitive to both CDW and SC (e.g. High-P Raman spectroscopy, ARPES, Second Harmonic Generation, etc) are desirable in order to fully understand this interesting interplay.




We acknowledge useful discussions with S. H. Simon, and experimental help from J. Strempfer. We are thankful to J. Analytis for supervising the sample synthesis efforts, and to the Analytis Lab for contributing facilities for sample synthesis and single crystal growth. We thank Roger Sommer for help with single crystal X-ray diffraction at the METRIC (Molecular Education, Technology, and Research Innovation Center) at North Carolina State University. We thank Sven Friedemann for providing unpublished data on 2H-NbSe_2 which is shown in the Supplemental Material Fig. S5. This work is based upon research conducted at the Cornell High Energy Synchrotron Source (CHESS) which is supported by the National Science Foundation under award DMR-1332208 and DMR-1829070. Work at Argonne was supported by the U.S. DOE Office of Science, Office of Basic Energy Sciences, under Award No. DE-AC02-06CH11357. F.F. acknowledges support from the Astor Junior Research Fellowship of New College, Oxford. Z.S., W.M.S., S.D., and S.H. acknowledge support provided by funding from the Powe Junior Faculty Enhancement Award, and William M. Fairbank Chair in Physics at Duke University. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida.


S.H. conceived and supervised research; T.H. grew the sample; S.K., J.L., J.R., D.H., and S.H.
conducted X-ray diffraction study at ambient P; Z.S., W.S., S.D., G.F., D.H., and S.H.
conducted X-ray diffraction study at high P; Z.S., D.G., and S.H. conducted dHvA
oscillation measurements; Z.S., F.F., and S.H. wrote the paper.


Z. S. and S. K. contributed equally to this work.






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Peierls1955 R. Peierls, Quantum Theory of Solids. Oxford University Press, London (1955).

Monceau2012 P. Monceau, Electronic crystals: an experimental overview. Adv. Phys. 61, 325 (2012).

Chen2016 C.-W. Chen, J. Choe, and E. Morosan, Charge density waves in strongly correlated electron systems. Rep. Prog. Phys. 79, 084505 (2016).

Gruner1988 G. Grüner, The dynamics of charge-density waves. Rev. Mod. Phys. 60, 1129 (1988).

Klintberg2012 L. E. Klintberg, S. K. Goh, P. L. Alireza, P. J. Saines, D. A. Tompsett, P. W. Logg, J. Yang, B. Chen, K. Yoshimura, and F. M. Grosche Pressure- and composition-induced structural quantum phase transition in the cubic superconductor (Sr,Ca)_3Ir_4Sn_13. Phys. Rev. Lett. 109, 237008 (2012).

Goh2011 S. K. Goh, L. E. Klintberg, P. L. Alireza, D. A. Tompsett, J. Yang, B. Chen, K. Yoshimura, and F. M. Grosche Enhancement of the superconducting transition temperature with hydrostatic pressure in Ca_3Ir_4Sn_13 single crystals. arXiv:1105.3941 (19 May 2011).

Johannes2008 M. D. Johannes and I. I. Mazin, Fermi surface nesting and the origin of charge density waves in metals. Phys. Rev. B 77, 165135 (2008).

Zhu2015 X. Zhu, Y. Cao, J. Zhang, E. W. Plummer, and J. Guo, Classification of charge density waves based on their nature. Proc. Natl. Acad. Sci. U.S.A. 112, 2367 (2015).

Flicker2015 F. Flicker and J. van Wezel, Charge order from orbital-dependent coupling evidenced by NbSe_2. Nat. Commun. 6, 7034 (2015).

Bardeen1957a J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic theory of superconductivity. Phys. Rev. 106, 162 (1957).

Wu2011 T. Wu, H. Mayaffre, S. Krämer, M. Horvatić, C. Berthier, W. N. Hardy, R. Liang, D. A. Bonn, and M.-H. Julien, Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa_2Cu_3O_y. Nat. Commun. 477, 191 (2011).


Wu2015 T. Wu, H. Mayaffre, S. Krämer, M. Horvatić, C. Berthier, W. N. Hardy, R. Liang, D. A. Bonn, and M.-H. Julien, Incipient charge order observed by NMR in the normal state of YBa_2Cu_3O_y. Nat. Commun. 6, 6438 (2015).


Comin2016 R. Comin and A. Damascelli, Resonant X-Ray scattering studies of charge order in cuprates. Annu. Rev. Condens. Matter Phys. 7, 369 (2016).

Moncton1975 D. E. Moncton, J. D. Axe, and F. J. DiSalvo, Study of superlattice formation in 2H-NbSe_2 and 2H-TaSe_2 by neutron scattering. Phys. Rev. Lett. 34, 734 (1975).

Monceau1976 P. Monceau, N. P. Ong, A. M. Portis, A. Meerschaut, and J. Rouxel, Electric field breakdown of charge-density-wave—induced anomalies in NbSe_3. Phys. Rev. Lett. 37, 602 (1976).

Gabovich2002 A. M. Gabovich, A.I. Voitenko, and M. Ausloos, Charge- and spin-density waves in existing superconductors: competition between Cooper pairing and Peierls or excitonic instabilities. Phys. Rep. 367, 583 (2002).

Norman2011 M. R. Norman, The challenge of unconventional superconductivity. Science 332, 196 (2011).

Fradkin2015 E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457 (2015).

Shi2020 Z. Shi, P. G. Baity, T. Sasagawa, and D. Popović, Vortex phase diagram and the normal state of cuprates with charge and spin orders. Sci. Adv. 6, eaay8946 (2020).

Shi2020a Z. Shi, P. G. Baity, J. Terzic, T. Sasagawa, and D. Popović, Pair density wave at high magnetic fields in cuprates with charge and spin orders. Nat. Commun. 11, 3323 (2020).

Lee2020 W. S. Lee, et al., Spectroscopic fingerprint of charge order melting driven by quantum fluctuations in a cuprate. Nature Phys. https://doi.org/10.1038/s41567-020-0993-7 (2020).

Gruner2017 T. Gruner, D. Jang, Z. Huesges, R. Cardoso-Gil, G. H. Fecher, M. M. Koza, O. Stockert, A. P. Mackenzie, M. Brando, and C. Geibel Charge density wave quantum critical point with strong enhancement of superconductivity. Nature Phys. 13, 967–972 (2017).

Jiao2014 W.-H. Jiao, Z. T. Tang, Y. L. Sun, Y. Liu, Q. Tao, C.-M. Feng, Y.-W. Zeng, Z.-A. Xu, and G.-H. Cao, Superconductivity in a layered  with PdTe_2 chains. J. Am. Chem. Soc. 136, 1284 (2014).


Helm2017 T. Helm et al., Thermodynamic anomaly above the superconducting critical temperature in the quasi-one-dimensional superconductor . Phys. Rev. B 95, 075121 (2017).

Du2015 Z. Du, D. Fang, Z. Wang, Y. Li, G. Du, H. Yang, X. Zhu, and H.-H. Wen, Anisotropic superconducting gap and elongated vortices with Caroli-De Gennes-Matricon states in the new superconductor . Sci. Rep. 5, 9408 (2015).


Jiao2016 W.-H. Jiao, L.-P. He, Y. Liu, X.-F. Xu, Y.-K. Li, C.-H. Zhang, N. Zhou, Z.-A. Xu, S.-Y. Li, and G.-H. Cao, Superconductivity in Ta_3Pd_3Te_14 with quasi-one-dimensional PdTe_3 chains. Sci. Rep. 6, 21628 (2016).


Pan2015 J. Pan, W. H. Jiao, X. C. Hong, Z. Zhang, L. P. He, P. L. Cai, J. Zhang, G. H. Cao, and S. Y. Li, Nodal superconductivity and superconducting dome in the layered superconductor . Phys. Rev. B 92, 180505(R) (2015).


Jo2017 N. H. Jo, L. Xiang, U. S. Kaluarachchi, M. Masters, K. Neilson, S. S. Downing, P. C. Canfield, and S. L. Bud'ko, Pressure induced change in the electronic state of . Phys. Rev. B 95, 134516 (2017).


Pang2018 G. Pang, M. Smidman, W. Jiao, L. Jiao, Z. Wang, W. Jiang, C. Guo, Y. Chen, G. Cao, and H. Yuan, Evidence for nodal superconductivity in a layered compound . J. Phys.: Condens. Matter 30, 055701 (2018).


Fan2015 Q. Fan et al., Scanning tunneling microscopy study of superconductivity, magnetic vortices, and possible charge-density wave in . Phys. Rev. B 91, 104506 (2015).

Li2016 Z. Li, W. H. Jiao, G. H. Cao, and G.-q. Zheng, Charge fluctuations and nodeless superconductivity in quasi-one-dimensional  revealed by ^125Te-NMR and ^181Ta-NQR. Phys. Rev. B 94, 174511 (2016).

Singh2014 D. J. Singh, Multiband superconductivity of  from Te p states. Phys. Rev. B 90, 144501 (2014).

Chen2015 D. Chen, P. Richard, Z. D. Song, W. L. Zhang, S. F. Wu, W. H. Jiao, Z. Fang, G. H. Cao, and H. Ding, Raman scattering investigation of the quasi-one-dimensional superconductor . J. Phys.: Condens. Matter 27, 495701 (2015).


Supplemental See Supplemental Material at [URL will be inserted by publisher] for details on experimental methods, dHvA quantum oscillation results, the CDW in , and additional discussions on CDWs in prototypical CDW materials. 

Borisenko2008 S. V. Borisenko, et al., Pseudogap and charge density waves in two dimensions. Phys. Rev. Lett. 100, 196402 (2008).

Yao2006 H. Yao, J. A. Robertson, E.-A. Kim, and S. A. Kivelson, Theory of stripes in quasi-two-dimensional rare-earth tellurides. Phys. Rev. B 74, 245126 (2006).

Chang2012b J. Chang, et al., Direct observation of competition between superconductivity and charge density wave order in YBa_2Cu_3O_6.67. Nature Phys. 8, 871–876 (2012).

Feng2015 Y. Feng, J. van Wezel, J. Wang, F. Flicker, D. M. Silevitch, P. B. Littlewood, and T. F. Rosenbaum, Itinerant density wave instabilities at classical and quantum critical points. Nat. Phys. 11, 865 (2015).


Giambiattista1990 B. Giambattista, C. Slough, W. McNairy, and R. Coleman, Scanning tunneling microscopy of atoms and charge-density waves in 1T-TaS_2, 1T-TaSe_2, and 1T-VSe_2. Phys. Rev. B 41, 10082 (1990).

Lee1973 P. A. Lee, T. M. Rice, and P. W. Anderson, Fluctuation effects at a peierls transition. Phys. Rev. Lett. 31, 462 (1973).

NunezRegueiro1992 M. Núñez-Regueiro, J.-M. Mignot, and D. Castello, Superconductivity at High Pressure in NbSe_3. Europhys. Lett. 18, 53 (1992).

NunezRegueiro1993 M. Núñez-Regueiro et al., Superconductivity under pressure in linear chalcogenides. Synthetic Metals 55-57, 2653 (1993).

Leroux2015 M. Leroux, I. Errea, M. Le Tacon, S.-M. Souliou, G. Garbarino, L. Cario, A. Bosak, F. Mauri, M. Calandra, and P. Rodière, Strong anharmonicity induces quantum melting of charge density wave in 2H-NbSe_2 under pressure. Phys. Rev. B 92, 140303(R) (2015).


Grasset2019 R. Grasset, Y. Gallais, A. Sacuto, M. Cazayous, S. Mañas-Valero, E. Coronado, and M.-A. Méasson, Pressure-Induced Collapse of the Charge Density Wave and Higgs Mode Visibility in 2H-TaS_2. Phys. Rev. Lett. 122 127001 (2019).


Freitas2016 D. C. Freitas et al., Strong enhancement of superconductivity at high pressures within the charge-density-wave states of 2H-TaS_2 and 2H-TaSe_2. Phys. Rev. B 93 184512 (2016).

Sipos2008 B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger, L. Forr, and E. Tutis, From Mott state to superconductivity in 1T-TaS_2. Nature Mater. 7, 960 (2008).


Wang2017 B. Wang et al., Pressure-induced bulk superconductivity in a layered transition-metal dichalcogenide 1T-tantalum selenium. Phys. Rev. B 95, 220501(R) (2017).

Kusmartseva2009 A. F. Kusmartseva, B. Sipos, H. Berger, L. Forró, and E. Tutič, Pressure induced superconductivity in pristine 1T-TiSe_2. Phys. Rev. Lett. 103, 236401 (2009).

Halbinger2019 J. Halbinger, D. Pimenov, and M. Punk, Incommensurate 2k_F density wave quantum criticality in two-dimensional metals. Phys. Rev. B 99, 195102 (2019).



