Quasi-one-dimensional superconductivity in the pressurized charge-density-wave conductor HfTe3

HfTe3 single crystal undergoes a charge-density-wave (CDW) transition at TCDW = 93 K without the appearance of superconductivity (SC) down to 50 mK at ambient pressure. Here, we determined its CDW vector q = 0.91(1) a* + 0.27(1) c* via low-temperature transimission electron microscope and then performed comprehensive high-pressure transport measurements along three major crystallographic axes. Our results indicate that the superconducting pairing starts to occur within the quasi-one-dimensional (Q1D) -Te2-Te3- chain at 4-5 K but the phase coherence between the superconducting chains cannot be realized along either the b- or c-axis down to at least 1.4 K, giving rise to an extremely anisotropic SC rarely seen in real materials. We have discussed the prominent Q1D SC in pressurized HfTe3 in terms of the anisotropic Fermi surfaces arising from the unidirectional Te-5px electronic states and the local pairs formed along the -Te2-Te3- chains based on the first-principles electronic structure calculations.


Introduction
Charge density wave (CDW) and conventional superconductivity (SC) are both caused by strong electron-phonon coupling and Fermi surface (FS) instabilities, and thus the interplay between these two collective electronic phenomena has been a subject of extensive investigations over the past decades [1]. The former usually occurs in the low-dimensional materials and involves a periodic modulation of the conduction electron density and the underlying lattice; this will open a gap over part of the FS and thus lead to reduced conductivity. In contrast, bulk SC is a macroscopic quantum phenomenon that requires phase coherence in three dimensions and it features a FS gap that results in infinite conductivity. As such, these two electronic orders are usually competing with each other. It is frequently observed in the low-dimensional CDW conductors that suppression of CDW order leads to a bulk SC. Here, we report on a rare case that an intrinsic quasi-one-dimensional (Q1D) filamentary SC emerges and persists in a relatively wide temperature/pressure range when the CDW order of HfTe3 along the same crystallographic direction is completely suppressed by pressure.
As shown in Fig. 1(a), HfTe3 crystallizes in the TiS3-type structure with monoclinic space group P21/m, featured by two kinds of chain-like units, i.e. the infinite prismatic (HfTe3)  chains along the b-axis and the almost linear chains of -Te2-Te3-with alternating short (2.80 Å) and long (3.07 Å) distances along the a-axis [2]. The twodimensional (2D) prism slabs in the ab plane are stacked along the c-axis through weak van der Waals (vdW) interactions. These quasi-1D and 2D structural characters have a profound impact on the physical properties of HfTe3 at ambient pressure. Resistivity measurements on high-quality HfTe3 single crystals reveal pronounced anisotropic behaviors and obvious anomalies in both a(T) and c(T) but a very weak kink-like anomaly in b(T) at around 93 K [2]. In analog with the sister compound ZrTe3 [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], the transport anomaly at TCDW = 93 K has been ascribed to the formation of a CDW order within the ac-plane [6]. Since the c is about two orders of magnitude higher than a, the CDW should have a dominate 1D character along the -Te2-Te3-infinite chains.
However, the wavevector of the CDW order has not been determined experimentally.
For high-quality single crystalline HfTe3 samples, no SC was detected down to 50 mK at ambient pressure [2]. In contract, the polycrystalline HfTe3 sample shows the coexistence of the CDW order below TCDW  80 K and filamentary SC with Tc  2 K at ambient pressure [2]. The resistivity anomaly around TCDW is much reduced in comparison with that of single crystal, implying that the emergence of SC should be correlated with the suppression of CDW, presumably due to the presence of chemical disorders or defects in the polycrystalline samples. However, the nature of the filamentary SC without magnetic and thermodynamic signatures and its relationship with CDW remains elusive so far. For the polycrystalline sample, Denholme et al. [20] have performed preliminary high-pressure (HP) resistivity measurements up to ~1 GPa; they observed pressure-induced enhancement of the CDW transition, but the evolution of SC and its relationship with respect to CDW was not reported [20]. Considering the strong anisotropy of the physical properties associated with the Q1D nature of the crystal structure, it is imperative to carry out a comprehensive HP study on singlecrystalline samples in order to unveil the nature of the filamentary SC and its interplay with CDW.
In this work, we first determined experimentally the CDW vector q = 0.91(1) a* + 0.27(1) c* of HfTe3 by low-temperature transmission electron microscope (TEM) at ambient pressure, and then performed a comprehensive HP transport study on the HfTe3 single crystals along three major crystallographic axes. Intriguingly, we observed the emergence of an intrinsic filamentary SC in Ra(T) at T  4-5 K when the CDW of HfTe3 is suppressed completely by P  5 GPa, whereas no clear sign of SC is observed down to 1.4 K in Rb(T) and Rc(T), demonstrating an extraordinary case of Q1D SC rarely seen in real materials. We have analyzed the excess conductivity according to the theory of Aslamazov and Larkin (A-L) [21] and compared our results with those of ZrTe3 [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]22]. Finally, we discussed the observed Q1D SC in terms of the anisotropic FSs and the local pairs formed along the -Te2-Te3-chains parallel to the a-axis based on the first-principles electronic structure calculations.

A. Characterizations at ambient pressure
The single-crystal XRD pattern of HfTe3 shown in Fig. 1(b) displays only the sharp (00l) diffraction peaks, which confirm that the single crystals are grown along the ab plane and stacked along the c-axis. The c-axis lattice constant is determined to be c = 10.0519 Å, in consistent with the previous report [2]. As shown in the inset of Fig. 1(b), the full width at half maximum (FWHM) of the (006) Bragg peak is about 0.2°, implying a high quality of the crystals. Within the plate-like crystal surface, the a and b axes are determined to be perpendicular and parallel to the crystal edges, respectively, by the Laue back diffraction. As shown in the inset of Fig. 1(c), the HfTe3 single crystal shows fringes extending along the b-axis, which is parallel to the crystal edge. Thus, the aand b-axis that are orthogonal with each other within the ab plane can be distinguished visually for the as-grown crystals. For this sample, we measured the temperature dependence of anisotropic resistance R(T) along the aand b-axis, respectively. As seen in Fig. 1(c), the a-axis resistance exhibits a pronounced hump anomaly at TCDW = 93 K associated with the CDW transition, while the b-axis resistance shows a very weak kink-like anomaly at TCDW without any trace of superconducting transition down to 1.8 K. These results agree well with the previous report [2]. The residual resistivity ratio, RRR  R(300 K)/R(2 K) along the b-axis is calculated to be 62, which also highlights the high quality of the studied single-crystal sample. We have thus selected samples with similar RRR values for the HP measurements as shown in the Supplementary Fig. 1.
Although it is expected that HfTe3 should form a similar CDW as that of ZrTe3 with q = 0.93 a* + 0.33 c*, the q-vector of HfTe3 has not been determined experimentally so far. Here we employed a low-temperature TEM to directly determine the CDW vector of HfTe3 single crystal. Figure 2(a) and (b) show the selected area electron diffraction (SAED) patterns of HfTe3 taken along the [001] zone axis at room temperature and 26 K, respectively. The pristine lattice at room temperature has a monoclinic structure with space group P21/m, and the satellite spots can only be identified below the CDW phase transition temperature. As shown in the inset of Fig. 2(b), the intensity of satellite spot goes up as the scattering angle increases and reaches a maximum at 17.5 mrad, indicating that the structural modulation vector q has both a* and c* components, i.e., q = qa a* + qc c*. The extra Laue zones (indicated by blue arrows) presented in the [001] CBED pattern, Fig. 2(c), further confirm that the modulation vector does not lie in the ab-plane.
In order to determine the value of qa and qc, the intensity distributions of CDW spots/rings in Fig. 2(b) and (c) are summarized in Fig. 2(d) and (e), respectively. Figure  2(d) shows the line intensity profile from spot (6 ̅ 00) to (10 ̅̅̅̅ 00) , in which the intervals between adjacent Bragg spots (LBragg) and the distance from one Bragg spot to the next-nearest CDW neighbor (LCDW) are labeled. Using qa = LCDW/LBragg, qa is calculated to be 0.91 (1). On the other hand, the value of qc can be estimated with the aid of the geometric configuration presented in Fig. 2 To further understand the temperature-dependent behavior of the structural modulation, we acquired SAED patterns at several different temperatures, as shown in Supplementary Fig. 2. It is evident that the superlattice spots (indicated by blue arrows) become dim at 81 K and totally disappear at 88 K, showing good agreement with the CDW phase transition temperature determined from the resistivity measurements. The result suggests that the superlattice spots observed in diffraction patterns are directly related to the formation of CDW phase.
It is worth noting that the satellite spots appeared in the electron diffraction patterns of ZrTe3 at room temperature. Eaglesham et al. [6] believed that the satellite spots at room temperature indicated the composition modulation of ZrTe3, which were caused by offstoichiometry. This is also considered to be a strong evidence that the quality of ZrTe3 crystals is difficult to control. For HfTe3 single crystal, we can see that the SAED patterns do not show any satellite spots at room temperature, which indicates that the crystal structure of HfTe3 single crystal is perfect and closer to the stoichiometry.  Fig. 4. At ambient pressure, Ra(T) displays a hump anomaly due to the CDW formation at TCDW = 93 K, which can be determined from the minimum of dRa/dT as shown in the inset of Fig. 3(a). With increasing pressure, TCDW first moves to higher temperatures, reaching the maximum of ~128 K at 1.5 GPa, and then decreases gradually to 122.4 K at 2.01 GPa. The jump of Ra(T) around TCDW is reduced gradually accompanying the suppression of CDW transition. Such a non-monotonic variation of TCDW(P) has also been observed in ZrTe3, [12] and thus should have an origin in common. The initial enhancement of TCDW at low pressures indicates that the lattice compression first prompts the CDW formation, presumably due to the enhancement of electron-lattice coupling or the optimization of the nesting condition for the parallel FSs along the a*-axis. Nonetheless, the CDW should be eventually suppressed by pressure as observed in ZrTe3 [12].

B. Effect of pressure on the a-axis resistance
The evolution of the CDW transition at P > 2 GPa is further investigated by measuring Ra(T) of a second HfTe3 crystal (#2) with a CAC. The obtained Ra(T) and the selected dRa/dT curves are displayed in Fig. 3(b). The Ra(T) at 2.4 GPa is similar to that measured in PCC, and the TCDW has been further reduced to 93.8 K. When the pressure is increased to 4 GPa, the TCDW is quickly lowered to 43.3 K and the resistance anomaly at TCDW is significantly suppressed, as shown in Fig. 3(b). At 5.4 GPa and above, no resistance anomaly can be discerned any more at T > 5 K, implying the complete suppression of the CDW transition in HfTe3. In this pressure range, the magnitude of Ra(T) at high temperatures decreases gradually with increasing pressure while that at low temperatures keeps almost constant.
Interestingly, we observed a gradual resistance drop at low temperatures for P > 5.4 GPa, signaling the possible emergence of SC accompanying the elimination of CDW in HfTe3. As shown in Fig. 3(c), the enlarged view of Ra(T) at T  6 K evidences a slight downturn feature below Tc onset  3 K at 5.4 GPa. Upon further increasing pressure, the drop of resistance becomes stronger and the Tc onset moves gradually to higher temperatures, reaching a nearly constant value of ~5 K at P  9.1 GPa. The zero resistance is finally achieved at Tc zero  2-2.5 K at P  11.4 GPa. We found that the resistance drop in Ra(T) can be suppressed by applying either high electric currents and/or external magnetic fields, as seen in Fig. 4(a) and (b). These observations are consistent with the occurrence of SC in HfTe3 at P  5.4 GPa when the CDW transition along the a-axis is suppressed completely. It should be noted that the resistance data became too noisy under magnetic fields higher than 0.3 T, which prevented us from determining precisely the upper critical field.
In order to ensure the reproducibility of the experimental results, we performed HP resistance measurements on a third HfTe3 (#3) single crystal along the a-axis. As shown in Fig. 3(d, e), the CDW transition at 93 K is clearly observed for Ra(T) at 2.5 GPa and is completely suppressed at 5 GPa; the SC transition is also confirmed at low temperature for P  5 GPa. It is noted that the observed Tc for this sample is slightly lower than that of sample #2, implying that the pressure-induced SC in HfTe3 is sensitive to the sample quality. As shown in Supplementary Fig. 5, the RRR values of sample #2 are indeed larger than those of sample #3 in the pressure range when CDW is suppressed completely. Accordingly, the Tc onset of sample #2 is also higher than that of sample #3. Nevertheless, the results of sample #3 are in general agreement with those of the #2 sample. Furthermore, we also measured the low-temperature resistance at 11 and 12.5 GPa under different electric currents and external magnetic fields. As shown in Fig

C. Effect of pressure on the b-axis resistance
Because the samples #2 and #3 measured above are always disassembled into many filaments after decompression from high pressures, presumably due to the special sample microstructure and the presence of residual stress upon compression. This issue prevents the subsequent measurements along the b-axis direction on the identical samples. Under such a circumstance, we selected another two HfTe3 samples #4 and #5 for measurements of the b-axis resistance Rb(T) in two separated runs in the pressure range 7-12.5 GPa, where zero resistance can be achieved in Ra(T). The results of Rb(T) for these two samples are shown in Fig. 5(a). As seen clearly, the Rb(T) curves of both samples do not exhibit any anomaly at the temperatures of ~4-5 K where Ra(T) starts to show the superconducting transition. For sample #5, we observed a slight downturn of Rb(T) starting at T < 2 K under P  10 GPa, Fig. 5(b), which is likely caused by some misalignment of the electrical contact so that some a-axis component was picked up.
These results are striking. First of all, they immediately exclude the possibility that the pressure-induced resistivity drop in Ra(T) arises from a secondary phase. Secondly, the observed SC in Ra(T) is very likely a Q1D phenomenon. As shown below, our simulations with a 2D resistor matrix can well reproduce the above Rb(T) results. To substantiate this conclusion, we further measured the c-axis resistance Rc(T) under various pressures.
Here, we want to point out that it is quite challenging experimentally to collect merely the b-axis resistance without any a-axis contribution for the resistance measurements on a bulk sample. This is particularly difficult given the fact that the a-axis resistance exhibits a pronounced drop around 4-5 K whereas the b-axis resistance almost levels off. Therefore, a slight misalignment of the voltage leads V+/V-with respect to the aaxis, i.e., they should be perfectly parallel with the a-axis, will inevitably introduce the a-axis feature, overshadowing the intrinsic response of b-axis resistivity. Under such a circumstance, the chance to observe a resistivity drop along the b-axis is high, but it should be convincing if we can observe no obvious resistivity drop by chance. To overcome this problem, we tried to first pattern the electrical contacts by evaporating the gold pads on the sample surface so as to make them as parallel to the a-axis as possible, see Supplementary Fig. 4(b), then we attached the electrical leads with silver paste. In this way, we are able to observe no any resistance drop for sample #4 and a tiny drop for sample #5 as shown in Fig. 5(b).

D. Effect of pressure on the c-axis resistance
We employed the configuration shown in the inset of Fig. 5(c) to measure precisely the Rc(T) of the plate-like HfTe3 crystal (#6). The electric current flows through the sample via the U-shaped I+/I-pads on the opposite sides of the crystal, while the voltage leads V+/V-are attached to the center pads on both sides. At ambient pressure, Rc(T) exhibits an upturn upon cooling through TCDW [2]. As seen in Fig. 5(c), the resistance upturn below TCDW is quickly suppressed by pressure, and the Rc(T) at 2.4 GPa displays a metallic behavior in the whole temperature range with a clear kink-like anomaly at TCDW  96.5 K, which is consistent with that determined from Ra(T) in Fig. 3(b). The CDW transition cannot be discerned any more at P  4.2 GPa, implying the complete suppression of CDW by pressure. In the investigated pressure range, we observed no emergence of possible SC down to 1.4 K. Interestingly, the Rc(T) curves at P  5.8 GPa even exhibit an unusual upturn below around 2.5 K, as shown in Fig. 5(d). The upward trend of Rc(T) below 2.5 K is more obvious when the pressure increases to 11.2 GPa. As shown below, these results can be well accounted for by turning on a Q1D SC in the resistor-matrix model. These above results thus unambiguously demonstrate the emergence of an extremely anisotropic SC in HfTe3 under pressures.

E. T-P phase diagram of HfTe 3
The pressure dependencies of TCDW and Tc determined from Ra(T) are shown in Fig. 6, which depicts clearly the competition between CDW and the possible Q1D SC along the a-axis. With increasing pressure, TCDW initially increases, reaching a broad maximum at 1.5 GPa, then decreases quickly and vanishes completely at about 5 GPa, above which the Q1D SC emerges abruptly. The Tc onset increases gradually and reaches a nearly constant value of ~ 5 K at least up to 12 GPa. The zero resistance is eventually realized at Tc zero  2 K under P > 11 GPa, however, the diamagnetic response cannot be obtained due to the lack of phase coherence along the b-and c-axis.
Therefore, our present work not only demonstrates unambiguously a competitive nature between CDW and SC but also reveals an intriguing anisotropic SC in the pressurized HfTe3, i.e., the Q1D SC emerges along the a-axis in a relatively broad temperature/pressure range after the CDW is completely suppressed, whereas the phase coherence between the superconducting chains cannot be realized down to 1.4 K along the other directions. One noteworthy feature of the phase diagram shown in Fig. 6 is that SC emerges along the a-axis only after the CDW is suppressed completely by pressure without showing any coexistence region. In contrast, the polycrystalline HfTe3 sample with at a lower TCDW  80 K shows the coexistence of CDW and filamentary SC at ambient pressure [2]. Although both the chemical disorders/defects and the physical pressure can suppress CDW and then induce SC, the above comparison actually highlights their distinct roles in tuning the CDW and SC states. In comparison with the "clean" tuning of physical pressure, the chemical disorders/defects may introduce extra charge carriers. According to the theoretical study by Imada et al. [23], the relationship between CDW and SC in a Q1D system depends sensitively on the band filling, i.e. they are separated for half-filled band but can coexist for quarter filling. They also pointed out that the coexistence of SC and CDW in a non-half-filled system is a general feature of anisotropic systems. It is plausible that the Q1D bands of HfTe3 along the a-axis is close to half filling, leading to a complete competitive relationship between SC and CDW, while the deviation from the half-filling in the polycrystalline samples gives rise to the coexistence of CDW and SC. Our simple calculations on the density of states (DOS) distribution of the px/py/pz type Wannier orbits on the -Te2-Te3-atoms seem to support such a scenario, i.e., the filling numbers of the px-type Wannier orbits along the 1D atom chain direction are about 0.55, Supplementary Fig. 14. It should be noted that the valence state of Te ions in HfTe3 is complex, and more experimental and theoretical investigations are needed to understand the interplay between CDW and SC.

F. Crystal structure under high pressure
To study the structural stability of HfTe3 under compression, we also measured the highpressure XRD on the pulverized HfTe3 crystals at room temperature and found no structure transition up to at least 14 GPa. As shown in Fig. 7, the lattice parameters a, b, c, and volume V contract smoothly with increasing pressure, implying that the crystal structure is stable in the studied pressure range. The pressure dependence of volume V(P) can be well fitted with the Birch-Murnaghan (B-M) equation of state as shown by the solid line in Fig. 7(b). By fixing B1 = 4, we obtained a bulk modulus B0 = 45.3(7) GPa and V0 = 228.7(4) Å 3 . In addition, there is no obvious anomaly for the lattice parameter a at Pc  5 GPa, where the CDW disappears and the Q1D SC emerges at low temperatures. The axial ratios c/a and b/a exhibit the monotonic decrease with the pressure gradually increasing (see Supplementary Fig. 7), indicating the b and c axis is more compressible than a-axis. The smooth evolutions of the axial ratios c/a and b/a up to ~ 10 GPa also indicate that the crystal structure is stable within the studied pressure range. Therefore, we can safely conclude that pressure-induced SC around 5 GPa should be correlated with an electronic transition rather than a structural transition, i.e., it is linked directly with the elimination of the CDW order. In this sense, the pressure-induced SC along the a-axis can be regarded as a new electronic phase, which shares similar crystal structure as the ambient phase but has no CDW instability.

Discussions
In order to simulate the above anisotropic transport behaviours at low temperatures under HP, we designed a simplified 2D resistor matrix consisting of 45 resistors as shown in Supplementary Figs. 8-11. We first simulate the change of b-axis resistance Rb with a perfect alignment upon entering the superconducting state along the a-axis, manifested by a reduction of the resistors' resistance from 5 to 0.1 Ω along this direction. As shown in Supplementary Fig. 8, the voltage drop across V+/V-along the b-axis (or equivalently Rb) remains unchanged when the resistance of resistors along the a-axis are reduced from 5 to 0.1 Ω, consistent with the experimental results for sample #4 in Fig. 5(b). However, when the alignment of voltage leads is not perfect, some a-axis component will be picked up as simulated in Supplementary Fig. 9. In this case, the voltage drop across V+/V-is found to decrease slightly when the resistors along the aaxis are reduced from 5 to 0.1 Ω. This reproduces well the observed resistance drop of sample #5 shown in Fig. 5(b), confirming that the change of Ra will affect the Rb sensitively if the alignment of voltage leads is imperfect.
We further simulated the influence of resistance drop of Ra and the misalignment of V+/V-on the c-axis resistance Rc. As illustrated in Supplementary Figs. 10 and 11, interestingly, the voltage drop across V+/V-along the c-axis (or equivalently Rc), no matter aligned perfectly or not, was both found to rise progressively when the resistance of the resistors along the a-axis is reduced gradually, in excellent agreement with the experimental results shown in Fig. 5(d). In contrast, the voltage drop will decrease if the superconducting transition takes place in the c-axis direction. These above simulations not only reproduce the experimental observations, but also support strongly the occurrence of a Q1D SC in the pressurized HfTe3 single crystal.
To further substantiate the occurence of Q1D SC along the a-axis, we then analyzed the excess condcutivity (∆ = − 0 ) around Tc according to A-L model [21], viz.
where D is the dimension and Tc is the mean-field transition temperature, which equal to Tc zero . For D =1, the excess conductivity due to 1D superconducting fluctuations is given by: Here, e is the electron charge, d is the diameter of a superconducting filament, and (0) is the coherence length at T = 0 K. As illustrated in Fig. 8, the excess conductivity ∆ from 9 to 12.5 GPa for sample #3 measured at a smaller current 0.1 mA indeed follows a nice linear relationship as a function of [( − )/ ] −3/2 ; here the ∆ is calculated from data shown in Fig. 3(e) and 0 is the conductivity at 6 K for I//a with 0.1 mA current. These observations indicate that strong 1D superconducting fluctuations persist even up to 12.5 GPa. It is also noteworthy that the slope of the fitting curve decreases gradually with increasing pressure with a ratio of 9 GPa : 11 GPa : 12.5 GPa = 2.4: 1.4: 1. As shown in Fig. 3(e), the Tc in the investigated pressure range only changes slightly, and thus the change of (0) should not dominate the significant decrease of the slope, or the pre-factor of the above Eq. (2). Instead, the reduction of the slope should be mainly attributed to the increase of the diameter d of the superconducting filaments under pressure. This is conceivable considering the fact that the superconducting filaments are getting closer under HP. It can be expected that the superconducting state will gradually evolve into three dimensions under higher pressure or lower temperatures where the Josephson coupling between filaments are strengthened.
Before discussing the possible origins for the observed Q1D SC in the pressurized HfTe3, it is impertive to understand the nature of the CDW transition and its interplay with SC. A side-by-side comparison between HfTe3 and ZrTe3 is also instructive. At ambient pressure, ZrTe3 and HfTe3 exhibit a CDW transition at TCDW = 63 K and 93 K, respectively. The determiend CDW vector of HfTe3, i.e. q = 0.91(1) a* + 0.27(1) c* is very similar to that of ZrTe3, i.e. q = 0.93 a*+ 0.33 c* [6]. In general opinion, the CDW vector is related with both the FS nesting vector and the electron-phonon coupling distribution. As shown in Fig. 9, the calculated electronic susceptibility ′ of HfTe3 demonstrates an obvious peak at 0.90 a*, which is in excellent agreement with the experimental result determined from the TEM measurements. Nevertheless, there is no obvious peak along the c* direction, so it is difficult to determine the c* component of q merely based on the information of FS nesting. We suspect that the electron-phonon coupling in HfTe3 may play a dominant role in the c* component of its CDW vector. In comparison, the calculated electronic susceptibilities of ZrTe3 under 0 and 5 GPa are shown in Supplementary Fig. 13. The red regions were drawn by the contour line of calculated electronic susceptibility ′, while the experimentally determined CDW vectors q were also labeled. Obviously, the c* components of experimental q do not correspond to the peaks of calculated ′, while their a* components seem to be well matched. This is similar to the above case of HfTe3.
In specific, a starts to decrease at Tc onset  4 K and reaches zero at Tc zero  2 K with a relatively broad superconducting transition, while b exhibits a much narrower transition from Tc onset  2 K to Tc zero  1.5 K. Based on the specific-heat measurements, the observed anisotropic superconducting transition has been ascribed to a gradual crossover from a filamentary SC (between 4 and 2 K) induced by local pairs along the a-axis to a bulk SC (below 1.5 K) when the superconducting filaments form phase coherence [13]. Such a mixed bulk-filament SC observed in ZrTe3 has been attributed to the peculiar electronic structures characterized by the presence of Q1D and 3D FSs [10,13,19]. In striking contrast to ZrTe3, no SC was detected down to 50 mK for the high-quality HfTe3 single crystals [2]. Their distinct low-temperature behaviors can be rationalized by comparing to the effects of physical pressure on the CDW and SC of ZrTe3. At ambient conditions, the cell volume of HfTe3 (228.70 Å 3 ) is about 1.35% smaller than that of ZrTe3 (231.83 Å 3 ), [8,20] which corresponds to the application of 0.84 GPa hydrostatic pressure on ZrTe3 given a compressibility of 0.016 GPa -1 [14]. At this pressure, it was found that the CDW transition of ZrTe3 is enhanced to ca. 100 K while the SC is suppressed completely [19,22], which are similar to the situations seen in HfTe3 at ambient pressure.
These above simple arguments indicate that the coexistent CDW and SC in ZrTe3 at ambient pressure are actually competing with each other, which becomes more evident under pressure. For ZrTe3, a reentrant SC emerges above 5 GPa when the CDW disappears abruptly; the observed SC was believed to be bulk in nature [17], but the diamagnetic evidence is still lacking to date as mentioned above [12]. In the present case, we observed similar pressure-induced SC in Ra of HfTe3, but the absences of SC along the other directations and the absence of diamagnetic signals point to an intrinsic filamentary character for the observed SC. Our work thus calls for a reexamination of the anisotropic properties of ZrTe3 under high pressures. Our DFT calculations also provide some clues for the suppression of CDW under pressure. As shown in Fig. 9(c, d), the distances between the Te2 and Te3 atoms in the -Te2-Te3-chain along the a-axis becomes more uniform at high pressures. In addition, there are more localized electrons between the Te2 and Te3 atoms from different prismatic (HTe3)  chains. Finally, we discuss the possible origins for the observed exteremly anistorpic or Q1D SC in the pressurized HfTe3 according to the proposal by Yamaya et al. [13] Similar to ZrTe3 [8,9], our electronic structure calculations show that HfTe3 owns both 3D FSs around the BZ center and Q1D FSs around the BZ boundary, as shown in Fig. 10(a). The former ones are dominated by Hf 5d orbitals in the (HfTe3)  prism, while the latter ones are mainly contributed by unidirectional Te-5px orbitals within the -Te2-Te3-Te2-Te3-chain along the a-axis, Fig. 1(a). At the intersection area between these two types of FSs, there are also Q1D+3D FSs as indicated by the black arrows in Fig. 10(a). On these FSs, the Fermi velocities are displayed with the color scales, where the red and blue colors represent the highest and zero Fermi velocities, respectively. In particular, the low Fermi velocities near the Q1D+3D FSs with narrow band feature may result in van Hove singularity (vHs) in the density of states. It is known that the presence of a vHs at EF can induce FS instabilities and lead to SC or CDW states. In the narrow bands that construct the Q1D+3D FSs, electrons can form local pairs that interact attractively with each other over short distances [13]. With increasing pressure, as shown in Fig.  10(b), the 3D FSs of HfTe3 around the Brillouin zone center expand, which enhance their overlapping with the Q1D FSs and reduce the composition of the Q1D bands at the Fermi level. As a result, the FS nesting of Q1D FSs along the a* direction is weakened under pressure, which may suppress the CDW order and induce the SC. Since the narrow-band electron of HfTe3 also has a dominant Te-5px characteristic originating in the -Te2-Te3-chains as in ZrTe3, the local pairs must be dominantly formed along the a-axis, leading to the Q1D SC transition in the a-axis direction [19]. In comparison, the 3D FSs of ZrTe3 are larger than those of HfTe3 at 0 GPa and even become the openorbital pockets along the a* direction at 5 GPa, as shown in Fig. 10 (c-d). This indicates that the CDW order due to the Q1D FSs nesting in ZrTe3 is not so strong as in HfTe3, which is consistent with the tendency of the experimental TCDW of ZrTe3 (63 K) [6] versus HfTe3 (93 K). Therefore, the anistropic SC property in HfTe3 can maintain in a wider temperature/pressure range, which makes HfTe3 being an ideal platform to study the local pairs in the Q1D superconducting system.
In summary, we have measured the anisotropic resistances of the CDW conductor HfTe3 under high pressures. We found that a Q1D SC emerges in Ra(T) at T < 4~5 K when the CDW is suppressed completely by P > 5 GPa, whereas no clear sign of SC is observed down to 1.4 K in Rb(T) and Rc(T). Our results demonstrate an intimate interplay between CDW and SC in HfTe3, which are particularly interesting given their Q1D characters. We have compared our results with those of ZrTe3 and discussed the pressure-induced Q1D SC in pressurized HfTe3 in terms of the peculiar electronic structures and the local pairs formed along the -Te2-Te3-chains parallel to the a-axis.

Single-crystal growth and characterization
High-quality HfTe3 single crystals used in the present study were grown with the iodine chemical-vapor-transport method as described elsewhere [2]. Before HP measurements, the crystals have been carefully characterized at ambient pressure by single-crystal Xray diffraction (XRD), energy dispersive spectroscopy (EDS), TEM, and resistivity measurements. The CDW vector q at ambient pressure was determined by TEM measurements at 26 K. The specimens for TEM observation were prepared by mechanical exfoliation at room temperature and then transfer onto the Cu grid with the aid of crystalbond and acetone. In-situ low-temperature TEM experiments were performed using a liquid-helium-cooled specimen holder (GATAN, HCHDT3010) in a JEM-2100F electron microscope equipped with a charge-coupled device (CCD) camera system and operated at 200 keV.

Details of high-pressure measurements
The temperature dependences of resistance R(T) under various pressures were measured with the standard four-probe method by using either a self-clamped pistoncylinder cell (PCC) [24] in the pressure range 0 < P  2 GPa or a palm-type cubic anvil cell (CAC) [25] in the pressure range 2 < P  12 GPa. In order to improve the electrical contact, we first evaporated four gold pads on the sample surface in vacuum and then attached four gold wires of 20 m in diameter on top of the gold pads with the silver paste. AC magnetic susceptibility under pressure has been measured with the mutual induction method. The excitation and pick-up coils are made of enameled cooper wires of 25 m in diameter, and they were manually wound around a Teflon sheet with a rectangular cross section of 0.3 mm × 0.5 mm. The Teflon sheet can be removed easily and the rectangular space ensures a large filling factor for the susceptibility measurements. Each coil has two layers and about 40 turns in total. The HfTe3 sample and a piece of lead (Pb) with similar size of about 0.3 × 0.3 × 0.25 mm 3 were put inside the pick-up coil. An excitation current of 1 mA with frequency of 1117 Hz was applied to the excitation coil and the output signal as picked up with a Stanford Research SR830 lock-in amplifier. For these HP measurements, the Daphne 7373 or glycerol was employed as the pressure transmitting medium. The pressure in PCC was determined from the superconducting transition temperature of Pb, while the pressure in CAC was estimated from the calibration curve at low temperatures [26]. High-pressure synchrotron powder XRD (λ = 0.6199 Å) was performed at room temperature at 4W2 beamline, Beijing Synchrotron Radiation Facility (BSRF). Silicon oil was used as the transmitting medium. The Le Bail method was employed to fit the XRD data with the General Structure Analysis System (GSAS) program package program. [27] The ruby fluorescence method was used to determine the pressure.

Theoretical calculations
To investigate the electronic structures of HfTe3 under pressure, we carried out density functional theory (DFT) calculations with the projector augmented wave method [28,29] as implemented in the VASP package [30][31][32]. The generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) type [33] was adopted for the exchange-correlation functional. The kinetic energy cutoff of the plane wave basis was set to 300 eV. A 16 × 16 × 8 k-point mesh was used for the Brillouin zone (BZ) sampling. The Fermi surface was broadened by the Gaussian smearing method with a width of 0.05 eV. The spin-orbit coupling effect was included as both Hf and Te are heavy elements. The vdW interaction was described by the optB86b-vdW functional [34]. In the structural optimization, both cell parameters and internal atomic positions were allowed to relax until the forces on all atoms were smaller than 0.01 eV/Å. The Fermi surface (FS) and Fermi velocity of HfTe3 were studied with the Wannier90 [35] and FermiSurfer [36]

Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.           3. The Ewald sphere can only intersect the CDW relrod centers at relatively large scattering angles, which is accompanied by the dimmed Bragg spots. As a result, the intensity decreases monotonously from Bragg spot (6 00) to (10 00), while the CDW intensity reaches its maximum in the vicinity of (7 00), as shown in Fig. 2(d) of the main text.
Supplementary Figure 3. Schematic of the Ewald sphere and diffraction conditions.
The intensity recorded depends on the distance of the relrod center to the intersection with the Ewald sphere. The black and gray ellipses represent the Bragg and CDW relrods, respectively.

Supplementary Notes2: High-pressure measurements
The picture of a single crystal HfTe3 shown in Supplementary Fig. 4. shows fringes extending along the b-axis, which is parallel to the crystal edge. Then, the a-axis is perpendicular to the crystal edge. For our resistivity measurements along the a-or baxis, we have cut the sample in such a way that the longest dimension of about 0.5 mm is along the axis to be measured. To improve the electrical contact, we first evaporated four parallel gold pads on the top surface of the crystal, as indicated by the red dotted frames in Supplementary Fig. 4(b), and then attached 20 m gold wires on the pads with silver paste. Supplementary Figs. 4(b, c) show the photos of the samples measured along the b-and a-axis, respectively. The distances marked on the photos were measured directly by a stereomicroscope. It is noted that the sample along the a-axis, Supplementary Fig. 4(c), was cut a bit wider because a narrower sample would disassemble into many fibers due to the peculiar microstructure of the sample. We have employed the configuration shown in Supplementary Fig. 4