Superconductivity in Ta₃Pd₃Te₁₄ with quasi-one-dimensional PdTe₂ chains

Abstract

We report bulk superconductivity at 1.0 K in a low-dimensional ternary telluride Ta₃Pd₃Te₁₄ containing edge-sharing PdTe₂ chains along crystallographic b axis, similar to the recently discovered superconductor Ta₄Pd₃Te₁₆. The electronic heat capacity data show an obvious anomaly at the transition temperature, which indicates bulk superconductivity. The specific-heat jump is ΔC/(γ_nT_c) ≈ 1.35, suggesting a weak coupling scenario. By measuring the low-temperature thermal conductivity, we conclude that Ta₃Pd₃Te₁₄ is very likely a dirty s-wave superconductor. The emergence of superconductivity in Ta₃Pd₃Te₁₄ with a lower T_c, compared to that of Ta₄Pd₃Te₁₆, may be attributed to the lower density of states.

Introduction

Superconductivity (SC) in low-dimensional systems attracts sustained attention in SC community. The discovery of first layered cuprate superconductor (La, Ba)₂CuO₄¹, has set off a wave of exploring high-T_c superconductors. Since after, a number of new superconductors with low dimensional structures, such as quasi-two-dimensional (Q2D) strontium ruthenate², ferroarsenides³, bismuth oxysulfides⁴, quasi-one-dimensional (Q1D) transition-metal chalcogenides⁵, ternary tellurides^6,7 and newly discovered chromium-based compounds⁸, were reported to display the features of novel SC. The spin (charge) fluctuations^9,10, strong electron-electron correlations¹¹, or metal-insulator boundaries¹² among low-dimensional systems constitute the newly strategic prerequisites to explore high-T_c superconductors. Generally, the presence of some transition metal elements among them, which bear strong electron correlations, are believed to play a significant role in producing the exotic pairing glue.

Owing to the inherent nature of transition metal chalcogenides¹³, the low-dimensional structures and rich physical properties, e.g., density-wave instability¹⁴, thermoelectricity¹⁵ and SC⁵, are prevalent among them. The tellurides, as compared with sulfides or selenides, are quite special in terms of its structures and properties because of the diffuse nature of the tellurium orbitals and thus far rarely studied. Recently, we reported the observation of SC with T_c = 4.6 K in a ternary telluride Ta₄Pd₃Te₁₆ with Q1D PdTe₂ chains⁶. The detailed studies of its pairing symmetry were followed in applying the techniques of scanning tunneling microscopy¹⁶, low-temperature heat capacity¹⁷ and thermal conductivity¹⁸. The results indicate an anisotropic gap structure with the possible presence of nodes, although electronic structure calculations show the contributions of Pd 4d electrons to the density of states at Fermi level are pretty small¹⁹.

From a crystal-structure viewpoint, Ta₄Pd₃Te₁₆ also belongs to a layered compound resulting from the condensation of Pd-based octahedral chains, Ta-based bicapped trigonal prismatic chains and Ta-based double octahedral chains^6,20. If the Ta-based double octahedral chains are replaced by Ta-based single octahedral chains, the condensation of the three different types of chains would form the atomic layer of a new compound Ta₃Pd₃Te₁₄, which was firstly synthesized by Liimatta and Ibers in 1989²¹. The major difference between them in structure is well reflected from Fig. 1(f), which shows the projection view of one atomic layer of Ta₃Pd₃Te₁₄ and Ta₄Pd₃Te₁₆ along the b axis. The structural details of Ta₃Pd₃Te₁₄ are discussed below. Then, considering the close structural relationship of this material with the superconductor Ta₄Pd₃Te₁₆, a natural question is whether the former is as well a superconductor.

Figure 1

Sample characterization and crystallographic structure of Ta₃Pd₃Te₁₄.

(a) Morphology of a batch of the as-grown Ta₃Pd₃Te₁₄ crystals under an optical microscope (left panel) and photographs of the crystals on a millimeter-grid paper (right panel). (b) Single-crystal X-ray diffraction pattern. The inset shows the third inflections in X-ray diffraction pattern for both Ta₃Pd₃Te₁₄ and Ta₄Pd₃Te₁₆. (c) A typical energy-dispersive X-ray spectrum with electron beams focused on the selected area (marked in the inset) of the crystals. A small amount of the element Al comes from the sample holder. (d) Crystal structure of Ta₃Pd₃Te₁₄ projected along the [010] direction. An individual layer of Ta₃Pd₃Te₁₄ is shown in the left panel of (e). The right panel of (e) is a piece of needle-like Ta₃Pd₃Te₁₄ crystal under an optical microscope, from which the layered morphology can be clearly identified. (f) Projection view of one atomic layer of Ta₃Pd₃Te₁₄ (left) and Ta₄Pd₃Te₁₆ (right) along the b axis.

In this paper, we report the observation of SC with T_c = 1.0 K in layered ternary telluride Ta₃Pd₃Te₁₄ with Q1D PdTe₂ chains. The bulk SC was identified by the electronic heat capacity data, which shows an obvious anomaly at the transition temperature. The specific-heat jump ΔC/(γ_nT_c) ≈ 1.35 indicates Ta₃Pd₃Te₁₄ may be a weakly coupled superconductor. In addition, the result of low-temperature thermal conductivity measurements of Ta₃Pd₃Te₁₄ crystal down to 80 mK suggests a dirty s-wave superconducting gap. We summarize our results by discussing the similarities and differences between the closely related superconductors of Ta₃Pd₃Te₁₄ and Ta₄Pd₃Te₁₆ and compiled an extended list of their physical properties.

Results

Single crystals of Ta₃Pd₃Te₁₄ were grown using a self-flux method, rather than the vapor transport method previously used²¹. Shiny flattened needle-like crystals with a typical size of 2 × 0.15 × 0.1 mm³ were harvested, as shown in Fig. 1(a). The X-ray diffraction (XRD) pattern at 298 K by a conventional θ-2θ scan for the crystals lying on a sample holder is shown in Fig. 1(b), in which we can observe only multiple peaks arising from the diffraction from ( 0 1) planes, consistent with the layered crystal structure of Ta₃Pd₃Te₁₄. Ta₃Pd₃Te₁₄ crystalizes in space group P2₁/m with a monoclinic unit cell of a = 14.088(19) Å, b = 3.737(3) Å, c = 20.560(19) Å and β = 103.73(5)° at 123 K²¹. As seen in Fig. 1(d), the layered slabs compose of successively six different chains of three different types. The three types of chains are Ta-based bicapped trigonal prismatic chains, Pd-based octahedral chains and Ta-based octahedral chains, respectively. The arrangement of the chains, in such a way that every Pd-based chain has two adjacent Ta-based chains and vice versa, constitute the layered slab as clearly depicted in Fig. 1(e). For simplicity, hereafter we define the a^* axis as to be parallel to the[1 0 1] direction and the c^* axis as to be perpendicular to the ( 0 1) plane. The interplane spacing at room temperature is determined to be 6.418 Å and this value is well consistent with the calculated one of 6.397 Å using the above mentioned parameters at 123 K, when taking into account the temperature difference. To compare the obvious difference of the interlayer spacing of Ta₃Pd₃Te₁₄ and Ta₄Pd₃Te₁₆, we plot the third reflection together, namely (−606) and (−309) peaks, in the inset of Fig. 1(b), from which one can easily find the interplane spacing of Ta₃Pd₃Te₁₄ is ~2% smaller than that of Ta₄Pd₃Te₁₆ and full width at half-maximum is only 0.05°, indicating the high quality of the crystals. The chemical composition determined by an energy-dispersive X-ray spectroscopy (EDS), are collected in a number of crystals and the average results confirm that composition of the crystals is the stoichiometric Ta₃Pd₃Te₁₄ within the measurement errors. The SEM image, shown in the upper right corner of Fig. 1(c), has a morphology with stripes along the b axis (chain direction), consistent with the preferential crystal growth along the chain direction.

Figure 2(a) shows temperature dependence of electronic resistivity along the b axis (ρ_b) for the Ta₃Pd₃Te₁₄ crystal (Sample 1). The larger room temperature resistivity (1.18 μΩ m), than that (0.61 μΩ m) of Ta₄Pd₃Te₁₆⁶, indicates Ta₃Pd₃Te₁₄ is less conductive, consistent with the previous reports^21,22. The temperature dependence of resistivity shows a metallic behavior without any obvious anomaly down to T_c = 1.0 K, at which a sharp superconducting transition appears, as clearly depicted in Fig. 2(c). The value of T_c is 3.6 K less than that of Ta₄Pd₃Te₁₆. The onset, midpoint and zero-resistance temperatures are 1.02 K, 0.94 K and 0.81 K, respectively and the superconducting transition width ΔT_c is 0.13 K. The ρ_b(T) data between 2 and 25 K can be well fitted by to ρ_b = ρ₀ + ATⁿ, giving a residual resistivity ρ₀ = 5.13 μΩ cm and n = 2.83 [Fig. 2(b)]. The value of n more than 2 was also observed in Ta₄Pd₃Te₁₆, which was attributed to the phonon-assisted s-d interband scattering¹⁸. The residual resistivity ratio (RRR) is estimated to be RRR = ρ_b(300 K)/ρ₀ ~ 23, similar to that of Ta₄Pd₃Te₁₆ (see Table 1).

Table 1 Comparison of some physical parameters of the superconductors Ta₃Pd₃Te₁₄ (present work and²⁰) and Ta₄Pd₃Te₁₆^6,17,19,20.

Figure 2

Electrical transport and superconducting phase-diagram.

(a) Temperature dependence of electronic resistivity of Ta₃Pd₃Te₁₄ crystal (Sample 1) along the b axis. (b) shows the power-law fit to ρ_b = ρ₀ + ATⁿ in the data range of 2 and 25 K. (c) zooms into the low-temperature range to clearly show the superconducting transition. (d) The low-temperature resistivity in fields H || c^* up to 0.1 T, from which the upper critical field (H_c2) is derived. (e) The red dashed line represents the Werthamer-Helfand-Hohenberg (WHH) fitting. (f) The extracted upper critical field H_c2 of Ta₄Pd₃Te₁₆ crystal (Sample 2) for different field orientations.

Figure 2(d) plots the low-temperature resistivity of Ta₃Pd₃Te₁₄ crystal (Sample 1) for H || c^* up to 0.1 T. Upon increasing the field, the superconducting transition is suppressed to lower temperature. The extracted upper critical fields H_c2(T) for H || c^*, determined by using 90% criterion, i.e., the field at which ρ_b reaches 90% of the normal state resistivity, are shown in Fig. 2(e). We applied the isotropic one-band Werthamer-Helfand-Hohenberg (WHH) formalism to roughly estimate H_c2²³. As can be seen in Fig. 2(e), H_c2 for H || c^* is estimated to be 0.075 T at zero temperature with the derived Maki parameter α = 1.7 and spin-orbit coupling parameter λ_so = 1.2. However, by employing the orbital limiting field (0) = −0.69μ₀ = 0.1 T in WHH model and the BCS Pauli-limiting field  = 1.84T_c = 1.84 T, the Maki parameter α = (0)/ is calculated to be 0.077. This inconsistence between the calculated value of α and the fitted one may originate from the anisotropic effect in Ta₃Pd₃Te₁₄. The extracted H_c2 with fields applied along a^*, b and c^* directions for Sample 2 are shown in Fig. 2(f) and the resistivity data of Sample 2 are not shown here. By roughly linear extrapolations, the anisotropic H_c2 at zero temperature are estimated to be 0.21, 0.27 and 0.086 T for the a^*, b and c^* directions. Using the Ginzburg-Landau formula, the superconducting coherence length ξ are calculated to be 545, 703 and 223 Å for a^*, b and c^* directions, respectively, which are much larger than those of its analog Ta₄Pd₃Te₁₆¹⁷. The SC in Ta₃Pd₃Te₁₄ is anisotropic but as well three-dimensional in nature, similar to other superconductors with Q1D characteristics, e.g., Ta₄Pd₃Te₁₆¹⁷, Nb₃Pd_0.7Se₇²⁴ and Nb₂Pd_xSe₅²⁵, since the interchain coherence length and are much larger than the distance between two arbitrarily adjacent chains.

The low-temperature specific heat data of Ta₃Pd₃Te₁₄ crystals, plotted as C/T vs T, are shown in Fig. 3(a). We fit the normal-state data from 1.2 to 6.5 K, employing the usual formula C/T = γ_n + βT², which is represented as the red dashed line. The fitting yields an electronic heat capacity coefficient γ_n = 28.2 ± 0.9 mJ mol⁻¹ K⁻² and a phononic coefficient β = 11.14 ± 0.04 mJ mol⁻¹ K⁻⁴. The calculated Debye temperature Θ_D = 151.6 K is close to the value of Ta₄Pd₃Te₁₆, consistent with the fact that the structures of two tellurides are closely related. However, the value of extracted coefficient γ_n is nearly 40% smaller than that of Ta₄Pd₃Te₁₆¹⁷. Using the relation N(E_F) =  for noninteracting electron systems, where k_B is the Boltzmann constant, we estimated the density of states at the Fermi level N(E_F) to be about 11.9 ± 0.8 eV⁻¹ fu⁻¹, which is 3.5 times that of the bare density of states N_bs(E_F), obtained from the previous band-structure calculations²⁰. Therefore, the larger renormalization factor [N(E_F)/N_bs(E_F) = 1 + λ], than that for Ta₄Pd₃Te₁₆, suggests much stronger electron-electron correlations in Ta₃Pd₃Te₁₄, although the recent band-structure calculations are concluded with a higher N_bs(E_F) = 9.6 eV⁻¹ fu⁻¹ for Ta₄Pd₃Te₁₆, thus resulting in a much lower renormalization factor¹⁹. To extract the electron-nonphonon coupling strength λ_nph in λ, we estimate the electron-phonon coupling constant λ_ph by employing the McMillan formula²⁶, λ_ph = [1.04 + μ^*ln(Θ_D/1.45T_c)]/[(1 − 0.62μ^*)ln(Θ_D/1.45T_c) − 1.04], where the Coulombic repulsion parameter μ^* is empirically set to be 0.13. The estimated value of λ_ph is 0.51, a little bit smaller than that of the superconductor Ta₄Pd₃Te₁₆⁶. However, the resultant constant λ_nph = λ − λ_ph = 1.99 is much larger than that of Ta₄Pd₃Te₁₆, possibly indicating much larger electron correlations in the former compound.

Figure 3

Temperature dependence of specific heat.

(a) C/T vs T, in which the red dashed line represents the fit with the formula C/T = γ_n + βT² for the normal-state data from 1.2 to 6.5 K. (b) The electronic specific heat divided by temperature C_el/T in the superconducting state, where C_el = C − βT ³.

Discussion

We discuss the electronic heat capacity C_el of Ta₃Pd₃Te₁₄ crystals in low-temperature range, obtained by C_el = C − βT ³. As can be seen in Fig. 3(b), a characteristic superconducting jump (ΔC_el) shows up around ~1 K, confirming the bulk SC. The ΔC_el/T_c is estimated to be 38.0 mJ mol⁻¹ K⁻² and the midpoint temperature of the thermodynamic transition is 1.0 K, consistent with the superconducting transition in low-temperature resistivity. The dimensionless specific-heat jump can be calculated to be 1.35, smaller than the theoretical value (1.43) of the well-known BCS theory, indicating Ta₃Pd₃Te₁₄ may be a weakly coupled superconductor. Unfortunately, due to the insufficient data points, we are unable to fit ΔC_el(T) with standard gap functions to give valuable information about the gap symmetry.

To shed light on the superconducting gap structure, we measured the thermal conductivity of Ta₃Pd₃Te₁₄ single crystal (Sample 2) in zero and magnetic fields (along c^* direction), the results of which are plotted as κ/T vs T in Fig. 4(a). Since all the curves presented in Fig. 4(a) are roughly linear as previously reported in Ta₄Pd₃Te₁₆¹⁸ and some iron-based superconductors^27,28, we fit all the curves to κ/T = a + bT^α−1 by fixing α to 2. The two terms aT and bT^α represent contributions from electrons and phonons, respectively^29,30. From the curve in magnetic field H = 0.09 T, which is close to the critical field H_c2(0) for H || c^*, one can see that the obtained κ₀/T roughly meets the normal-state Wiedemann-Franz law expectation κ_N0/T = L₀/ρ₀ = 2.44 mW K⁻² cm⁻¹. Here, L₀ is the Lorenz number 2.45 × 10⁻⁸ W Ω K⁻² and ρ₀ = 10.04 μΩ cm is the residual resistivity of Sample 2. The verification of the Wiedemann-Franz law in the normal state demonstrates that our thermal conductivity measurements are reliable. For the curves in H = 0 and 0.01 T, however, the linear fittings give two negative values, κ₀/T = −0.24 and −0.16 mW K⁻² cm⁻¹, respectively. These negative κ₀/T have no physical meaning, just because the temperature of our measurement is not low enough, comparing to the T_c. Down to lower temperature, the curve in zero field should deviate from the linear behavior.

Figure 4

Low-temperature thermal conductivity data.

(a) Low-temperature thermal conductivity of Ta₃Pd₃Te₁₄ crystal (Sample 2) in zero and magnetic fields applied along c^* direction. The dashed lines are fits to the formula κ/T = κ₀/T + bT. The black dashed line is the normal-state Wiedemann-Franz law expectation L₀/ρ₀. (b) The field dependence of κ/T at 0.1 K. (c) Normalized residual linear term κ₀/T as a function of normalized field H/H_c2 for the clean s-wave superconductor Nb³⁶, the dirty s-wave superconducting alloy InBi³², the multi-band s-wave superconductor NbSe₂³⁷ and an overdoped d-wave cuprate superconductor Tl-2201³⁸.

Since we can not extrapolate κ₀/T at low field, we plot the field dependence of κ/T at T = 0.1 K, well below T_c, in Fig. 4(b) to get more information about the superconducting gap structure of Ta₃Pd₃Te₁₄³¹. One can see that the increase of κ/T at low field is rather slow and the curve is similar to that of dirty s-wave superconduting alloy InBi³², which is shown in Fig. 4(c). By using the estimated value of the coherence length ξ along b direction, the formula ξ = 0.18ħυ_F/k_BT_c gives the Fermi velocity υ_F = 5.11 × 10⁴ m s^{−1 33}. Then, according to the relationship κ_N0/T = γ_nυ_Fl_e/3, the electron mean free path is estimated to be l_e = 322 Å, which is much smaller than the b-direction ξ. This result indicates Ta₃Pd₃Te₁₄ is indeed in the dirty limit. Therefore, it is concluded that Ta₃Pd₃Te₁₄ is very likely a dirty s-wave superconductor.

It is instructive to compare the physical properties of the two structural closely related compounds of Ta₃Pd₃Te₁₄ and Ta₄Pd₃Te₁₆, which we summarize in Table 1. Both of the two tellurides show Q1D characteristic with Q1D PdTe₂ chains. The larger RRR could account for the much sharper superconducting transition for Ta₃Pd₃Te₁₄. Although T_c of Ta₃Pd₃Te₁₄ is 3.6 K less than that of Ta₄Pd₃Te₁₆, the former compound show much stronger electron correlations, verified by the larger renormalization factor and larger electron-nonphonon coupling strength λ_nph. The small values of both ΔC/(γ_nT_c) and λ_ph indicate Ta₃Pd₃Te₁₄ is a weakly coupled superconductor. In addition, if assuming the Drude model, in which the electron-electron interactions are neglected, is applicable, a superconductor is expected to have a lower T_c for a lower value of Sommerfield coefficient γ_n, which in this case signifies the lower density of states at Fermi level. This simple conclusion is compatible with the general trend for the above two superconductors. Therefore, the lower T_c in the title compound, may be attributed to the lower density of states at Fermi level. In this sense, by tuning the Fermi level of Ta₃Pd₃Te₁₄ or Ta₄Pd₃Te₁₆ by the way of doping or proper intercalations, the value of T_c may be enhanced. By the way, in recently discovered PdTe or PdS chains based superconductors, there have been several pieces of work that show the evidences of two-gap SC^17,34,35. However, our results presented above indicate the new superconductor Ta₃Pd₃Te₁₄ with PdTe₂ chains is very likely a fully gapped s-wave one. We have previously reported that Ta₄Pd₃Te₁₆ is possibly a two-gap superconductor with a gap symmetry of s + d waves¹⁷. Thus, if assuming the reduced part of electronic states at the Fermi level in Ta₃Pd₃Te₁₄, compared to that in Ta₄Pd₃Te₁₆, is primarily due to the reduced contribution from the Pd d states, it would be reasonable to see only a s-wave gap left in the former compound.

Methods

Powders of the elements Ta (99.97%), Pd (99.995%) and Te (99.99%) with a ratio of Ta : Pd : Te = 2 : 3 : 10 were thoroughly mixed together, loaded and sealed into an evacuated quartz ampule. The sample-loaded quartz ampoule is then heated to 1223 K, held for 24 h and cooled to 723 K at a rate of 5 K/h, followed by furnace cooling to room temperature. The above procedures are similar to that in growing Ta₄Pd₃Te₁₆ crystals⁶. The chemical composition is checked by an EDS with an AMETEK EDAX (Model Octane Plus) spectrometer, equipped in a field-emitting scanning electron microscope (SEM, Hitachi S-4800). It is worthy to mention that, although one may speculate the as-grown Ta₃Pd₃Te₁₄ crystals should be mixed by the Ta₄Pd₃Te₁₆ crystals in the same batch, our results do not support this speculation. The reasons are as follows: Using this method and atomic ratio of Ta : Pd : Te = 2 : 3 : 10 to grow Ta₃Pd₃Te₁₄ crystals, Ta₄Pd₃Te₁₆ crystals were occasionally harvested. However, our results show that the two kinds of crystals are never mixed with each other in the same batch. This conclusion is drawn by the fact that once the crystals, randomly picked up from the final product in one batch, are checked by both XRD and EDS to be Ta₃Pd₃Te₁₄, not a single piece of Ta₄Pd₃Te₁₆ crystals can be identified in the same batch and vise versa.

The magnetoresistance (MR) measurements were carried out in a ³He cryostat down to 0.27 K by a stand four-probe technique with current applied along the b axis. The specific heat for a bundle of shiny needle-like crystals with a total mass m = 1.30(2) mg was measured by a long relaxation method utilizing a commercial ³He microcalorimeter (Quantum Design PPMS-9). The thermal conductivity was measured in a commercial dilution refrigerator, using a standard four-wire steady-state method with two RuO₂ chip thermometers, calibrated in situ against a reference RuO₂ thermometer. The chemical composition of the crystals employed for above measurements have also been checked to be Ta₃Pd₃Te₁₄ by both XRD and EDS.