Enhanced upper critical fields in a new quasi-one-dimensional superconductor Nb₂Pd_xSe₅

Figure 1(a) shows the neutron powder diffraction pattern and the Rietveld refinement results for the polycrystalline Nb₂Pd_1.2Se₅ at room temperature. Without including any additional magnetic diffraction peaks, we could successfully refine the measured Bragg peaks by a monoclinic (space group C12/m) structure with lattice parameters a = 12.788 Å, b = 3.406 Å, c = 15.567 Å and β = 101.63°. From the independent x-ray powder diffraction measurements and the Rietveld refinement, we could obtain almost similar lattice parameters within 1%.

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Figure 1(b) displays several features of the refined crystal structure of Nb₂PdSe₅: (i) one-dimensional stacks of the edge-sharing Nb(1)–Se and face-sharing Nb(2)–Se trigonal prisms form the most conductive chains along the b-direction. (ii) The Pd(1) sites located at the center of the Se squares also constitute a part of the conductive b-chain. (iii) The planar connection along the [1 0 2] direction between the two trigonal prisms and the Pd(1)–Se squares results in the (− 2 0 1) planes. (iv) The Pd(2) site is sandwiched between the (− 2 0 1) planes. These structural features turned out to be well reflected in the facets and the crystallographic orientations of the single crystal; a needle-shaped crystal was grown along the b-axis and easily cleaved along the (− 2 0 1) plane (figure 1(c)), of which orientation was determined by the electron backscattering diffraction. For convenience, we define hereafter a', b and c* directions as [ − 2 0 1], [0 1 0] and [1 0 2], respectively, in a setting of the conventional unit cell.

The actual atomic composition of the single crystal was determined to be Nb₂Pd_0.74Se_5.11 by the EDX measurements. The proximity to ∼0.75 of the Pd composition implies that nearly one quarter of the Pd sites are vacant. According to the structural features described above, it is likely that the defects are formed more easily in the Pd(2) site than the Pd(1) site because the former has a longer bond length and weaker bonding strength with the neighboring Se. The weaker Pd(2)–Se bonding seems to be also consistent with the cleaved (− 2 0 1) plane. Previous reports on Nb₂Pd_0.71Se₅ and Ta₂Pd_0.89S₅ have also indicated that there existed point defects in the Pd(2) sites [22, 23]. Therefore, the presence of the Pd(2) defects is likely to be a common feature in this q1D material.

Figures 2(a) and (b) present the resistivity, magnetic susceptibility (4πχ) data of the polycrystalline sample, all of which provide evidence of the superconducting transition. The resistivity curve starts to drop near 5.9 K and becomes zero at 5.5 K. The magnetic susceptibility measured at H = 10 Oe after zero-field-cooling develops the onset of the Meissner effect at 5.5 K and an almost full shielding below 5.0 K, whereas the field-cooled susceptibility exhibits only a small drop, possibly due to a strong flux pinning effect.

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Figure 3 shows the electronic specific heat (C_el) divided by temperature. Fitting the normal state specific heat just above T_c to the form of C/T = γ + βT² yielded the Sommerfeld coefficient γ = 15.7 mJ mol⁻¹ K⁻² and a phononic coefficient β = 0.874 mJ mol⁻¹ K⁻⁴. C_el was then obtained by subtracting the phonon contribution from the measured specific heat data. It is noticed in figure 3 that the C_el/T data display a clear jump, thereby constituting evidence for bulk superconductivity. As the temperature is further lowered, C_el/T is almost exponentially suppressed but remains at a finite value of γ₀ = 2.8 mJ mol⁻¹ K⁻² below 0.6 K, which indicates the presence of a non-superconducting metallic region. In this case, the Sommerfeld coefficient of the superconducting volume becomes γ_n = γ − γ₀ = 12.8 mJ mol⁻¹ K⁻², which corresponds to about 80% of the total volume. The normalized specific heat jump ΔC_el/γ_nT_c is estimated to be 1.63 ± 0.13, of which error bars depend on the choice of a linearly varying region in the C_el/T data below T_c. The ΔC_el/γ_nT_c = 1.63 is larger by about 13% than the weakly coupled BCS value of 1.43.

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Figure 3 also shows the fitting curves of the BCS α-model plus γ₀, in which α = 2Δ₀/k_BT_c denotes coupling strength, Δ₀ is the size of the superconducting gap and k_B is the Boltzmann constant [24]. The one-band α-model with α = 4.0 seems to roughly explain the specific heat jump and exponentially vanishing behavior at low temperatures. Consistent with the enhanced ΔC_el/γ_nT_c value, the coupling strength α = 4.0 is also larger than the BCS prediction, 3.54. Both results thus suggest that Nb₂Pd_xSe₅ is a BCS superconductor located beyond the weak coupling regime.

Although the one-band α-model describes the experimental C_el/T data reasonably well, several features are still worth being discussed in figure 3. Firstly, we note that the hump feature around 3.5 K cannot be easily explained by the one-band α-model. Although the two-band α-model was also employed, the hump feature could not be explained without introducing the volume fraction with lower T_c. Therefore, the deviation of the data from the fit at  ∼ 3.5 K might be due to the presence of a superconducting volume fraction with lower T_c. To be consistent with this conjecture, specific heat data measured above 2 K in other batches of the polycrystal specimens showed the smaller hump feature near 3.5 K. Secondly, except the hump feature, we could also fit experimental data better, particularly at low temperatures below 2.5 K upon employing the two-band α-model (see figure 3) [25]. The coupling strengths of each bands were α₁ = 4.1 with the relative fraction x_rel = 0.9 and α₂ = 2.1 with the remaining fraction 1 − x_rel = 0.1, respectively. However, the difference between the one-band and the two-band fitting results was not big so that a type of bands with α ≈ 4.0 seems to dominate the thermodynamic properties in the superconducting state. Thirdly, although the jump and shape of the C_el/T curve are roughly consistent with the one-band α-model, it should be reminded that another q1D superconductor Li_0.9Mo₆O₁₇ with a presumably p-wave and triplet pairing symmetry has shown a similar consistency with the one-band α-model [26]. Thus, the agreement with the α-model alone does not necessarily support the s-wave pairing symmetry of Nb₂Pd_xSe₅.

To investigate the normal state transport properties, the MR and the Hall effect were first measured in the polycrystalline samples. At overall temperatures, MR defined as Δρ_xx(H)/ρ_xx(0 T) = (ρ_xx(H)–ρ_xx(0 T))/ρ_xx(0 T) showed variation nearly in proportion to H² which is a usual behavior expected in conventional metallic systems (figure 4(a)). On the other hand, its slope showed rather a strong increase below T* =  ∼ 50 K. As a result, MR estimated at 9 T started to increase around T* to reach as high as 6% at low temperatures while it remained small (⩽ 1.5%) above T* (figure 5(a)). The transverse resistivity (ρ_xy) (figure 4(b)) showed almost linear variation up to 9 T in a broad temperature window below 300 K, indicating that the multiband effect is not clearly observable in the ρ_xy data. On the other hand, the resultant Hall coefficient R_H = ρ_xy/H (figure 5(a)) clearly showed peculiar temperature dependence; upon temperature being lowered, R_H decreases continuously to reach a minimum at T* =  ∼ 50 K and increases again below T*. Moreover, the Hall mobility (μ_H ≡ R_H/ρ_xx) also shows a very steep increase near T*. Consistent with these transport data, ρ(T) data for both single crystal and polycrystal showed rather a steep decrease around 50 K so that dρ/dT curves show a characteristic, broad maximum around T*. In analogy to the heavy fermion materials showing a broad dρ/dT maximum near a coherent temperature [27, 28], the maximum of dρ/dT indicates the presence of an energy scale T* =  ∼ 50 K, below which a new electronic ground state is stabilized. Therefore, all of the three independent transport properties presented in figure 5 coherently support the existence of a crossover from a high- to a low-temperature metallic phase at T*.

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It should be noted in the single crystal that the resistivity along the conductive b-direction shows a slight increase below  ∼ 20 K (dρ_b/dT < 0), while that of the polycrystalline sample becomes more or less saturated. It was found that MR of the single crystal becomes negative near the temperature where dρ_b/dT < 0 is observed (see e.g. the inset of figure 7(c)). Thus, the resistivity upturn may be associated with the weak localization effect, possibly coming from the Pd deficiency. On the other hand, the resistivity upturn along the conductive chain direction is also reminiscent of the case of the q1D Li_0.9Mo₆O₁₇, in which an insulating ground state was realized below  ∼ 30 K [29, 30]. Therefore, the origin of the insulating behavior in Nb₂Pd_xSe₅ should be further investigated to address whether it is one of the common features in the q1D materials or just an outcome of the atomic defects.

Resistivity measurements under hydrostatic pressure were performed on another piece of a single crystal showing an even steeper resistivity increase at low temperatures below  ∼ 50 K. As shown in figure 6(a), the resistivity upturn was gradually suppressed upon increasing pressure to induce the metallic ground state down to T_c. In each pressure, T* was determined by an intersection temperature of the two linearly extrapolated lines in the ρ(T) curve below and above the transition and T_c was determined at which 50% of the normal state resistivity is realized. Figure 6(b) depicts the summarized phase diagrams for T_c and T* with respect to pressure variation. Upon increasing pressure, T_c =  ∼ 4.7 K at ambient pressure increases with an initial rate of  ∼ 1.8 K GPa⁻¹, and then becomes nearly saturated to  ∼ 5.8 K around 2 GPa. On the other hand, T* =  ∼ 50 K at ambient pressure systematically decreases to 27 K at 2.1 GPa. The evolutions of T_c and T* suggest that the superconducting ground state competes with the electronic state whose energy scale is characterized by T*.

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To determine H_c2(T) in the single crystal, we measured the temperature-dependent resistivity at fixed magnetic fields below 9 T (not shown here) and the field-dependent resistivity up to 35 T at fixed temperatures (figure 7). To avoid ambiguity due to the surface superconductivity effect and superconducting fluctuation, H_c2 was chosen as the critical field at which 50% of the normal resistivity value was achieved in the superconducting transition. Figure 8(a) summarizes the resultant H_c2(T). The H_c2 slopes near T_c are determined as −11.5, −4.78 and −3.70 T K⁻¹ for H//b, c* and a' respectively. Based on the anisotropic Ginzburg–Landau (GL) relation

$$\begin{equation} \frac{{H_{{\rm{c}}2}^i }}{{H_{{\rm{c}}2}^j }} = \frac{{\sqrt {\sigma _i } }}{{\sqrt {\sigma _j } }} = \frac{{\sqrt {\rho _j } }}{{\sqrt {\rho _i } }}, \end{equation} \tag{ 3.1 }$$

the anisotropies of H_c2 near T_c predict the resistivity ratio ρ_b:ρ_c*:ρ_a' = 1:5.8:9.7. This result is consistent with the expectation that the charge transfer would be most effective along the Nb–Se conductive chain, i.e. b-direction. On the other hand, the transport anisotropy ratio is much smaller than that of the organic superconductors (1:10³:10⁵) [31, 32] but comparable to that of Li_0.9Mo₆O₁₇ [15, 29, 33].

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The extrapolated H_c2 values at zero temperature, H_c2(0), are 10.5, 35 and 22 T for H//a', b and c*, respectively. If H_c2(0) is mainly determined by the orbital pair breaking effect originating from kinetic energy of super-current around vortices, a coherence length is given by the GL relation, H_c2ⁱ(0) = ϕ₀/2πξ_j(0)ξ_k(0), where ϕ₀ is the flux quantum and ξ_i(0) is the GL coherence length for the i-direction. The calculated ξ(0) values representing a characteristic size of a Cooper pair are 21, 68 and 43 Å for a', b and c*, respectively. Because all of ξ(0) are clearly longer than the lattice constants, this observation indicates that the superconducting order parameter extends over the several conductive chains and planes.

Due to the lack of established theory to describe H_c2(T) in the q1D superconductors, we have tried to apply the Werthamer–Helfand–Hohenberg (WHH) formula for an isotropic BCS superconductor in a dirty limit [34]. In figure 8(a), we could fit H_c2(T) for H//b with the WHH formula. The Maki parameter $\alpha _{\rm{M}} = \sqrt 2 H_{{\rm{c}}2}^{{\rm{orb}}} (0)/H_{\rm{P}}^{{\rm{BCS}}} = 6.1$ is calculated from the orbital limiting field H_c2^orb(0) = −0.69|dH_c2/dT|_TcT_c = 39.8 T and the BCS Pauli limiting field, H_P^BCS = 1.84T_c = 9.2 T. The spin–orbit coupling constant λ_so was an adjustable parameter to fit the H_c2(T) data. The obtained λ_so = 55 is unusually large, and qualitatively indicates that the large spin–orbit scattering is necessary to describe the H_c2(T) behavior.

The angular-dependent H_c2, H_c2(θ), was also measured at 0.3 K in the a'–c* and a'–b planes as shown in figure 8(b). For a three-dimensional (3D) anisotropic superconductor, H_c2(θ) is expressed by the 3D GL model as

$$\begin{equation} H_{{\rm{C2}}} (\theta ) = \frac{{H_{{\rm{c}}2} (0^\circ )}}{{\sqrt {\gamma ^2 \sin ^2 \theta + \cos ^2 \theta } }} \end{equation} \tag{ 3.2 }$$

for a given anisotropic ratio γ = H_c2(0°)/H_c2(90°). On the other hand, the 2D Tinkham model is applicable for superconductors when their coherence lengths perpendicular to layers is much shorter than the interlayer spacing. In this case, H_c2(θ) is expressed as [35]

$$\begin{equation} \left| {\frac{{H_{{\rm{c2}}} (\theta ){\rm sin}\,\theta }}{{H_{{\rm{c2}}} (0^\circ )}}} \right| + \left( {\frac{{H_{{\rm{c2}}} (\theta ){\rm cos}\,\theta }}{{H_{{\rm{c2}}} (90^\circ )}}} \right)^2 = 1, \end{equation} \tag{ 3.3 }$$

where θ denotes a tilted angle away from a 2D plane. We found that the observed H_c2(θ) curves are well explained by the 3D GL model with the anisotropy ratio 3.35 and 2.15 for the a'–b and a'–c* plane, respectively. However, the cusp-like behavior predicted by the 2D Tinkham model is not consistent with the measured data. Therefore, both the estimated coherence lengths and the H_c2(θ) features suggest that the superconducting order parameter is coherent across the several conductive chains. As a consequence, the field-induced dimensional crossover that allows the Cooper pairs to sustain the orbital pair breaking effect is not likely to occur in this material.

The band calculations were performed for the stoichiometric compound although the actual single crystal and polycrystal might have excessive or deficient Pd. Because the Pd deficiency or excess can be easily generated in the weakly bound Pd(2) sites sandwiched between the main Nb₄PdSe₁₀ slabs, the population change in the Pd(2) site is not expected to modify the topology of the band structure significantly but only shift the Fermi energy level. Therefore, it is likely that our band calculations for the stoichiometric compound could capture an essential feature of the electronic structure in Nb₂Pd_xSe₅.

Figure 9 depicts the calculated Fermi surfaces in a conventional reciprocal unit cell. The contribution of the d-orbitals of Nb and Pd yields the multi-sheet Fermi surfaces consisting of a set of electron-like flat sheets, closed pockets and hole-like corrugated cylinders. Such characteristic Fermi sheets and cylinders are related to the charge transfers along the 1D Nb–Se chain and the conducting plane on the Nb₄PdSe₁₀ slab, respectively. The multiple Fermi surfaces also suggest the possible stabilization of multiple superconducting gaps in Nb₂Pd_xSe₅. The linearly increasing H_c2 for H//c* down to the lowest temperature can be regarded as an experimental hallmark of the multiband superconductivity, as similarly observed in MgB₂ and iron-based superconductors [36, 37].

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The resistivity upturn behavior observed in the single crystal may indicate the weak localization effect—a quantum interference of conducting electrons due to defects or disorder [38]. However, regardless of the resistivity upturn, both single-crystal and polycrystal data consistently show the resistivity slope change near T*. Moreover, the weak localization effect, possibly coming from the Pd defects, cannot easily account for the sharp superconducting transition observed in both single crystal and polycrystal and the systematic changes of the Hall coefficient/mobility and MR below T* observed in the polycrystal. Furthermore, the observation of the systematic suppression of T* and enhancement of T_c with pressure suggests that the low-temperature metallic state is clearly competing with superconductivity. The hydrostatic pressure generally increases an electron hopping and leads to an effective 3D transport. Therefore, it is expected that the electronic state stabilized below T* should have a low-dimensional Fermi surface topology, such as a sheet or cylinder-like shape.

One appealing scenario for explaining the observed anomalies is to assume a q1D Fermi surface (sheet-like) which is already formed far above T* and whose effective Fermi energy is quite small. The Femi surface may then allow a coherent electron transport below T* while the transport at higher temperatures can be incoherent. In this scenario, the q1D Fermi surface is not likely to favor superconductivity because the superconducting state is competing with the energy scale of T*. Superconductivity in Nb₂Pd_xSe₅ is then expected to favor the 2D or 3D like Fermi surface topology.

An alternative scenario for the energy scale T* would be a gradual stabilization of a weak CDW order. Note that the crystal structure of Nb₂Pd_xSe₅ possesses the same 1D Nb–Se chain as in NbSe₃, a renowned CDW material. The band structure calculations of Nb₂Pd_xSe₅ consistently predict the parallel sheets of the Fermi surfaces, of which the nesting feature may naturally allow the density wave formation. The systematic suppression of T* and enhancement of T_c with pressure may then indicate that the density wave order gets weaker while the superconducting order gets stronger. The competition between the density wave order and superconductivity has been often observed in low-dimensional superconductors. The role of the applied pressure is then expected to change the Fermi surface shape into a more 3D like one to destabilize a nesting feature required to form a density wave order.

On the other hand, it should be noticed that the observed anomalies near T* does not clearly show a localization feature observed in conventional CDW materials, in which the resistivity upturn or a clear sign changing behavior of Hall coefficients are observed [39]. Moreover, the thermodynamic evidence for the density wave transition such as a specific heat jump near T* has not been observed so far. In this context, we postulate that the density wave order, even if it exists, might have rather an incommensurate and short-ranged character as indicated by the transport properties in incommensurate CDW systems [40, 41].

When a Fermi surface of a superconductor has a strong electronic anisotropy, the orbital pair-breaking effect becomes quite weak in a magnetic field applied along highly conductive directions, giving rise to a large H_c2. In such a case, the usual Pauli limiting effect, originated from the competition between the Zeeman energy and the superconducting condensation energy in a Cooper pair, should be considered. This behavior is expected from the Chandrasekhar–Clogston relation [42]

$$\begin{equation} \textstyle\frac{1}{2}(\chi _{\rm{N}} - \chi _{\rm{S}} )H_{\rm{P}}^2 = \textstyle\frac{1}{2}N(E_{\rm{F}} )\Delta _0^2 , \end{equation} \tag{ 4.1 }$$

where χ_N (χ_S) is a spin susceptibility of a normal (superconducting) state, N(E_F) is a density of state at the Fermi level and Δ₀ is a superconducting gap at 0 K. In a weakly coupled BCS limit, this relation provides the Pauli limiting field as H_P^BCS = 1.84 T_c. Remarkably, all H_c2(0) are larger than H_P^BCS (= 9.2 T) in Nb₂Pd_xSe₅. In particular, H_c2 (0) for H//b reaches to 35 T, which is nearly four times larger than H_P^BCS. Therefore, it remains to be understood how superconductivity is robust against the spin paramagnetic effect. It is also worthwhile to remind that the unusually high H_c2(0) beyond H_P^BCS have been found in many other q1D organic and inorganic materials so that the phenomenon seems to be quite generic.

Equation (4.1) suggests that the superconducting gap enhancement due to the strong coupling effect may give rise to a large H_c2 beyond H_P^BCS [43]. Although the observed specific heat jump indicated the α ≈ 4.0, the enhancement of 13% over the BCS limit (α ≈ 3.54) is insufficient to explain the four-fold enhancement of H_c2. Moreover, an unusual enhancement of H_c2(0), often observed in the FFLO state due to a spatial modulation of a superconducting order parameter, is not likely to occur here because the hysteresis expected in the first-order transition was not observed and the shapes of our H_c2(T) curves is almost saturated along both H//b and H//c* directions.

One explanation for the enhanced Pauli limiting field is the enhancement of the spin–orbit scattering. As implied by (4.1), the finite χ_s generated by the spin-flip scattering should reduce the Zeeman energy term and result in a larger H_P than H_P^BCS. This mechanism has been widely suggested to explain the observation of the large H_c2(0) above H_P^BCS in the superconductors containing a large Z number atom [43, 44]. The spin–orbit scattering becomes a significant portion of the transport scattering for the large Z number as known from the Abrikosov relation τ_tr/τ_so = (Z/137)⁴ where 1/τ_tr (1/τ_so) is transport (spin–orbit) scattering rate, which reaches as high as 0.013 for Z_Pd = 46 [45].

By using the Drude model with the measured Sommerfeld coefficient and the single-crystal resistivity value, we estimated that the mean free path (l_tr) is  ∼ 5 Å and 1/τ_tr is ∼10¹⁴ s⁻¹ [46]⁶. This estimation implies that this superconductor is in a dirty limit (l_tr < ξ_b = 68 Å). Furthermore, based on the spin–orbit scattering constant λ_so = 2ℏ/3πk_BT_cτ_so obtained from the fit in figure 8(a), 1/τ_so becomes  ∼ 10¹² s⁻¹, which is about 100 times smaller than 1/τ_tr. Therefore, although it is valid for one-band isotropic superconductors, the fit in figure 8(a) seems to satisfy the WHH assumption (1/τ_so < 1/τ_tr) and the ratio of τ_tr/τ_so seems to roughly meet the Abrikosov relation. Therefore, the large H_c2 might arise from the presence of large spin–orbit scattering in a dirty-limit superconductor, particularly related with the presence of ions with a large Z number such as Pd.

Although the presence of Pd or its deficiency might be a main origin to cause the enhanced λ_so in this system, it seems still worthwhile to consider a mechanism based on the common feature of such q1D electronic structure because the greatly enhanced λ_so within the WHH scheme seems to be generic in many q1D superconductors [15, 47]. In this context, an alternative route to increase the spin–orbit scattering is the breaking of inversion symmetry, which might be associated with an inherent nature of the low-dimensional electronic structure [48]. In thin film superconductors, the inversion symmetry breaking at the interface is known to result in the Rashba-type spin–orbit coupling and gives rise to the enhanced H_c2 beyond H_P^BCS as well as the anomalously enhanced MR [49, 50]. In the q1D materials like NbSe₃, an incommensurate CDW introduces the breaking of inversion symmetry [51, 52]. In analogy with those q1D systems, the density wave order, possibly stabilized below T* in Nb₂Pd_xSe₅ might play a role in increasing the spin–orbit coupling. If this scenario is valid, a local inversion symmetry breaking induced by a CDW-like order should not only result in an enhanced H_c2 but also point to a stabilization of a peculiar superconducting ground state with an admixture of the spin singlet and spin triplet components [12, 53]. Such an exotic superconducting state has been claimed to exist in other q1D superconductors such as (TMTSF)₂PF₆ and Li_0.9Mo₆O₁₇ [2, 15]. It should also be noted that those two superconductors share a common feature of the large H_c2(0) beyond the conventional Pauli limiting field as observed in this study. Therefore, understanding the nature of a possible density wave order and searching for evidence for the spin triplet pairing state seem to be all important ingredients in elucidating the true origin of the greatly enhanced H_c2(0) with large spin–orbit scattering in Nb₂Pd_xSe₅.