Band dependence of charge density wave in quasi-one-dimensional Ta₂NiSe₇ probed by orbital magnetoresistance

Low-dimensional transition-metal chalcogenides (TMCs) garnered great interest due to their rich physical properties,^1,2 including the recent discovery of valley dependent transport³ and superconductivity⁴ in MoS₂, extremely large magnetoresistance (MR) in WTe₂ (Ref. 5), and the stunning topological phases.⁶ The low dimensionality and high structural symmetry make TMC ideal for studying the structure-property relationship. Uniquely, the electron-electron interaction in one-dimensional (1D) metals causes strong perturbations, in sharp contrast to the case in the 2D and 3D Fermi-liquid counter parts, and leads to Luttinger liquid (LL) behavior.⁷ On the other hand, the charge-density wave (CDW) in the quasi-1D metal can be viewed as a classical analogue of a LL state.⁸ These facts raise special interest in the study of CDW in 1D metallic systems for exploring emergent behaviours. In addition, strong nonlinear electrical transport due to CDW sliding in 1D chain systems leads to high dielectric constants⁹ and narrow-band noise,¹⁰ both of which may find interesting use in applications.

Ta₂NiSe₇ is a quasi-1D ternary TMC showing an incommensurate CDW.¹¹ Its quasi-1D structure is illustrated in Fig. 1(a). Similar to FeNb₃Se_10,¹² the unit cell of Ta₂NiSe₇ consists of double rows of tantalum atoms (Ta1) in bicapped trigonal prismatic selenium coordination and the other double rows of tantalum atoms (Ta2) in octahedral selenium coordination; nickel atoms are in highly distorted octahedral coordination. Band structure calculations showed that the Fermi surface consists of contribution from Ta2 d orbitals in the octahedral chains and Se2 p orbitals from trigonal prismatic columns.¹³ At around 52 K, an incommensurate CDW occurs. Interestingly, the CDW formation mechanism was suggested not to be the Fermi surface nesting effect, but rather through the charge transfer between the two Ta2 chains.¹⁴ However, the X-ray diffraction (XRD) experiment pointed out that all Ta2 atoms are equivalent in symmetry in the CDW state.¹⁵ So far, this discrepancy is not fully resolved but is mitigated by the observation of two independent CDWs, with modulation wave vectors 2k_F and 4k_F, each corresponding to the transverse displacement of Ni and Se2¹⁵ and the longitudinal modulation of Ta2,¹⁶ respectively.

The CDW in Ta₂NiSe₇ is found to be sensitive to defects.^11,17 The typical role of defects on CDW includes scattering carriers, pinning the CDW domains and jeopardising the formation of long-range structural order. In 1D systems, the situation can be more complicated. The spacial distribution of defects can be driven by the CDW order to show a spacial distribution with the same modulation wave vector, which is the so-called defect quasiregularity induced by CDW.¹⁸ The consequence, such as the thermal hysteresis of the CDW transition,¹⁹ was indeed found in Ta₂NiSe_7.¹¹ CDW gap only removes part of the Fermi surface, so that the system remains semimetallic below CDW transition temperature (T_CDW).^13,14

Regarding the transport property of Ta₂NiSe₇, so far only a small kink in the temperature dependence of resistivity R(T) was found to correspond to the CDW.¹¹ The transport related to the 1D character and the underlying electronic states are not explored. Here, we report the result of the MR measurement on high-quality Ta₂NiSe₇ single crystals. In addition to a sizable MR up to 35% at low temperatures, we find that Kohler's scaling is valid at high temperatures but fails in the CDW state. A clear change of curvature in the field dependence of the MR upon entering the CDW state is also observed. The behavior is well interpreted using a two-band orbital MR model, which shows that the hole density is strongly suppressed in the CDW state, while the electron density is less affected. Our results provide transport evidences showing that CDW occurs mostly in hole-like band from Se p orbitals.

Ta₂NiSe₇ single crystals were prepared by using the flux method. X-ray diffraction (XRD) was performed on a Bruker D8 diffractometer. XRD on selected samples showed a space group of (C2/m) and lattice constants of $a=13.84 Å, b=3.48 Å, c=18.60 Å, α=γ=90°, β=108.8°$⁠, consistent with a previous report.²⁰ Single crystal morphology and elemental analyses were carried out by scanning electron microscopy and energy dispersive X-ray spectroscopy, respectively. Crystals used in our measurement are highly selected, with a residual resistance ratio of RRR > 7, which is the highest among those reported in the literature. The typical size of the crystal is $200×50×20 μm3$⁠. R_b, the resistance along the b axis, was measured by using the standard four terminal method in a Quantum Design Physical Property Measurement System (PPMS) with a 14 T magnet, with a rotator for controlling the relative orientation between the magnetic field and crystal.

Ta₂NiSe₇ shows a metallic behavior in the temperature range of 2–300 K, as shown in Fig. 1(b). T_CDW, defined by the peak temperature in dR/dT, is at around 62.5 K. This value is nearly 10 K higher than the 52.5 K reported in previous literatures^11,14,15 and comparable to the highest reported so far.¹⁷ The large difference in T_CDW is attributed to different sample qualities due to the existence of impurities. Typically, this can be characterised using RRR. We studied several samples grown in different batches and obtained the relationship between T_CDW and RRR in Fig. 1(c). The value from a previous study¹¹ is also included in the plot. It is clear to see that T_CDW increases monotonically with increasing RRR. In this paper, we focus on the sample with the highest T_CDW of 62.5 K, which corresponds to the lowest impurity concentration. Another interesting feature in R(T) is the hysteresis loop below T_CDW. One should note that the resistivity upon warming is higher than that upon cooling. This feature was attributed to the defect quasiregularity induced by CDW, as described above.

Now, we focus on the MR of Ta₂NiSe₇. Shown in Fig. 2 is the MR measured along the b axis under a rotating magnetic field in the bc plane. The polar plot of R(θ) in Fig. 2(a), measured at 10 K under a 9 T field, shows a pronounced two-fold symmetry, with the symmetry axis along b/c direction. In this configuration, different parts of the Fermi surface are probed when the magnetic field rotates along different directions. To explore the nature of the MR at low temperatures, the field dependence of the MR at 10 K is shown in Fig. 2(b). The MR is the largest with a field perpendicular to the current (θ = 0), reaching 35% at 14 T, and decreases progressively to below 4% when the field is parallel to the current. Clearly, the orbital MR (also known as the ordinary MR) is dominating. We further illustrate this point in Fig. 2(c) by comparing two curves: ρ_b(H) for the field along the c axis and ρ_b vs. 14 × cos(θ) under a rotating field of 14 T. For the latter, 14 × cos(θ) is essentially the field component perpendicular to the current direction. The two curves overlap rather well, meaning that the field perpendicular to current is critical, which further proves that the effect is dominated by the orbital MR. For the large θ region in ρ_b vs. 14 × cos(θ) curve, the field is oriented close to the b axis, which in turn probes a different area of the Fermi surface, as shown in Fig. 2(a). This leads naturally to the difference between the two curves in the large θ region [small field region in the ρ_b(H) curve].

In an in-plane magnetic field perpendicular to current, the field dependence of the MR at different temperatures is shown in Fig. 3(a). One major observation is that MR(H) shows concave and convex curvatures for temperatures above and below T_CDW, respectively. The slope change in the field dependence of the ordinary MR signifies a multiband effect, consistent with the semimetallic band structure found in the previous calculation.¹³ At high temperatures, MR features a parabolic field dependence, again indicating that the orbital MR is dominating, a mechanism the same as that at low temperatures.

We find that MR data above T_CDW follows Kohler's rule,²¹ namely, Δρ/ρ₀ = f(H/ρ₀). Here, ρ₀ is the zero-field resistivity at a given temperature, and f(H/ρ₀) stands for a function of H/ρ₀. Kohler's rule typically holds well for materials with carrier density insensitive to temperature and with isotropic scattering. In Fig. 3(b), data in Ta₂NiSe₇ for T > T_CDW fall on a master curve of Kohler scaling, with an exponent of 1.99, as expected for an orbital MR. However, when the temperature is lowered through T_CDW, a significant deviation develops, leading to the scattered curves in Kohler's plot, as shown in the right panel of Fig. 3(b). In this multi-band system, the breakdown of Kohler's rule in the CDW states indicate that either the ratio of the electron and hole carrier changes strongly or the scattering becomes anisotropic. While both possibilities may be relevant, the former appears to be more tempting since a direct consequence of a CDW order is the removal of carriers.

Here, we attempt to provide a more quantitative understanding of the MR using the semiclassical two-band resistivity,

Here, e is the electron charge, n_e and n_h are the electron and hole density, and μ_e and μ_h are the electron and hole mobility, respectively. With this equation, we can fit the MR at different temperatures and extract the transport quantities. Unfortunately, because of the inclusion of four parameters, it is difficult to get a reliable fitting if we set all the four parameters independent. Now with the implication of the above discussions, especially the clue we obtained in the Kohler scaling, we intentionally set the mobility μ of the electron and hole to be identical to facilitate an easier fitting. Figure 4(a) shows the representative MR data and the fitting curve. One can see that the two-band ordinary MR equation describes the data quite well, for temperatures both above and below T_CDW. The resultant fitting parameters are summarized in Fig. 4(b). One will also note that the electron and hole are entirely symmetric in the two-band resistivity model; therefore, it is impossible to differentiate the two solely from the model. For this, we rely on the Hall resistivity data of Ta₂NiSe₇ and found that the hole dominates over the entire temperature range (data not shown here), which allows us to distinguish n_e and n_h in Fig. 4(b). Consistently, the resultant hole density is several times higher than the electron density over the entire temperature range. We see that the carrier density shows clearly different behavior for electron-like and hole-like bands: the electron density increases only slightly upon cooling, while the hole density decreases significantly when entering the CDW state. This contrasting behavior shows unambiguously that the CDW gap opens predominately in the hole-like band. Recalling that the Fermi surface of Ta₂NiSe₇ consists of an electron-like band from Ta2 d orbitals in the octahedral chains and a hole-like band Se2 p orbitals from trigonal prismatic columns,¹³ our results indicate that the carrier transport is dominated by the hole-like band from the p orbitals from Se2 atoms, which finds good consistency with previous synchrotron X-ray diffraction results that the primary part of CDW resides on Se2 and Ni.¹⁵ 

In summary, we report our magneto resistivity measurement on the quasi-1D transition-metal chalcogenide Ta₂NiSe₇. Kohler's rule is valid at high temperatures but breaks down in the CDW state. A clear change in curvature in the field dependence of magnetoresistivity upon entering the CDW state is observed, which is fully accounted for by a two-band ordinary magnetoresistivity model. The two-band fitting shows that the hole carrier density is strongly suppressed in the CDW state, indicating that CDW takes place in the hole-like band, mostly the p orbitals from Se2 atoms.

The work at SJTU was supported by MOST (Grant No. 2015CB921104) and NSFC (Grant Nos. 91421304 and 11474198), at Penn State by NSF (Grant No. EFMA1433378), and at Tulane supported by the U.S. Department of Energy under EPSCoR Grant No. DE-SC0012432 with additional support from the Louisiana Board of Regents (support for a graduate student, materials, travel to NHMFL).