Topological materials have been researched extensively due to their exotic physical phenomena, and electrical transport is widely investigated for these materials for its use in observing the behavior of bands near the Fermi level[1, 2]. In particular, when combined with spin–orbit coupling, some spin states of topological materials could break the time reversal symmetry of the bands under the protection. Ferromagnetic Fe₃Sn₂ and antiferromagnetic Mn₃Sn could be excellent examples because of their physical phenomena as magnetic topological materials, where Dirac and Weyl bands produce large anomalous Hall effects through interaction with their innate spin state[3, 4]. This is likely to be a case when a spin becomes unsettled by the nearest neighbors of the lattice[5]. Gerard Toulouse introduced this concept of geometrical frustration, and topics on crystals consisting of triangles and hexagons of 3d transition metals have been actively investigated.

Meanwhile, some crystals can possess topological properties from structural transformation. For instance, some IV–VI semiconducting crystals, such as SnTe, can transform from a cubic to a rhombohedral phase by decreasing the synthesis temperature[6]. Though symmetry is reduced for such crystals, they are still able to generate topological semimetallic properties such as nodal lines or Weyl points. However, although such method of distortion could break symmetry or produce new topological properties that can lead potentially intriguing research fields, it is difficult to find proper samples to be investigated.

It has been reported that many T–Sn binary phases can reduce symmetry via synthesis conditions. For instance, cubic Cu₃Sn undergoes transformation to an orthorhombic phase around 650 °C. Here, orthorhombic Cu₃Sn could generate large magnetoresistance because of the broken symmetry[7]. Likewise, Ni₃Sn₂ can have hexagonal (P6₃/mmc) and orthorhombic (Pnma) structures that are temperature dependent as shown in Table 1[8]. The hexagonal structure contains random vacancies at two sites of the unit cell, which should require high-activation energy of migration. In contrast, when synthesizing at lower temperatures, an orthorhombic distortion is permitted by assigning one site as fully filled while leaving others vacant; they have been named HT-Ni₃Sn₂ and LT-Ni₃Sn₂.

Table 1 Lattice parameters of HT-Ni₃Sn₂ and LT-Ni₃Sn₂, excerpted from [8].

	HT-Ni3Sn2	LT-Ni3Sn2
Space group	P63/mmc	Pnma
a [Å]	0.41045	8.1541
b [Å]	0.41045	7.1247
c [Å]	0.51831	5.1954
α	90°	90°
β	90°	90°
γ	120°	90°
V[Å3]	75.62	301.84

HT-Ni₃Sn₂ grown via chemical vapor transport (CVT) is known to exhibit anisotropic and enhanced paramagnetic behavior and unusual temperature dependence of resistivity that are proportional to the power law[9]. In fact, such dependence has frequently been observed in various materials consisting of 3d transition metals with geometrical frustration[10, 11]. What is fascinating is that Ni₃Sn₂ possesses such cusp from a relatively high temperature. In contrast, LT-Ni₃Sn₂ is less known as it has not been synthesized as a single-crystalline phase so far. Together with its ortho-rhombically distorted structure, the fully filled and other fully vacant sites could break structural symmetry enough.

Furthermore, spin heptamers may occur when the vacant site is filled, where hexagonal isostructural Fe_1.3Sb exhibits a topological Hall effect via scalar spin chirality derived from the Dzyaloshinskii–Moriya interaction of the spin heptamer[12]. Similarly, stacked spin heptamers in LT-Ni₃Sn₂ function as spin clusters and exhibit intriguing physical properties, concerning geometrical frustration, when compared to HT-Ni₃Sn₂.

Here, we report that the synthesis of HT-Ni₃Sn₂ and LT-Ni₃Sn₂ using the flux method is possible, and we discuss some distinguishable properties observed in LT-Ni₃Sn₂ from the perspective of resistivity measurements.

All Ni₃Sn₂ single crystals in the study were synthesized using the flux method. We determined synthesis conditions using the Ni–Sn binary phase diagram. Ni, Sn, and Bi elements were mixed and placed in an alumina crucible. The silica ampoule containing the crucible was then sealed under partial Ar atmosphere. They were initially heated up to 1000 °C to ensure completely melting, HT-Ni₃Sn₂ single crystals were obtained at 800 °C, whereas LT-Ni₃Sn₂ single crystals were obtained at 450 °C via annealing for 7 days. Unlike the HT-Ni₃Sn₂ single crystals that were grown as long hexagonal rods, the as-grown LT-Ni₃Sn₂ single crystals resembled needles.

X-ray diffraction (Empyrean, Malvern Panalytical, United Kingdom) was conducted to identify the Miller indices of the single crystals, and energy-dispersive X-ray spectroscopy (S-4800, Hitachi, Japan) was used to determine its nominal composition. Electrical transport was measured using a home-built 14-T magnet (Oxford Instruments, United Kingdom). Magnetoresistance was defined via the formula MR = 100 × [ρ(H)−ρ(T=0)/ρ(T=0)] and corrected using the equation ρ(|H|)=[ρ(H)+ρ(−H)]/2.

Figure 1(a) shows HT-Ni₃Sn₂ with one hexago-nal layer of a fully occupied Ni(I) site and another hexagonal layer of a half-filled Ni(II) and Sn sites. Meanwhile, lowering the synthesis temperature changes the unit cell orthorhombically by aligning the Ni(II) sites, as shown in Fig. 1(b). Here, the top-view schematics of Figs. 1(c) and 1(d) represent the emergence of spin clusters, as the ordered Ni(II) sites create a line of spin heptamers onto the b-axis (equivalent to the hexagonal c-axis). Considering the distorted trimers of Ni(I) nets arising in broken symmetry, this could produce interesting phenomena. As-grown single crystals are shown in Figs. 1(e) and 1(f).

Figure 1. (Color online) (a, b) Crystal structure of (a) HT-Ni₃Sn₂ and (b) LT-Ni₃Sn₂. Red, green, and black balls represent Ni(I), Ni(II) and Sn, respectively. (c, d) Top-view schematic of (c) HT-Ni₃Sn₂ and (d) LT- Ni₃Sn₂. (e, f) Optical image of HT-Ni₃Sn₂ and LT-Ni₃Sn₂.

Figure 2 displays the X-ray diffraction patterns analyzed to determine the orientation of the planes of HT-Ni₃Sn₂ and LT-Ni₃Sn₂, which are shown in Figs. 1(e) and 1(f), respectively. Their sharp peaks guaranteed the crystal’s growth. Still, since HT-Ni₃Sn₂ and LT-Ni₃Sn₂ were almost indiscernible during the analysis, we intentionally synthesized the Ni₃Sn₂ crystals by pulling the ampoule at 450 °C without annealing. As they showed a mixed phase where the peaks were separated, our observation strongly suggests that the HT-Ni₃Sn₂ and LT-Ni₃Sn₂ phases, grown under distinct temperatures, had not indented each other. The energy-dispersive spectroscopy (EDS) result for each sample was nominally Ni_2.98Sn₂ and Ni_3.24Sn₂. This implies that Ni₃Sn₂ crystals could have nonstoichiometric Ni_3+δSn₂ or Ni_3-δSn₂, which is consistent with the Ni–Sn binary phase diagram. The Bi element had not been identified, implying that the flux was not indented or contaminated the crystal.

Figure 2. (Color online) (from the top) XRD patterns of HT-Ni₃Sn₂, mixed Ni₃Sn₂ and LT-Ni₃Sn₂. The insets show zoomed-in graphs between 80° and 83°.

Figure 3 depicts resistivity as a function of temperature. We found that the residual resistivity (ρ5K) of HT-Ni₃Sn₂ was 73.5 μΩ cm. Since ρ300K was 88 μΩ cm, the residual resistivity ratio (RRR = ρ300K/ρ5K) achieved was 1.27. This defective property was previously reported in isostructural Fe₅Sn₃, which had ρ5K 60 μΩ cm and an RRR of <2[11]. It is comparable to the out-of-plane resistivity measured at CVT-grown HT-Ni₃Sn₂ that exhibited ρ2K reaching 450 μΩ cm[9]. This may suggest the presence of anisotropy in the resistivity. The isostructural Fe_1.3Sb showed semiconducting behavior in the same out-of-plane direction measurement and metallic behavior in the in-plane direction measurement in the same way as our configuration[12].

Figure 3. (Color online) Temperature dependence of resistivities and power-law fittings of HT-Ni₃Sn₂ and LT-Ni₃Sn₂. Red solid lines are results of the power law fitting for each crystal.

Fascinatingly, the order of the resistivity was similar in LT-Ni₃Sn₂, where ρ5K and ρ300K were 98.3 and 124 μΩ cm, respectively. The RRR achieved 1.20, indicating that the bad metal-like properties still existed. Therefore, although LT-Ni₃Sn₂ arranges randomly occupied Ni atoms, the resistivity was in a similar order against HT-Ni₃Sn₂. This is analogous to the Cu₆Sn₅, which also has hexagonal isostructural Cu₆Sn₅ and the monoclinic distortion η8-Cu₅Sn₄[13]. Thus, randomly distributed vacancy protects the structural symmetry, and its alignment may not enhance the metallic behavior.

The resistivity of metals is usually described using Matthiessen’s rule, ρ=ρd+ρp, where ρd is from the electron–defect scattering and ρp is from the electron–phonon scattering[14]. ρd is independent of temperature while ρp is linearly dependent on temperature. Therefore, the power law model ρ=a+bTn is applicable to describe the behavior of normal metals with n = 1. However, for magnetic metals with long-range ordering, n is greater than 1[15,16], whereas for metals with short-range ordering caused by geometrical frustration, n is less than 1[9,10].

We used the power law model to analyze HT-Ni₃Sn₂ over a temperature range between 50 and 300 K and found that n of HT-Ni₃Sn₂ was 0.89. When the fitting range became restricted to the intervals of 80 and 200 K, n was modified to 0.85. The measured value of n along the c-axis in Fig. 1(a) exhibits a substantial deviation from n = 0.07 that was previously measured for the CVT-grown crystal perpendicular to the c-axis[9]. Both measurements resulted in n < 1, which suggests that the crystal exhibits a short-range ordering that is different in each direction. The spin heptamer from the occupancy of Ni(II) atoms would make a spin cluster, which induces the anisotropic short-range ordering. The same analysis can be applied to LT-Ni₃Sn₂, which resulted in a value of 0.79 for n along the b-axis in Fig. 1(b), which is equivalent to the c-axis in HT-Ni₃Sn₂. By narrowing the fitting range from 80 to 200 K, n changed to 0.75. The smaller value of n observed for LT-Ni₃Sn₂ may indicate the presence of stronger short-range excitations in the crystal. It can be because of the ordered spin clusters via stacked heptamers, in contrast to the HT-Ni₃Sn₂, which can possess randomly distributed heptamers and clusters.

Notably, HT-Ni₃Sn₂ and LT-Ni₃Sn₂ also exhibited different behaviors in magnetoresistance, as depicted in Fig. 4. In Fig. 4(a), we found no clear dependence of magnetoresistance against the magnetic field for the HT-Ni₃Sn₂, in contrast to the reference[9]. This could be interpreted that our flux-grown HT-Ni₃Sn₂ may contain fewer Ni atoms or it has highly anisotropic transport behavior.

Figure 4. (Color online) (a, b) Transverse magnetoresistance graph of (a) HT-Ni₃Sn₂. and (b) LT-Ni₃Sn₂. Magnetic field was applied in ab-plane in HT-Ni₃Sn₂, and in ac-plane in LT-Ni₃Sn₂. Current was applied parallel to c-axis in HT-Ni₃Sn₂ and to b-axis in LT-Ni₃Sn₂.

In contrast, Fig. 4(b) shows the magnetoresistance of LT-Ni₃Sn₂, which is related to the increase in the magnetic field. It increased abruptly under 1 T with decreasing temperature and achieved 1.18% at 14 T and 5 K. Such a dip might imply fast short-range spin correlations via the occurrence of spin clusters in LT-Ni₃Sn₂. It could simply be a signal of the weak antilocalization derived from strong spin–orbit coupling, as demonstrated using several topological materials[17]. This behavior should be investigated further to elucidate the issue.

We successfully synthesized both single crystals of HT-Ni₃Sn₂ and LT-Ni₃Sn₂ using the flux method. The observed X-ray diffraction peaks imply that the phases are not merged and that each crystal shows that, as with the EDS results, they are not contaminated by flux. We found that both crystals are metallic with similar resistivity orders of magnitude. However, our analysis using the power law suggests LT-Ni₃Sn₂ may have stronger short-range excitations than HT-Ni₃Sn₂. In magnetoresistance, while HT-Ni₃Sn₂ was almost independent against the applied magnetic field, LT-Ni₃Sn₂ showed a strong dip at lower temperatures and weaker magnetic fields. This phenomenon might have something to do with its hidden nature in innate spin clusters. We hence conclude that LT-Ni₃Sn₂ could possess unusual physical properties, and further investigation would be required to elucidate speculations.

This work was supported by the DGIST institution specific program (21-BRP-07) and the faculty startup funds from DGIST.