We have studied magnetotransport properties of a topological insulator material Ru_{2}Sn_{3}. Bulk single crystals of Ru_{2}Sn_{3} were grown by a Bi flux method. The resistivity is semiconducting at high temperatures above 160 K, while it becomes metallic below 160 K. Nonlinear field dependence of Hall resistivity in the metallic region shows conduction of multiple carriers at low temperatures. In the high-temperature semiconducting region, magnetoresistance exhibits a conventional quadratic magnetic-field dependence. In the low-temperature metallic region, however, high-field magnetoresistance is clearly linear with magnetic fields, signaling a linear dispersion in the low-temperature electronic structure. Small changes in the magnetoresistance magnitude with respect to the magnetic field angle indicate that bulk electron carriers are responsible mainly for the observed linear magnetoresistance.

Dirac fermion systems have recently attracted much attention in spintronics,^{1} because of their potential high efficiency in the interconversion between spin currents and charge currents. One of the most prominent examples of Dirac fermion systems is topological insulators (TIs).^{2–4} In TIs, strong spin-orbit coupling opens a charge gap in the bulk states, but 2D Dirac fermion systems appear on the surfaces. Dirac fermions form a Dirac cone in the band structure near the Fermi level; transport phenomena are governed by the Dirac fermions if the bulk conduction is sufficiently suppressed. On topological insulators *e.g.* Bi_{2}Se_{3}^{5} and $\alpha $-Sn,^{6} efficient interconversion between spin currents and charge currents was reported at room temperature.

In spintronics studies on TIs, a Bi_{2}Se_{3} family has frequently been used.^{7–15} In Bi_{2}Se_{3} systems, bulk-insulating samples are obtained by chemical substitution, *e.g.* Sb substitution in Bi sites and Te substitution in Se sites. However, Bi, Sb, Se, and Te are toxic and volatile. Moreover, to synthesize high-quality TI film samples, high-cost MBE (molecular beam epitaxy) systems have been required; this should be a big barrier to realization of commercial spintronics devices based on TIs. Hence, material search of other topological insulator materials is still important.

In this letter, we focus on a topological intermetallic compound Ru_{2}Sn_{3}: a rare strong 3D topological insulator caused by band inversion between Ru 4*d* and Sn 5*p* branches due to strong spin-orbit coupling.^{16} Ru_{2}X_{3} (X=Si, Ge, Sn) belongs to a family of Nowotny chimney-ladder compounds.^{17,18} Ru_{2}Si_{3} and Ru_{2}Ge_{3} are semiconducting, while metallic resistivity was reported in Ru_{2}Sn_{3}^{18,19} despite the same number of valence electrons. Band calculations for Ru_{2}Sn_{3} show a small indirect band overlap leading to a semi-metal state,^{16,18} while angle-resolved photoemission spectroscopy (APRES) data^{16} clearly shows that Ru_{2}Sn_{3} has a full band gap with surface states formed inside the gap. The Fermi energy in the as-prepared Ru_{2}Sn_{3} crystals falls clearly within the bulk band gap,^{16} which is suitable for the observation of surface Dirac transport.

Though an ARPES study at 1 K revealed topological surface states of Ru_{2}Sn_{3} near the Fermi level,^{16} there has been no follow-up study on the surface properties. Our magnetotransport study on bulk single crystals of Ru_{2}Sn_{3} shows that linear magnetoresistance is observed at low temperatures. Since the linear magnetoresistance emerges in line with a structural phase transition, the present results provide the signature of a linear dispersion in the low-temperature electronic structure. Chemical doping is thereby not necessary to observe the linear magnetoresistance in Ru_{2}Sn_{3}, in contrast to Bi_{2}Se_{3} systems.^{2–4} Since Ru and Sn are not so toxic or volatile, and since the non-trivial topological state can be pushed up to room temperature by an applied pressure,^{20} Ru_{2}Sn_{3} could be promising for spintronics applications.

The crystal structure of Ru_{2}Sn_{3} at room temperature is illustrated in Fig. 1(a). It is known that Ru_{2}Sn_{3} undergoes a phase transformation from a tetragonal non-centrosymmetric high-temperature structure to an orthorhombic centrosymmetric low-temperature structure.^{17,19} At room temperature, Ru_{2}Sn_{3} crystallizes in its tetragonal phase; Ru atoms form a $\beta $-Sn type structure with 4-fold helical order along the *c* axis, and Sn atoms intervene in the voids of the Ru helices and compose another 3-fold helical arrangement along the *c* axis [Fig. 1(a)].^{18} Below room temperature, the atomic displacements occur gradually over a wide temperature range, resulting in the gradual structural phase transition.^{17,19} The orthorhombic structure realized in Ru_{2}Sn_{3} at low temperatures is similar to that found in Ru_{2}Si_{3} and Ru_{2}Ge_{3} at room temperature.^{17}

Bulk single crystals of Ru_{2}Sn_{3} were grown by a Bi flux method.^{16} Powder of Ru (99.95%), Sn (99.99%), and Bi (99.99%) elements was mixed in a molar ratio of Ru : Sn : Bi= 2 : 3 : 67^{16} and sealed in an evacuated quartz tube. The powder was then heated up to 800 °C and kept at 800 °C for 12 hours, followed by a slow cooling to 100 °C for 70 hours. Numerous submillimeter-size Ru_{2}Sn_{3} single crystals were obtained by dipping the ingot into dilute HNO_{3} acid. Longitudinal and Hall resistivity was measured for a single-crystalline sample being 0.5-mm long, 0.2-mm wide, and 0.05-mm thick. The widest plane was determined to be a (100) plane in the tetragonal phase [Fig. 1(b)]. Electrical contacts were formed by fixing gold wires on the sample using conductive paste. The measurements of magnetoresistance and Hall effects were performed in a 9 tesla Physical Property Measurement System (Quantum Design, Inc.). An external magnetic field was applied perpendicular to the sample plane in the Hall effect measurements, while in the magnetoresistance measurements, the magnetic field direction was changed so that the magnetic field is always perpendicular to the current direction [see Fig. 4(a)].

We show temperature (*T*) dependence of the longitudinal resistivity, $\rho $, in Fig. 1(c). $\rho $ increases with decreasing *T* down to 160 K, and then starts to decrease below 160 K. The temperature dependence of $\rho $ is consistent with the reported one on single crystals^{16} and on polycrystals.^{18} This unusual trend seems to reflect the broad tetragonal to orthorhombic phase transition.^{16} Here, it is noted that $\rho $ exhibits a clear drop at 3.7 K, as shown in the inset to Fig. 1(c). This drop is probably due to a superconducting transition of minor Sn-metal contamination observed even in Sn-deficient Ru_{2}$ Sn 3 \u2212 \delta $ samples.^{18} Since this superconductivity is limited in a very small *T*-*H* region, the impurity phase hardly affects the following magnetotransport studies of Ru_{2}Sn_{3}.

Figure 2(a) shows magnetic field (*H*) dependence of the magnitude of magnetoresistance in perpendicular-to-plane magnetic fields, MR, at various temperatures, where MR is defined as $ [ \rho ( H ) \u2212 \rho ( H = 0 ) ] / \rho ( H = 0 ) $. In the semiconducting regime above 160 K [Fig. 1(c)], overall *H* dependence of MR does not change with temperature; the magnetic-field dependence is quadratic, expected in standard conductors. In the metallic regime below 160 K, however, MR starts to increase with decreasing *T*, as shown in Fig. 2(a). At 5 K, magnetoresistance at high fields is linear with *H*, and MR reaches 13% at 9 T. Hence, in the metallic regime, the magnetoresistance in the high-field range is linear, not quadratic. The linear MR at low temperatures suggests that the low-*T* transport is linked to the linear dispersions, as observed in surfaces of topological insulators (Bi,Sb)_{2}(Se,Te)_{3}^{21–25} and SmB_{6},^{26} Dirac semimetals Cd_{3}As_{2}^{27–29} and Na_{3}Bi,^{30} and other 3D Dirac materials.^{31–33}

In Figs. 3(a) and 3(b), we compare the magnetoresistance at high temperatures [Fig. 3(a)] and at low temperatures [Fig. 3(b)]. As shown in Fig. 3(a), MR above 150 K is quadratic as a function of *H* in the entire *H* range from 0 to 9 T. From the quadratic *H* dependence, the effective mobility, $ \mu MR $ $ ( \u2261 M R / \mu 0 H ) $, is estimated to be 152 cm^{2}/Vs in the semiconducting region. In the low-*T* metallic state, by contrast, the magnetoresistance is clearly linear with magnetic fields, as shown in Fig. 3(b). Also remarkable is that, as shown in Fig. 3(c), MR at 9 T suddenly increases below the temperature of $ d \rho / d T = 0 $, suggesting the enhancement of the mobility in the metallic regime. The large linear magnetoresistance indicates that, with the structural transition from the high-*T* tetragonal phase to the low-*T* orthorhombic phase, Dirac fermion transport shows up.

In general, linear magnetoresistance can originate from several mechanisms.^{34,35} In the case of linear magnetoresistance in silver chalcogenides $ Ag 2 + \delta $Se and $ Ag 2 + \delta $Te,^{36} multiple scattering by impurities in inhomogeneous conductors is responsible for the giant linear magnetoresistance.^{37} On the other hand, linear magnetoresistance can come from quantum effects;^{38–40} for gapless semiconductors with a linear energy spectrum, magnetoresistance in perpendicular-to-plane magnetic fields becomes linear with magnetic fields at low temperatures.^{41} In recently found topological materials,^{21–30} linear magnetoresistance of this mechanism has been observed. In the present case of Ru_{2}Sn_{3}, since the linear dispersion of the topological surface state was observed near the Fermi level by ARPES measurements at 1 K,^{16} Dirac fermions on the topological surface state may be an origin of the linear magnetoresistance at low temperatures.

To reveal whether the origin of the linear magnetoresistance is 2D or 3D, one standard way is the angular dependence of the magnetoresistance;^{22} for 2D surface-state transport, the magnetoresistance will respond only to the perpendicular component of the magnetic field [$ H cos \theta $ in Fig. 4(a)]. Figure 4(b) shows the magnetic-field dependence of $\rho $ at 2 K measured in three field configurations of $ \theta = 0 \xb0 $ (perpendicular-to-plane *H*), $ \theta = 45 \xb0 $, and $ \theta = 90 \xb0 $ (in-plane *H*). The overall *H* dependence of $\rho $ is found to be similar among the different field configurations, which indicates that the linear magnetoresistance is not from 2D surface transport alone but from 3D bulk transport. Though the full-gap bulk state was observed in ARPES measurements,^{16} the surface transport is hardly observed in as-prepared bulk-form samples of Ru_{2}Sn_{3}. The situation may be similar to tetradymite topological insulators,^{21–25} where sufficiently small bulk volumes^{22–24} and/or bulk insulation achieved by bulk carrier compensation^{21,25} are necessary to observe the surface conduction.

Though the angle-dependent change in magnetoresistance is as small as 1.5 % at 2 K, the $ \rho ( \theta ) $ exhibits a characteristic angular dependence, as shown in Fig. 4(c). $ \rho ( \theta ) $ has broad peaks around $ \theta = 0 \xb0 $, $ 90 \xb0 $, and $ 180 \xb0 $, and dips around $ \theta = 45 \xb0 $ and $ 135 \xb0 $. Hence, $ \rho ( \theta ) $ seems to include $ cos ( 2 \theta ) $ and $ cos ( 4 \theta ) $ components. This angle dependence should reflect the anisotropic Fermi surface of Ru_{2}Sn_{3}, but looks inconsistent with the star shaped Fermi surface of the topological surface state.^{16} As temperature increases, the $\theta $ dependences of $\rho $ become smaller, and disappear above 160 K [Fig. 4(d)]. By fits to the $ \rho ( \theta ) $ data at each temperature in the metallic regime, $ \rho 2 \theta cos ( 2 \theta ) $ and $ \rho 4 \theta cos ( 4 \theta ) $ are separated, and the values of $ \rho 2 \theta $ and $ \rho 4 \theta $ are plotted against temperature in Fig. 3(e). $ \rho 2 \theta $ and $ \rho 4 \theta $ are less than 5 $ \mu \Omega cm $, which is 1000 times as small as $\rho $ $ ( \u223c 1 $ $ m \Omega cm ) $.

To further study the magnetotransport properties of Ru_{2}Sn_{3}, Hall resistivity, $ \rho H $, is shown as a function of magnetic fields at various temperatures in Fig. 2(b). In the semiconducting regime above 160 K, the Hall resistivity is positive and linear with magnetic fields. The transport in the semiconducting regime is thereby explained by conduction of thermally-excited holes. From the slope *R*_{H}[$ \u2261 d \rho H / d ( \mu 0 H ) $] at 300 K, the effective carrier density is estimated to be $ 7.6 \xd7 10 19 $ cm^{−3}. Also, the Hall mobility ($ \u2261 R H / \rho $) at 300 K is 71.1 cm^{2}/Vs, which is similar to the $ \mu MR $ value in the high-*T* range.

As *T* decreases to the orthorhombic metallic regime, the Hall resistivity clearly deviates from the linear magnetic-field dependence, signaling multi-carrier transport. Below 60 K, the slope *R*_{H} at low magnetic fields is still positive, but becomes negative at high magnetic-fields. In a classical expression for Hall effects, the Hall coefficient in the high-field limit reads 1/(*en*_{eff}),^{42} where *n*_{eff} is the effective carrier density which includes conduction electrons and holes. In Fig. 3(d), the *R*_{H} value at 9 T is shown as a function of *T*. As *T* decreases, *R*_{H} at 9 T starts to decrease at 240 K and shows a sign change around 100 K. The *p*-*n* crossover at low temperatures is in agreement with that observed in the *T* dependence of the Seebeck coefficient.^{16}

The electron carriers which dominate low temperature transport are expected to induce the linear magnetoresistance in the metallic regime [Fig. 3(b)]. In fact, in Figs. 3(c) and 3(d), the rapid increase in MR is observed at low temperatures where the sign of *R*_{H} is negative. Though the Dirac dispersion of the topological surface state was observed near the Fermi level,^{16} the topological surface state is unlikely to produce electron carriers; in an ARPES measurement,^{16} a single Dirac point originating from the band inversion is located above the Fermi level at $ \Gamma \xaf $ [Fig. 1(d)],^{16} which implies hole conduction on the surface states. The small $\theta $ dependence of linear magnetoresistance [Fig. 4(b)] also supports that the linear magnetoresistance results from the 3D bulk transport, whereas a linear dispersion has not been reported in the bulk electronic structure.^{16} Nevertheless, an expected semimetallic or narrow-gap semiconducting bulk-state is suitable for the observation of quantum linear magnetoresistance, if it has small pockets of the Fermi surface with a small effective mass.^{43} Since there is discrepancy between calculated electronic structure^{16,18} and ARPES data,^{16} future in-depth calculations are required for the full understanding of the magnetotransport results.

In summary, magnetoresitance and Hall effects were studied in a topological insulator Ru_{2}Sn_{3}. Magnetotransport properties are dramatically different between the high-temperature tetragonal phase and the low-temperature orthorhombic phase. Magnetoresistance shows classical quadratic magnetic-field dependence in the high-temperature semiconducting regime, while it becomes linear in the low-temperature metallic regime. As opposed to the reported data on the surface electronic structure near the Fermi level,^{16} the linear magnetoresistance at low temperatures seems to originate from the linear-like dispersion in the bulk state. In the Hall-effect measurement, a *p*-*n* crossover of dominant carriers was observed with decreasing temperature, which suggests that the electron carriers are expected to induce the linear magnetoresistance. In contrast to the tetradymite topological insulators, nanofabrication or bulk-carrier compensation is not necessary to observe linear magnetoresistance in Ru_{2}Sn_{3}, which may be promising for application in topological spintronics.

This work was supported by JST (ERATO “Spin Quantum Rectification project”), JSPS (KAKENHI No. 16H00977 and the Core-to-Core program “International research center for new-concept spintronics devices”), MEXT (Innovative Area “Nano Spin Conversion Science” (No. 26103005)), and Center for Spintronics Research Network (CSRN), Tohoku University, Japan.