Magneto-transport properties of proposed triply degenerate topologicalsemimetal Pd3Bi2S2

Shubhankar Roy, Arnab Pariari, Ratnadwip Singha, Biswarup Satpati, and Prabhat Mandal1

Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Calcutta 700 064,India

We report transport properties of single-crystalline Pd3Bi2S2, which has been predicted to host an uncon-ventional electronic phase of matter beyond three-dimensional Dirac and Weyl semimetals. Similar to severaltopological systems, the resistivity shows field-induced metal to semiconductor-like crossover at low tempera-ture. Large, anisotropic and non-saturating magnetoresistance has been observed in transverse experimentalconfiguration. At 2 K and 9 T, the MR value reaches as high as ∼1.1×103 %. Hall resistivity reveals thepresence of two types of charge carriers and has been analyzed using two-band model. In spite of the largedensity (> 1021 cm−3), the mobility of charge carriers is found to be quite high (∼ 0.75×104 cm2 V−1 s−1

for hole and ∼ 0.3×104 cm2 V−1 s−1 for electron). The observed magneto-electrical properties indicate thatPd3Bi2S2 may be a new member of the topological semimetal family, which can have a significant impact intechnological applications.

Topology protected electronic properties of materialshave opened up a new arena of research in modern con-densed matter physics1–7. These novel materials havegenerated immense research interest in fundamentalphysics of low-energy relativistic particles. In high-energy physics, the relativistic fermions are protectedby Poincare symmetry, while in condensed matter,they respect one of the 230 space group symmetries(SGs) of the crystal. The variation of crystal symmetryfrom one material to another escalates the potentialto explore free fermionic excitations such as Dirac,Weyl, Majorana and beyond. Besides fundamentalinterest, the discovery of topological insulators1,2, 3DDirac and Weyl semimetals3,4, and Majorana fermionsin superconducting heterostructures5–7 have plantedthe seed of technological revolution in research anddevelopment of electronic devices. Fabrication of fastelectronic devices, spintronics applications and faulttolerant quantum computing are the few among theseveral specific topics of interest. So, the predictionand experimental realization of new materials, whichhost such quasi-particle excitations, can have significantimpact on paradigm shifting in technological application.

Very recently, Bradlyn et al.8 have predicted the ex-istence of exotic fermions near the Fermi level in sev-eral materials, governed by their respective space groupsymmetry. Unlike two- and four-fold degeneracy in 3DWeyl/Dirac semimetals, these systems exhibit three-, six-, and eight-fold degenerate band crossing at high sym-metry points in the Brillouin zone. It has been proposedthat Pd3Bi2S2 with space group 199 hosts three-fold de-generate fermion as the quasi-particle excitation due tothe three-band crossing at the high symmetry P point,which is just 0.1 eV above the Fermi level8. So far, nomagnetotransport experiment has been done on this ma-terial. The results are important not only for the fun-damental research but also to manipulate them in tech-nological application. In the present work, we have syn-thesised the single crystal of Pd3Bi2S2 and performed

5 nm

(a)

20 40 60 80

Inte

nsity

(arb

. unit

)

2(in degree)

YobsYcalYobs-Ycal

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1.92 Å

(b)

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1-102-20

3-304-40

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[221]bcc

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Inte

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(d)

FIG. 1. (a) High resolution transmission electron microscopyimage, taken on a representative piece of Pd3Bi2S2 single crys-tal. (b) Fourier-filtered image of a small selected region in Fig.(a). (c) The selected area electron diffraction (SAD) patternstaken along [221]. (d) X-ray diffraction pattern of powderedsingle crystals of Pd3Bi2S2. Red open circles are experimen-tal data (Yobs), black line is the calculated pattern (Ycal),blue line is the difference between experimental and calcu-lated intensities (Yobs-Ycal), and green lines show the Braggpositions.

the magnetotransport experiments in field up to 9 T andtemperature down to 2 K.

Single crystal of Pd3Bi2S2 was synthesized via stan-dard self-flux method. At first, polycrystalline Pd3Bi2was prepared by melting the stoichiometric mixtureof high purity Pd pieces (Alfa Aesar 99.99%) and Bishot (Alfa Aesar 99.999%) in an arc furnace, underargon atmosphere. Next, Pd3Bi2 and stoichiometricamount of sulphur pieces were kept at 350C for 24 hand at 900C for 48 h in a vacuum sealed quartz tube.Finally, the quartz tube was slowly cooled down toroom temperature at the rate of 4C/h. Shiny plate-like

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crystals of typical dimensions 2.0 × 0.4 × 0.3 mm3

were mechanically extracted. Typical morphology anddifferent crystallographic directions of a representativesingle crystal of Pd3Bi2S2 have been shown in Fig. S1.Phase purity and the structural analysis of the sampleswere done using both high resolution transmissionelectron microscopy (HRTEM) in a FEI, Tecnai G2 F30,S-Twin microscope equipped with energy dispersivex-ray spectroscopy (EDX, EDAX Inc.) unit and thepowder x-ray diffraction (XRD) technique with Cu-Kα

radiation in a Rigaku x-ray diffractometer (TTRAX III).The transport measurements have been performed in a9 T physical property measurement system (QuantumDesign) using ac-transport option and 10 mA current.The I-V characteristic of the sample has been recordedwith current up to 40 mA to ensure that the electricalcontacts are Ohmic. For both the resistivity and Hallmeasurements, the electrical contacts were made infour-probe configuration using conducting silver paintand gold wires (diameter 50 µm). The HRTEM image ofa typical Pd3Bi2S2 single crystal is shown in Figs. 1(a)and (b). Very clear periodic lattice structure impliesthat there is no secondary phase, atom clustering ordisorder in the studied sample. Figure 1(c) showsthe selected area electron diffraction (SAD) patternrecorded along [221] zone axis. The periodic pattern ofthe spots in SAD implies high-quality single crystallinenature of the grown samples. The diffraction patternwas indexed using the lattice parameters of cubic unitcell. To determine the stoichiometry, the EDX has beenperformed on different regions of the single crystal. InFig. S2, we have shown one typical EDX spectrum.From the experimental results, the atomic ratio of Pd,Bi and S is found to be ∼2.93:2.28:1.79 throughoutthe sample with maximum experimental error ±5 %.However, for low Z element sulphur, the statisticalerror in the EDX spectra is expected to be little higher.Figure 1(d) shows the high-resolution x-ray diffractionpattern of the powdered sample at room temperature.Within the resolution of XRD, we did not see any peakdue to any impurity phase. Using the Rietveld profilerefinement, we have calculated the lattice parametersa=b=c=8.310(2) A with space group symmetry I213.

In Fig. 2(a), the zero-field resistivity (ρxx) of Pd3Bi2S2

crystal shows metallic behavior over the whole range oftemperature. ρxx exhibits a monotonic decrement withthe decrease in T and becomes very small (∼ 270 nΩ cm)at 2 K. The residual resistivity ratio (RRR), ρxx(300K)/ρxx(2 K), is 213, which signifies good metallicity andhigh quality of single crystal. This value of RRR is largerthan that reported (∼ 110) earlier9. RRR of Pd3Bi2S2

is comparable to that reported for several topologicalsemimetals and metals10–12. ρxx up to ∼13 K can befitted well with the equation, ρxx(T ) = a + bT 2 [insetof Fig. 2(a)]. This indicates pure electronic correlationdominated scattering in the low-temperature region13.The behaviour of ρxx(T ) curve at low temperature is

0 1 0 0 2 0 0 3 0 00

1 5

3 0

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6 0

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0 . 9

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0 . 2

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0 2 0 4 01 . 4

1 . 6

1 . 8

2 . 0

r xx (m

W cm)

T ( K )

9 T 7 T 0 T

( a )

r xx (mW

cm)

T ( K )

0 T F i t t i n g

( b )

MR (1

03 %)

B ( T )

2 K5 K1 0 K2 0 K5 0 K

Expo

nent

(m)

T ( K )

FIG. 2. (a) Temperature dependence of resistivity (ρxx) bothin presence and absence of external magnetic field. Current(I) is applied perpendicular to magnetic field (B). Inset showsthe low-temperature region fitted with ρxx(T ) = a+ bT 2. (b)Transverse magnetoresistance as a function of magnetic fieldat some representative temperatures. The current is appliedalong the crystallographic a-axis and the magnetic field isalong c-axis. Inset shows the exponent (m) as a function oftemperature, where MR at various temperatures have beenfitted with the power law behavior, MR∝Bm.

quite different from that reported for most of the Diracand Weyl semimetals10,12,14–17, where ρxx is found to beeither very weakly dependent on T or ρxx ∝ Tn withn≥3. A T 2 dependence of ρxx has only been observedin type-II Weyl semimetal WTe2

18. Almost linear T de-pendence of ρxx above ∼ 25 K suggests electron-phononscattering dominated transport at high temperature. Tdependence of ρxx for Pd3Bi2S2 crystal has a strikingsimilarity with conventional metal. With application ofmagnetic field, a small upturn in ρxx(T ) curve is seenin the low-temperature region. The field-induced metal

3

to semiconductor-like crossover and low-temperaturesaturation-like behavior in resistivity are the commonlyobserved phenomena in topological semimetals12,14–16.

The magnetoresistance of Pd3Bi2S2 in transverse ex-perimental configuration (I ‖a-axis, B ‖c-axis) has beenmeasured at various temperatures. As shown in Fig.2(b), MR, which is defined as [ρxx(B)-ρxx(0)]/ρxx(0), ispositive and large at low temperature and shows super-linear field dependence up to 9 T. At 2 K and 9 T, thevalue of MR is about 1.1×103 %, which is comparable tothat observed in several Dirac and Weyl materials19–24.With the increase in temperature, however, MR de-creases rapidly. MR at various temperatures follows thepower-law behavior, MR∝ Bm. Unlike conventional met-als and topological semimetals, exponent (m) is found tobe extremely sensitive to temperature. From the inset ofFig. 2(b), one can see that m increases almost linearlywith the increase in temperature. As a result of suchunusual behavior of MR, strong violation of the Kohlerscaling is observed, which is shown in Fig. 3(a). Thisviolation is ascribed to various aspects of charge conduc-tion mechanism such as the presence of more than onetypes of charge carrier and different temperature depen-dence of their mobilities (µ)17,18,25,26.

With the help of a sample rotator, we have studied theangular dependence of MR. In this experimental set up,the current direction has been kept fixed along crystal-lographic a-axis on the largest flat plane, and the mag-netic field is rotated about a-axis, as illustrated in theschematic diagram of Fig. 3(b). The anisotropy in MRis evident from the experimental data at 2 K, which hasbeen plotted in Fig. 3(b) for two representative mag-netic field strengths 5 and 9 T. MR is maximum whenthe magnetic field is perpendicular to the plane of thesample (φ=0) and it decreases as the field is rotated to-wards inplane direction and becomes minimum at φ=90.The anisotropy ratio in MR is ∼ 2.2. Similar anisotropicbehavior of MR has been reported in several topologi-cal semimetals11,27,28. In a non-magnetic material, MRis mainly determined by the mobility of the charge car-rier in the plane perpendicular to the external magneticfield29. Again, the mobility is defined as the ratio of thescattering time (τ) to the effective mass of the chargecarrier (meff ), µ ∼ τ

meff. Both the quantities τ , which

depends on the cross-sectional area of the Fermi surfacein the plane perpendicular to magnetic field, and meff ,which depends on the electronic band curvature, are di-rectly related to Fermi surface. As a consequence, thenature and geometry of the Fermi surface play a crucialrole in determining the angle dependence of transversemagnetoresistance. The anisotropy in MR can be ex-plained by considering the anisotropy in the Fermi sur-face of Pd3Bi2S2. In a material with simple cubic crystalstructure, one should not expect any anisotropy in theFermi surface and as a consequence, the angular depen-dence in MR will be negligible. However, the cubic crys-

0 5 1 0 1 5 2 0 2 5 3 00 . 00 . 20 . 40 . 60 . 81 . 0

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03 %)

B / r 0 ( T m W - 1 c m - 1 )

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)

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( b )

( a )

FIG. 3. (a) MR at different temperatures are scaled usingKohler’s rule. (b) Angular dependence of MR for 2 K andmagnetic field 5 and 9 T. The magnetic field is rotated aboutthe crystallographic a-axis.

tal structure of Pd3Bi2S2 has chiral nature − absence ofcenter of inversion and mirror planes9. Due to this lowercrystal symmetry, one expects anisotropic Fermi surfaceand angular dependence in the orbital magnetoresistanceinduced by the broken four-fold symmetry in cubic struc-ture. Often, topological semimetals show negative longi-tudinal magnetoresistance (LMR), where current is par-allel to the magnetic field30–33. We have not observedany negative LMR in the present compound. On the con-

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FIG. 4. The field dependence of Hall resistivity (ρyx) at dif-ferent temperatures. A schematic of Hall measurement set upis shown in inset.

trary, we have observed a positive LMR, which has beenshown and discussed in supplementary material (see Fig.S3).

To determine the type of charge carrier and its mobil-ity, the Hall resistivity (ρyx) measurement has been per-formed. ρyx has been obtained by the operation, ρyx=[ρyx (positive field)-ρyx (negative field)]/2, to remove theresistivity contribution induced by the small unavoidablemismatch in Hall resistivity contacts on the opposite sideof the sample. In Fig. 4, the field dependence of ρyxis shown at some representative temperatures. At hightemperature, ρyx is almost linear in field and positive.However, with decreasing temperature, ρyx(B) becomesweakly non-linear. This implies the presence of morethan one types of charge carrier. In presence of ex-ternal magnetic field, the electrical conductivity (σxx)and Hall conductivity (σxy) are given by the expressions,σxx= ρxx

ρ2yx+ρ2xx

and σxy=ρyx

ρ2yx+ρ2xx

, following the tensorial

inversion of the resistivity matrix. In Fig. 5(a), we haveplotted σxy as a function of magnetic field for a repre-sentative temperature 2 K. The Hall conductivity showsa non-monotonic behavior, which can be explained fromthe field dependence of ρyx and ρxx. In absence of mag-netic field, as ρyx=0 and ρxx 6=0, σxy is zero. In low fieldregion, σxy is small and increases with increasing mag-netic field. With further increase in field strength, ρyxincreases monotonically but the denominator (ρ2yx+ρ2xx)becomes very large due to large enhancement of ρxx fromits zero field value. Hence, the field dependence of σxyshows a low field maximum and tends towards zero athigh fields. The field dependence of ρyx and 1

ρ2yx+ρ2xx

can

make this discussion more clear, which have been shownin Fig S5. Considering the semiclassical two-band model,the following expressions of σxy and σxx have been used

to fit the experimental data30.

σxy(B) = [nhµ2h

1

1 + (µhB)2− neµ2

e

1

1 + (µeB)2]eB (1)

σxx(B) = e[nhµh1

1 + (µhB)2+ neµe

1

1 + (µeB)2] (2)

where nh(ne) and µh(µe) are hole(electron) density andmobility, respectively. We have done the global fittingof σxy(B) and σxx(B) [where these two quantities havebeen fitted simultaneously using Eqs. (1) and (2)] forall temperatures and in Fig. 5(a), we have shown theglobal fitting at 2 K as a representative. As evidentfrom Fig. 5(b), the mobilities show strong tempera-ture dependence, while the density of the two types ofcharge carrier [inset of Fig. 5(b)] remains almost in-variant throughout the temperature range within the ac-curacy of our two-band analysis. These results suggestconsiderable contribution from both types of charge car-rier. Fig. S6 also clearly depicts that within the accu-racy of our two-band analysis and experimental error,the ratio nh

neremains almost same. Whereas, µh

µein-

creases to a large value (∼ 11) at higher temperaturesfrom ∼ 2.6 at 2 K. As hole and electron density arecomparable, mobility plays important role on charge con-duction. For this reason, the low temperature ρyx vs Bplot appears to be weakly non-linear with upward cur-vature, compared to the ρyx vs B plot at higher tem-peratures, where hole-type carriers dominate the trans-port properties. The obtained hole (electron) density ∼1.8×1021 cm−3 (1.6×1021 cm−3) is quite large comparedto that usually observed (∼ 1017-1019 cm−3) in Dirac andWeyl semimetals10–12,14,17. However, the carrier densityin Pd3Bi2S2 is comparable to that observed in topolog-ical nodal-line semimetals (n∼1020-1023 cm−3)34,35 andconventional metals. In spite of large carrier density, themobility of charge carrier in Pd3Bi2S2 is found to be quitehigh (∼ 0.75×104 cm2 V−1 s−1 for hole and ∼ 0.3×104

cm2 V−1 s−1 for electron) and comparable to that re-ported for different topological electronic materials. Onthe other hand, in conventional metal, the mobility ofcharge carrier is low and the MR obeys the Kohler scal-ing. It appears from the above discussion that some ofthe properties of Pd3Bi2S2 are similar to that of con-ventional metals, whereas some are similar to topologicalsemimetals.

In conclusion, we have synthesized a newly discoveredelectronic material Pd3Bi2S2 and performed the system-atic magneto-electronic transport experiments. A largenon-saturating transverse magnetoresistance has beenobserved. The large anisotropy ratio in MR indicatessignificant anisotropy in the Fermi surface. Hall measure-ment reveals high mobility of charge carriers (∼ 0.75×104

cm2 V−1 s−1 for hole and ∼ 0.3×104 cm2 V−1 s−1 forelectron), in spite of the unusually large carrier density(> 1021 cm−3). Similar to conventional metal, ρxx shows

5

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1 . 4

1 . 6

1 . 8

2 . 0

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2

4

6

8

0 2 4 6 80

1 0 0

2 0 0

3 0 0

4 0 0

n h, n e(1

021 c

m-3 )

T ( K )

n h n e

m h , me (1

03 cm2 V-1 s-1 )

T ( K )

m h m e

( b )

s xy , s

xx (1

06 W-1 m-1 )

B ( T )

A t 2 K s x y s x x F i t t i n g

( a )

FIG. 5. (a) Global fitting of the Hall conductivity (σxy) andconductivity (σxx) using two-band model [Eqs. (1) and (2)].(b) Temperature dependence of electron and hole carrier mo-bility obtained from global fitting. In inset electron and holecarrier density have been shown as a function of temperature.

T 2 dependence at low temperature and T dependence athigh temperature. However, MR does not obey Kohlerscaling. The present results suggest that Pd3Bi2S2 mayhost unconventional electronic band structure beyondthree-dimensional Dirac/Weyl semimetals and can be apossible candidate for several technological applications.

Supplementary Material

In the supplementary material, we have shown the crystalstructure and image of a typical single crystal. Detailedinformation regarding the EDX, longitudinal magnetore-sistance and scaling of magnetoresistance have been pro-vided. We have also included additional discussions andplots regarding the field dependence of Hall conductivityand mobility ratio of the two types of carriers, in thissection.

1M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045-3067 (2010).2X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).3Z. K. Liu, J. Jiang, B. Zhou, Z. J.Wang, Y. Zhang, H. M. Weng, D.Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. Kim,M. Hoesch, Z.

Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain and Y. L. Chen,Nat. Mater. 13, 677 (2014).

4S. -Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang,R. Sankar, G. Chang, Z. Yuan, C. -C. Lee, S. M. Huang, H. Zheng,J. Ma, D. S. Sanchez, B. Wang, A. Bansil, F. Chou, P. P. Shibayev,H. Lin, S. Jia, and M. Z. Hasan, Science 349 (6248), 613 (2015).

5R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105,077001 (2010).

6V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers,and L. P. Kouwenhoven, Science 336, 1003 (2012).

7S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A.H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602(2014).

8B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J.Cava, and B. Andrei Bernevig, Science 353 (6299), aaf5037 (2016).

9M. Kakihana, K. Nishimura, Y. Ashitomi, T. Yara, D. aoki, A. Naka-mura, F. Honda, M. Nakashima, Y. Amako, Y. Uwatoko, T. Sakak-ibara, S. Nakamura, T. Takeuchi, Y. Haga, E. Yamamoto, H. Harima,M. Hedo, T. Nakama, Y. nuki, J. ELEC. MAT. 46, 3572 (2017).

10R. Singha, A. Pariari, B. Satpati, and P. Mandal, Proc. Natl. Acad.Sci. USA 114, 2468 (2017).

11Z. Yuan, H. Lu, Y. Liu, J. Wang, and S. Jia, Phys. Rev. B 93, 184405(2016).

12C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leer-makers, U. Zeitler, Y. Skourski, J. Wosnitza, Z. Liu, Y. Chen, W.Schnelle, H. Borrmann, Y. Grin, C. Felser, and B. Yan, Nat. Phys.11, 645 (2015).

13J. M. Ziman, Electrons and Phonons: The Theory of TransportPhenomena in Solids (Classics Series, Oxford University Press, NewYork, 2001).

14F. F. Tafti, Q. D. Gibson, S. K. Kushwaha, N. Haldolaarachchige,and R. J. Cava, Nat. Phys. 12, 272 (2016).

15Z. J. Yue, X. L. Wang, and S. S. Yan, Appl. Phys. Lett. 107, 112101(2015).

16S. Sun, Q. Wang, P.-J. Guo, K. Liu, and H. Lei, New J. Phys.18,082002 (2016).

17R. Singha, A. Pariari, B. Satpati, and P. Mandal, Phys. Rev. B 96,245138 (2017).

18Y. L. Wang, L. R. Thoutam, Z. L. Xiao, J. Hu, S. Das, Z. Q. Mao,J. Wei, R. Divan, A. Luican-Mayer, W. Crabtree, and W. K. Kwok,Phys. Rev. B 92, 180402(R) (2015).

19A. Pariari, P. Dutta, and P. Mandal, Phys. Rev. B 91, 155139 (2015).20O. Pavlosiuk, P. Swatek, and P. Wisniewski, Sci. Rep. 6, 38691

(2016).21Y. Wang, J. Zhang, W. Zhu, Y. Zou, C. Xi, L. Ma, T. Han, J. Yang,

J. Wang, J. Xu, L. Zhang, L. Pi, C. Zhang, and Y. Zhang, Sci. Rep.6, 31554 (2016).

22A. Wang, I. Zaliznyak, W. Ren, L. Wu, D. Graf, V. O. Garlea, J. B.Warren, E. Bozin, Y. Zhu, and C. Petrovic, Phys. Rev. B 94, 165161(2016).

23S. Huang, J. Kim, W. A. Shelton, E. W. Plummer, and R. Jin, Proc.Natl. Acad. Sci. USA 114, 6256 (2017).

24M. Hirschberger, S. Kushwaha, Z. Wang, Q. Gibson, S. Liang, C. A.Belvin, B. A. Bernevig, R. J. Cava, and N. P. Ong,Nat. Mater. 15,1161-1165 (2016).

25R. H. McKenzie, J. S. Qualls, S. Y. Han, and J. S. Brooks, Phys.Rev. B 57, 11854 (1998).

26L. Wang, M. Yin, J. Jaroszynski, J.-H. Park, G. Mbamalu, and TimirDatta, Appl. Phys. Lett 109, 123104 (2016).

27K. Wang, D. Graf, L. Wang, H. Lei, S. W. Tozer, and C. Petrovic,Phys. Rev. B 85, 041101(R) (2012).

28Y. Zhao, H. Liu, J. Yan, W. An, J. Liu, X. Zhang, H. Wang, Y. Liu,H. Jiang, Q. Li, Y. Wang, X.-Z. Li, D. Mandrus, X. C. Xie, M. Pan,and J. Wang, Phys. Rev. B 92, 041104(R) (2015).

29A. Collaudin, B. Fauque, Y. Fuseya, W. Kang, and K. Behnia, Phys.Rev. X 5, 021022 (2015).

30X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang,M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, Phys. Rev. X 5,031023 (2015).

31H. Li, H. He, H.-Z. Lu, H. Zhang, H. Liu, R. Ma, Z. Fan, S.-Q. Shen,and J. Wang, Nat. Commun. 7, 10301 (2016).

32Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic, A. V.Fedorov, R. D. Zhong, J. A. Schneeloch, G. D. Gu, and T. Valla,Nat. Phys. 12, 550 (2016).

33A. Pariari, and P. Mandal, Sci Rep. 7, 40327 (2017).34J. Hu, Z. Tang, J. Liu, X. Liu, Y. Zhu, D. Graf, K. Myhro, S. Tran,

C. N. Lau, J. Wei, and Z. Mao, Phys. Rev. Lett. 117, 016602 (2016).35R. Sankar, G. Peramaiyan, I. P. Muthuselvam, C. J. Butler, K. Dim-

itri, M. Neupane, G. N. Rao, M.-T. Lin, and F. C. Chou, Sci. Rep.

7, 40603 (2017).