3D Weyl metallic states realized in the Bi 1-x Sb x alloy and BiTeI. Heon-Jung Kim Department of Physics, Daegu University, Korea

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1 3D Weyl metallic states realized in the Bi 1-x Sb x alloy and BiTeI Heon-Jung Kim Department of Physics, Daegu University, Korea

2 Content 3D Dirac metals Search for 3D generalization of graphene Bi 1-x Sb x alloy and 3D Weyl fermions Zero-gap state in Bi 1-x Sb x Topological phase transition, topological critical point 3D Weyl fermions in Bi 1-x Sb x Topological EB term Adler-Bell-Jackiw (chiral) anomaly Anomalous transport phenomena BiTeI : another 3D Weyl metal Helical Fermi liquid system with a Weyl point Extreme disparity of mobility between IFS and OFS Hall resistivity scaling & antilocalization Prospect and summary

3 Dirac fermions in the flatland Graphene 3D generalization of graphene or TI Dirac fermions in the 3D world Search for 3D Dirac metals

4 Dirac vs. Weyl fermion A particle whose state is described by Dirac theory Massless case 1) Dirac fermions need 4-component spinor Particle in the positive E Hole in the negative E particle hole 2) Weyl fermions need 2-component spinor has definite chirality Massless Dirac fermion with well-defined chirality Effectively described by a 2-compoent spinor Neutrino may be a Weyl particle if its mass is zero

5 What is a candidate material of a 3D Dirac/Weyl metal? Q1) Zero-gap semiconductor with linear band dispersion? Bi 0.97 Sb 0.03 at magnetic field Q2) Does it have any properties related with nontrivial topological structure? Chiral anomaly

6 Bismuth (Bi) wikipedia Rhombohedral lattice Space group : R3m (166) Typical semimetal e and h Fermi surfaces at L and T e-pocket at L Brillouin zone h-pocket at T

7 Phase diagram of Bi 1-x Sb x alloy Band insulator Topological insulator x < 3 % : band insulator x ~ 3-4 % : zero-gap semiconductor or 3D Dirac metal? x > 7 % : 3D TI with inverted band

8 Theoretical predictions for topological critical point & topological phase transition with I or TR symmetry breaking S. Murakami, 2007 I or TR symmetry breaking Splitting of a single Dirac band into two bands with well defined chirality Weyl semimetal

9 Weyl fermions in the Bi 1-x Sb x alloy Energy -k x Trigonal Binary Bisectrix 3~4 : Inversion of the band at L x (%) T L +k x ~ 3-4%: Dirac metal in 3+1 d Quantum critical point B x External B field breaks TR symmetry

10 What is special about Weyl fermions in 3D? (nontrivial topological structure) Berry curvature (single Dirac cone) : antilocalization correction Adler-Bell-Jackiw or chiral anomaly +k x -k x Chiral anomaly : The anomalous non-conservation of a chiral current in the momentum space Anomalous transport phenomena [H. B. Nielsen and M. Ninomiya, Physics Letters 130B, 389 (1983)] Chiral anomaly is expressed by a topological term ( r ) E B

11 Adler-Bell-Jackiw anomaly (1d) N N R R N e E & N 2 L L 0 & N R e E 2 N L Q 5 e E

12 Adler-Bell-Jackiw anomaly (3d) EB e Q N N N N EB e N EB e N L R L R L R & 4 & 4 Ultra-quantum region, quasi-classical region, T. D. Son and B. Z. Spivak, PRB 88, (2013) Essentially 1D due to Landau levels

13 Chiral anomaly and the nontrivial topological properties is represented by a topological term ( r ) E B This also leads to topological magnetoelectric effect and axion electrodynamics

14 Topological Magnetoelectric effect Multiferroics Local curvature effect Transverse or longitudinal depending on mechanism Topological magnetoelectric effect Global curvature effect Longitudinal from the chiral anomaly Axion electrodynamics Axion electrodynamics

15 Spatial dependence of (r) 3 0 3

16 One consequence of axion electrodynamics (chiral anomaly) is anomalous electrical transport 1. Anomalous Hall effect 2. Chiral magnetic effect 3. Negative longitudinal MR (We observed this in Bi 0.97 Sb 0.03 )

17 MR MR = 90º (a) (b) = 0 = 90º Static B field = 0 B (T) Magnetic field B θ Current I T = 2K B cos ( 3D WAL LMR Phys. Rev. Lett. 111, (2013) Phys. Rev. B 89, (2014) Pulsed B field B (T) T = 4.2 K TMR

18 Origin of negative longitudinal MR for B//E : chiral anomaly The main consequence of the increasing distance between the paired Weyl cones : increase of Negative MR ( 1/ ) Ultra-quantum or quasi-classical region? Theory is necessary to determine all these quantitatively

19 Theoretical results in the semi-classical region (3) L // -MR, xx B x (4) B yx x For the single Weyl cone E B

20 Conductance ( (a) Transverse MC WAL + n Analysis based on semiclassical formula B (T) Conductance ( 76 Longitudinal MC (b) (1+CB 2 ) WAL + n B (T)

21 Analysis of different T data (L-MR) (ks/cm) K a = 1.2 b = 61 c = 0.10 d = 25 e = (ks/cm) K a = 2.3 b = 57.5 c = 0.14 d = 25 e = (ks/cm) K a = 3.0 b = 54.8 c = 0.15 d = 25 e = (ks/cm) B (T) 30 K a = 4.0 b = 53.5 c = 0.15 d = 25 e = (ks/cm) B (T) 50 K a = 6.0 b = 56.5 c = 0.12 d = 25 e = 0.04 (ks/cm) B (T) 70 K a = 9.7 b = 65.8 c = 0.22 d = 25 e = (ks/cm) B (T) B (T) 100 K a = 8.0 b = 68 c = 0.10 d = 25 e = B (T) C W (T -2 ) CW (T -2 ) B (T) (d) (1-T/Tc) T (K)

22 BiTeI : another 3D Weyl system Search for physics of a 3D Dirac/Weyl metal It is known as a bulk Rashba system But we believe it is another 3D Weyl metal Topological structure, chiral anomaly, etc Difference and similarity with graphene and BiSb?

23 Giant Rashba material BiTeI Is BiTeI a 3D Weyl metal? Does it have any nontrivial topological structure?

24 BiTeI : another 3D Weyl metal Electronic structure of BiTeI in comparison with BiSb The first kind (BiSb) The second kind (BiTeI) A pair of Weyl points at the same E but at different k A pair of Weyl points at the same k but at different E

25 Graphene Disordered BiTeI Disordered BiTeI may have a groundstate different from clean one BiTeI Spinless fermions in two Dirac cones at K and K Valley degeneracy Single Dirac cone governs physics No helicity (No chirality) Disorder Intra- and inter-valley scattering Two different bands: IFS & OFS Definite helicity (spin-chirality) Disorder Intra- and inter-valley scattering Smearing of helicity through Inter-valley scattering

26 Getting information of impurity scattering Experimental method : MR and Hall For the case of single charge-carriers Hall resistance Magnetoresistance

27 Tuning of Fermi energy E F E F Weyl point tuning (mcm) #1 [x0.05] #2 [x0.2] #3 [x0.3] #4 [x0.3] #5 # T (K) 6 BiTeI crystals with different E F

28 MR & Hall resistance MR (%) #1 #2 #3 #4 #5 #6 [x0.5] B (T) H (cm) 100 #1 [x0.5] #2 #3 50 #4 #5 [x10] #6 [x10] B (T)

29 MR and Hall resistivity are quite unusual MR (%) #1 #2 #3 #4 #5 # B (T) H (cm) #1 #2 #3 [x0.5] #4 [x0.2] #5 [x5] #6 [x10] B (T) Possible reason : two different types of charge carriers Two-carrier model: One with larger mobility and the other with smaller

30 Analysis : two-carrier model 1. Hall resistance Hall coefficient is given by Hall coefficient in the two-band model In the limit that the magnetic field is small with l << h H 1 nhec 1 Β 2 B n B lec h

31 Hall resistance H (cm) #1 [x0.5] #6 [x10] H (cm) #2 #3 #4 [x2] #5 [x5] B (T) B (T) H 1 nhec 1 Β B 2 h n h > 0 for #1 ~ #3 (hole type) n h < 0 for #4 ~ #6 (electron type)

32 2. Magnetoresistance (MR) total OFS WF OFS 1 2 out out out out WAL out WAL out out WF 1 2 in in in in WAL in WAL in in OFS out out WAL outout N out out WAL WF in in WAL inin N in in WAL N out A B A B N 2 B 1 in out out out out out in A in total c N out N in WAL out in WAL WAL WAL ( aout ain ) B

33 Decomposition of MR with different conduction channels b

34 Extreme disparity of mobility between IFS and OFS Divergent IFS mobility at the Weyl point IFS ( WF ) ~ m 2 /Vs ( OFS ) ~ m 2 /Vs Hall resistance : IFS (WF ) /( en ) ( OFS ) 1 OFS 0 MR : / IFS Ain in / ( OFS ) Aout out

35 Origin for extremely large IFS mobility Reduced phase space due to small density of state near the Weyl point and spin-chirality (helicity) Suppressed IFS intra-valley scattering near the Weyl point Theoretical consideration Self-consistent Born approximation (SCBA) in the Rashba model Calculation of the IFS and OFS mobility Weak inter-valley scattering (preserved helicity)

36 IFS and OFS mobility as a function of E F Solid lines : Weak inter-valley scattering Dashed lines : No inter-valley scattering Inter-valley scatter is weak or negligible? (ev)

37 Growing interest on 3D Dirac/Weyl metals Recent growing interest on 3D Dirac metals BiO 2, Cd 3 As 2, Na 3 Bi Young, S. M. et al. Phys. Rev. Lett. 108, (2012) [BiO 2, theory] Liu, Z. K. et al., Science 343, 864 (2014) [Na 3 Bi, Exp.] Neupane, M. et al., Nat. Commun. 5, 3786 (2014) [Cd 3 As 2, Exp.] Borisenko, S. et al. Experimental realization of a three-dimensional dirac semimetal. Phys. Rev. Lett. 113, (2014). [Cd 3 As 2, Exp.] Liu, Z. K. et al., Nat. Mater. 13, (2014). [Cd 3 As 2, Exp.] Xu, S.-Y. et al., Science 347, 294 (2015) [Na 3 Bi, Exp.] No clear evidence about nontrivial topology

38 Supporting results : negative MR in TaAs TaAs : proposed 3D Weyl metal in zero magnetic field Jia (Peking) and Hasan (Princeton), unpublished Chen (Chinese Academy of Science), unpublished Negative longitudinal MR This feature is exactly same as the one in BiSb

39 Summary A 3D Weyl metal is a new state of matter with nontrivial topological structure (chiral anomaly) Bi 0.97 Sb 0.03 in magnetic field (1 st kind) Two Weyl cones are not independent (chiral anomaly) Anomalous transport chiral anomaly (axion electrodynamics) BiTeI (2 nd kind) Huge difference between IFS and OFS mobility Divergent IFS mobility near the Weyl point Weak inter-valley scattering

40 Collaborators Prof. Ki-Seok Kim in POSTECH Prof. M. Sasaki in Yamagata University Thank you very much