Weyl semi-metal: a New Topological State in Condensed Matter

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1 Weyl semi-metal: a New Topological State in Condensed Matter Sergey Savrasov Department of Physics, University of California, Davis Xiangang Wan Nanjing University Ari Turner and Ashvin Vishwanath UC Berkeley Yekaterinburg, September 23, 2016

2 Chiral Quantum Hall States in Condensed Matter Klitzing, K.; Dorda, G.; Pepper, M. (1980). Phys. Rev. Lett. 45 (6): Bulk insulating state with zero conductivity σ = 0 A 2D quantum Hall state is like an insulator but has chiral edge state, quantized Hall conductivity σ 2 xy = ne / h Both states are differentiated by Chern number: i = 2 n d k uk k uk 2π BZ bands Berry Flux

3 Quantum Spin Hall Insulator (2D Topological Insulator) Bernevig, B. A., and S. C. Zhang, (2006) Phys. Rev. Lett. 96, Quantum Hall state permits conduction in one dimension along sample boundary Quantum Spin Hall state at zero magnetic field permits conduction in spin filtered edge states. (HgTe-CdTe quantum well structures) Conductivity is dissipationless since there is no state for the electron at the edge to back scatter on non-magnetic impurity!

4 3D Topological Insulators Hadj M. Benia, Chengtian Lin, Klaus Kern, and Christian R. Ast, PRL 107, (2011)

5 Simplest Realization of Topological Insulator The 4x4 Hamiltonian for 3D cubic lattice that shows protected surface states where α and β are 4x4 Dirac matrices.

6 Surface Band Structures Projected Bulk Spectrum Increasing the size of cell along z axis z shrinks the size of the BZ along kz axis Single Ten Cells M (001) Surface BZ ΓXM are surface TRIMs Band Structures for all k z =const planes are projected to k z =0 plane

7 Surface States - Slab Calculations z Vacuum layer M (001) Surface BZ ΓXM are surface TRIMs In Slab Calculation Surface(s) ) Spectrum appears on top of projected bulk E(k)

8 Surface states E F Projected Bulk Spectrum Slab Spectrum

9 From Topological lnsulator to Weyl Semimetal (Wan, Turner, Vishwanath, SS, PRB2011)

10 Dirac Points in 3D Band Structures A modified TB model with two orbitals per site shows 3D Dirac/Weyl points at Ef

11 Weyl Semi-Metal and its Berry Flux In the vicinity of Weyl Point: q=k-k D E ( ) ( ) 2 ± q =± vi q H ( q) = vi qσ i i= xyz The Berry curvature is evaluated to be i= xyz N occ 1 1 Ω ( k) = i kukn kukn εijk ( vi vj )( vk q) n= 1 ijk 2 ( ) (only Weyl point contributes) 2 ( v ) 3/2 i q i Integrating over small sphere surrounding Weyl point produces flux that is given by chiral charge c 1 c = ds ( k) sign( v1 v2 v3) 2π Ω = S Weyl point acts as a magnetic monopole at the origin: whose charge is given by chirality Ω ( q) = 1 q c 2 q 3

12 Weyl Semi-Metal: Formation of Surface States Construct a curve in the surface Brillouin zone encircling the projection of the bulk Weyl point. Consider H(λ, k z ) = H(k λ, k z ), the gapped Hamiltonian of the two dimensional subsystem defined by this curve. The Chern number of this two dimensional band structure is given by the Berry curvature integration which, corresponds to the monopole density enclosed within the torus, that is the chirality c of enclosed Weyl node. Then, the two dimensional subsystem defines a quantum Hall insulator with unit Chern number. When defined on the half space z < 0, this corresponds to putting the quantum Hall state on a cylinder, and hence we expect a chiral edge state. Its energy E(k λ ) spans the band gap of the subsystem, as k λ is varied. Hence, this surface state crosses zero energy somewhere on the surface Brillouin zone. Gap k λ k λ defines a 2D subsystem with non-zero C number: we expect a Chiral Edge State

13 Weyl Semi-Metal: Formation of Fermi Arc Fermi Arc connects Weyl points of opposite chirality

14 Tight-Binding Simulation of (001) Surface Band Structure Fermi Arcs connecting Weyl points of opposite chirality can be directly observed

15 Properties of Weyl semimetals Each Weyl point can be characterized by chiral charge determined in terms of electron velocities at this k W point: Anomalous Hall conductivity is characterized by Chern vector ν c (Haldane, 2004) 2 2 e ie σab = εabcνc = [ k u ] a k k u b k k u b k k u a k 2πh 2πh c c = s ign( v v v) Weyl points are topological: they can only be eliminated by recombining with each other. bands Yang, Lu, and Ran, PRB 2011: In general Weyl semimetal ν nodes = ( 1) ξi Pi where P i is momentum of each node and ξ i is its chirality. i No bulk conductivity as in topological insulators, only surface Fermi arcs conduct.

16 dk/dt = ev k x B These magnetic orbits involving Fermi arcs are distinguished from conventional magnetic orbits in ordinary 2D systems or in surface states of 3D systems by the dependence on the slab thickness, L. 2π εn ( n + γ) = μ T T t + t k / evb+ L/ v arc 1 πv 1/ B = ek0 ( n + γ ) L μ B k / L sat 0 slab 0

17 Chiral Anomaly and Large Negative Magnetoresistance If electric field E is applied along B, all states move along the field according to dk/dt= ee. Zeroth Landau level is chiral, i.e., it disperses only one way for each Weyl node. Therefore, motion of the states along E corresponds to electrons disappearing from right-moving band and reappearing in the left-moving one. In order to equilibrate the charge imbalance, large-momentum internode scattering is required. Such processes are weak, the internode scattering time τ is very large, causing a remarkable increase in conductivity:

18 Pyrochlore Iridates as Novel Topological Systems Geometrically Frustrated Pyrochlore Lattice A 2 Ir 2 O 7 where A is Yttrium or RE

19 Relativistic 5d electrons j=5/2 Γ 8 Ir 5d 5 Γ 7 j=3/2 A small Hubbard U produces Mott insulating behavior for Γ 7 band at half feeling

20 Electronic Structure Studies of Y 2 Ir 2 O 7 and Rare Earth Pyrochlore Iridates LDA+U studies using full potential LMTO method. Ir 5d electrons: 0<U<4 ev; Rare Earth 4f electrons: U= 6eV Spin Orbit Coupling Various Magnetic (Collinear and Non-Collinear) Configurations explored for frustrated pyrochlore lattice

21 Studies of Magnetic Configurations in Y 2 Ir 2 O 7 Collinear (111) Non-Collinear 2-in/2-out Non-Collinear All-in/out Collinear (001) Magnetic Non-Collinear All-In/Out Configuration has lowest energy!

22 Electronic Structure as Function of Moments Orientations: LDA+U=1.5 ev Collinear (001) Collinear (111) Non-Collinear All-in/out Large Fermi Surface Normal Metal! Small Fermi Surface Bad Metal Looks Like Insulator! (in fact depends on U) Metallic behavior appears when moments are aligned collinearly, easy to switch with applied magnetic field Magnetic Field Induced Insulator to Metal Transition!

23 Birth of Weyl semi metal phase at intermediate values of 1.0 ev<u<1.8 ev Calculated energy bands using LDA+SO+U=1.5 ev LEFT:within k z =0 plane of BZ RIGHT:for k z =0.3 plane of BZ

24 Weyl Points in Iridates There are positive Weyl points at locations k W and negative Weyl points at locations k W. Here is our case: Positive and negative Weyl points in BZ

25 Tight-Binding Simulation of (110) Surface Band Structure for Iridates Fermi Arcs connecting Weyl points of opposite chirality can be directly observed

26 Suggested Electronic Phase Diagram of Iridates

27 Osmium Spinels (d 5 ) Spinel compounds have B-sites forming a corner sharing tetrahedral network. It may be expected that spinel structure will be more tunable by pressure, external fields or by doping as compared to closely packed pyrochlore lattice. Valence arguments: AOs 2 O 4 (A=Mg, Ca, Sr, Ba) 3s band of Mg appears around the Fermi level, thus we do not consider MgOs 2 O 4 (Wan, Vishwanath, SS, PRL 2012)

28 Optimizing Lattice The Os-Os bond length of spinel osmates is shorter than that of Y 2 Ir 2 O 7, and one can expect that the U in CaOs 2 O 4 is smaller than in Y 2 Ir 2 O 7. We therefore believe that the U is in the range between 0.5 and 1.5 ev.

29 Magnetic configuration Collinear (001) Non-Collinear 2-in/2-out Non-Collinear All-in/out

30 Suggested Electronic Phase Diagram for CaOs 2 O 4 SrOs 2 O 4 shows the same rich phase diagram as Iridates!

31 Challenges in finding Weyl Semimetals So many compounds have negative MR, transport does not help!! Weyl nodes at arbitrary k points Where is the Fermi level? How to separate surface and bulk? Calculation/Prediction can NOT tell us that it will work, so many people predicted so many Weyl compounds

32 Recent Experimetal Discoveries

33 Bulk Electronic Structure of TaAs: : 24 Weyl Nodes Yan Sun, Shu-Chun Wu, and Binghai Yan, PRB 2015

34 Detecting Fermi Arcs in TaAs by ARPES

35 Conclusion Novel Weyl semimetals have been predicted: Weyl nodes are topological objects, they cannot be gapped unless by annihilating with each other. They give rise to protected surface states similar to topological insulators: Fermi arcs at surface BZ that connect projected bulk Weyl points of opposite chirality. They show a variety of interesting properties: a giant anomalous Hall effect, a large negative magnetoresitance and so on. Pyrochlore Iridates could be candidates for Weyl semimetals (also hypothetical Os spinels) via time reversal symmetry breaking due to magnetic ordering. Recent discoveries of inversion symmetry broken materials such as TaAs confirmed theoretical predictions.