Weyl fermions and the Anomalous Hall Effect

Transcription

1 Weyl fermions and the Anomalous Hall Effect Anton Burkov CAP congress, Montreal, May 29, 2013

2 Outline Introduction: Weyl fermions in condensed matter, Weyl semimetals. Anomalous Hall Effect in ferromagnets and its relation to Weyl semimetals. AHE in SrRuO3.

3 Dirac materials Graphene. Topological Insulators. Weyl semimetals. Ordinary metallic ferromagnets.

4 Weyl semimetals In 3D two nondegenerate bands can generically cross at isolated points in momentum space (Herring, 1937). This is because one needs to tune 3 real parameters to make two levels cross (von Neumann & Wigner): the 3 parameters are 3 components of crystal momentum. Each band crossing is described by Weyl Hamiltonian: H(k) =±v F σ k

5 Weyl nodes in 3D H(k) =±v F σ k Volovik, 2003 An isolated Weyl node (can be visualized as a hedgehog in momentum space) is absolutely indestructible. The only way to get rid of a Weyl node is to annihilate it with an antinode : a Weyl node of opposite chirality. Nielsen-Ninomiya theorem: Weyl fermions always appear in pairs of opposite chirality. Topological robustness results from separating such pairs in momentum space: Weyl semimetal.

6 Obstacle: Kramers degeneracy Kramers theorem: all bands doubly degenerate in the presence of TR and I symmetries. Degeneracy of band pairs requires fine-tuning. Need to break either TR or I to achieve Weyl point degeneracies in bandstructure. Weyl semimetal: need to further align Fermi level with the nodes.

7 Proposed realizations Pyrochlore iridates. Y 2 Ir 2 O 7 Wan, Turner, Vishwanath, Savrasov, 2011 HgCrSe. Xu, Weng, Wang, Dai, Fang, 2011 Any magnetic material near TI-NI transition. AB and Balents, 2011 TI NI

8 Weyl semimetal: 3D generalization of the quantum Hall effect Trivial generalization of QHE to 3D: a stack of 2D QH systems. B d σ xy = e2 hd = e2 h G 2π Kohmoto, Halperin, Wu, 1992

9 Weyl semimetal: 3D generalization of the quantum Hall effect Trivial generalization of QHE to 3D: a stack of 2D QH systems. Ε Kohmoto, Halperin, Wu, G k z σ xy = π/d π/d 1 2 dk z 2π e 2 h C(k z)= e2 h G 2π Quantized 3D Hall conductivity, associated with chiral 2D surface states.

10 Weyl semimetal: 3D generalization of the quantum Hall effect Nontrivial generalization: Weyl semimetal. Chern number changes as a function of kz. Ε 0.5 K k z 0.5 AB, Balents, 2011 Non-quantized 3D Hall conductivity, but still associated with chiral 2D surface states ( Fermi arcs ). σ xy = K/2 K/2 dk z 2π e 2 h C(k z)= e2 h K 2π

11 Anomalous Hall Effect AHE is a contribution to the Hall resistivity of a ferromagnetic metal, that does not vanish even when the external magnetic field is absent. ρ xy = R 0 B +4πR s M FIG. 1. The Hall effect in Ni data from Smith, From Pugh and Rostoker, 1953.

12 Intrinsic Anomalous Hall Effect Broken TR and SO interactions are needed, otherwise many distinct sources of AHE are possible. Recent work shows that topological electronic structure properties likely very important in many materials ( intrinsic AHE ). Broken TR and SO interactions lead to nontrivial momentumdependent spinor structure of Bloch wavefunctions, which may be thought of as analog of Aharonov-Bohm effect in momentum space: Berry connection: Berry curvature: Anomalous velocity: A nk = ink k nk Ω nk = k A nk v nk = e E Ω nk

13 Intrinsic Anomalous Hall Effect Intrinsic anomalous Hall conductivity is given by integral of Berry curvature over occupied states. σ xy = e2 n d 3 k (2π) 3 n F ( nk )Ω z nk Sundaram & Niu, 1999 Jungwirth, Niu, MacDonald, 2002

14 Intrinsic AHE: Fermi surface property? View electronic structure of 3D ferromagnet as a stack of 2D electronic structures, parametrized by crystal momentum along the magnetization direction (z-direction). σ xy = π/a π/a dk z 2π σ2d xy (k z ) σ 2D xy (k z )= e2 n d 2 k (2π) 2 n F [ nk (k z )][ k A nk (k z )] z Haldane, 2004

15 Intrinsic AHE: Fermi surface property? Use Stokes theorem to rewrite as integral over Fermi surface: σ 2D xy (k z )= e2 n d 2 k (2π) 2 n F [ nk (k z )][ k A nk (k z )] z σ 2D xy (k z )= e2 2πh n dk A nk (k z ) Haldane, 2004

16 Intrinsic AHE: Fermi surface property? Conclusion: AHE is a Fermi surface property, like all transport properties of metals, modulo quantized contribution, coming from completely filled bands. σ xy = e2 h G 2π

17 Intrinsic AHE: Fermi surface property? Conclusion: AHE is a Fermi surface property, like all transport properties of metals, modulo quantized contribution, coming from completely filled bands. This is typically regarded as exotic (unrealistically large magnetization needed) and ignored. σ xy = e2 h G 2π Haldane, 2004

18 Intrinsic AHE: NOT a Fermi surface property! The above reasoning ignores Weyl nodes, which act as quantum Hall transition points in momentum space, changing the Chern numbers. This gives a nonquantized contribution to AHE, which is not associated with Fermi surface, but is associated with Fermi arc surface states. Existence of Weyl nodes does not require unrealistic conditions, they are likely present in the bandstructure of any ferromagnet.

19 AHE in SrRuO3 Perovskite structure, typically slightly distorted, but this is not important for us. Relatively simple electronic structure, consisting of 6 bands, derived mostly from the t2g orbitals of Ru. Chen, Bergman, AB, 2013

20 AHE in SrRuO3 Perovskite structure, typically slightly distorted, but this is not important for us. Relatively simply electronic structure, consisting of 6 bands, derived mostly from the t2g orbitals of Ru. H = k a kσ δ ab δ σσ + f ab k δ σσ + iλ abc τ c σσ d kaσ d kbσ

21 AHE in SrRuO3 Et k z (a) λ Et k z (b) FIG. 1. (Color online). (a) Bandstructure of SrRuO al several of the bands. (b) Same bandstructure but with λ/t 0.4. While some of the crossings are eliminated, most clea 3 the k x = k y =0linewithλ/t 1 =0,t 2 /t 1 = 0.2, f/t 1 0.2, and m/t 1 = 1. Nonzero m induces crossings betw survive. SO interaction does not eliminate band crossings! understand the important role, played by the Weyl no in the intrinsic AHE, is to realize that these nodes as magnetic-monopole-like sources of the Berry cur

22 AHE in SrRuO3 E t X M Z R A Z A M X R Total of 22 pairs of Weyl nodes!

23 AHE in SrRuO3 Weyl nodes are quantum Hall transition points in momentum space: C 2 k z σ 2D xy (k z )= k z 1 e2 2πh n 2 (a) dk A nk (k z )+ e2 h n C n (k z ),

24 AHE in SrRuO3 Integral of the Chern number term over kz gives a nonquantized contribution to Hall conductivity, associated with σ edge xy Fermi arc surface states: = e2 h Σ xy (a) π Ε F t 1 π dk z 2π n C n (k z ) Σ xy Σ xy (a) Ε F t mt 1 (b) FIG. 4. (Color online). Total anomalous Hall conductivity

25 Conclusions Range of materials with topologically-nontrivial electronic structure goes far beyond topological insulators. (Almost) any ferromagnetic metal has chiral Fermi arc edge states, which carry part of its anomalous Hall conductivity. Would be interesting to find a way to separate out this contribution experimentally (perhaps orientation dependence?).

26 Collaborators Yige Chen Doron Bergman