Jung Hoon Han ( S ung K yun K wan U , Suwon)

1. Skyrmions and Anomalous Hall Effect in a Dzyaloshinskii-Moriya Magnet Jung HoonHan (SungKyunKwanU, Suwon) • Su Do Yi, Jin Hong Park SKKU • Shigeki Onoda RIKEN • Naoto Nagaosa U of Tokyo

3. What is MnSi ? • Nearly ferromagnetic metal • Spiral spins with a long modulation period l~180A below Tc~29.5K • Dzyaloshinskii-Moriya (DM) interactionis found responsible for spirality • 4 Mn & 4 Si atoms in a unit cell • Curie-Weiss fit moment ~ 2.2mB • Ordered moment ~ 0.4mB Nakanishi et al. SSC 35, 995 (1980)

4. The local moment electrons and itinerant electrons co-exist (probably inseparable), theoretically treated as separable objects • Aspects of spiral magnetism has been addressed a long way back

5. Per Bak’s Model of Spiral Spins • Due to lack of inversion symmetry, a term with a single gradient is allowed in GL theory; a spiral spin structure with nonzero modulationvector k is stabilized Bak & Jensen J Phys C 13, 881 (1980)

6. k k

7. PhaseDiagram of MnSi (ambient pressure) Muhlbauer et al. Science 323, 915 (2009)

8. Bragg spots of hexagonal symmetry found in A-phase Muhlbauer et al. Science 323, 915 (2009) • Neutron Bragg spots of hexagonal symmetry • Interpreted as the triangular lattice of anti-Skyrmions

9. Simple GL argument • Interaction effects in GL theory gives rise to • With uniform magnetic field, this interaction becomes • Three vectors form a closed triangle -> crystal of hexagonal symmetry k3 k2 k1

10. Hall effect of topological origin in A-phase MnSi • A-phase= • Hexagonal Skyrme crystal= • AHE Neubauer et al. PRL 102, 186602 (2009)

11. Hall effect of topological origin in MnSi Lee et al. PRL 102, 186601 (2009)

12. Skyrmion number and Anomalous Hall Effect (AHE) • A number of ideas relating the topologicalspin texture to AHE appeared in the past decade • In a model of coupled local and itinerant moments, the spin texture of the underlying moments acts as an effective magnetic field with the strength given by • A nonzero Skyrmion number = nonzero uniform B Jinwu Ye et al. PRL 83, 3737 (1999) Chun et al. PRL 84, 757 (2000); PRB 63, 184426 (2001) Bruno et al. PRL 93, 096806 (2004) Binz &Vishwanath, Physica B 403, 1336 (2008)

13. Single Skyrmion e • An electron zipping through such a spin texture • “feels” a flux quantum h/e

14. AHE experiments • Ong’s group finds AHE under pressure + mag. field • Pfleiderer’s group finds AHE under mag. field alone • Both groups say AHE is due to nonzero spin chirality (due to Skyrmion condensation) Bext Bind >Bext Bext= 0 Bind

15. A Two-step Strategy • Choose a classical spin Hamiltonian to obtain spin configuration (Monte Carlo) – on a lattice • Choose sd Hamiltonian with local moment Sr from previous classical spin Hamiltonian; diagonalizeHsd • Use Kubo formula for transverse conductivity

16. Hamiltonian - Classical • All three terms (J<0) appear in the superexchange calculation with spin-dependent (spin-orbit-mediated) hopping • J~ l0, K~ l1, A2~ l2 , l=spin-orbit energy

17. PhaseDiagram (2D, fixed J & K, T=0) • A1 plays a minor role • Small Zeeman and A2 gives spiral spin (SS) • Large compass term A2 gives Skyrme crystal (SC2) • Large Zeeman gives hexagonal Skyrme crystal (SCh)

22. There is a (visual) analogy to Abrikosov vortex lattice physics in type-II superconductors • I do not think there is a theory that allows us to interprete the Skyrme crystal lattice in strict analogy with Abrikosov lattice • In Abrikosov lattice inter-vortex distance is set by magnetic field • In Skyrme crystal lattice inter-vortex distance is still set by K/J, which also sets the spiral spin period

24. Skyrmion Textures • Spin texture analyzed by FT: Sk=SrSreik*r • Spin texture analyzed by local Skyrmion number, • or spin chiralitycr

25. A Two-step Strategy • Choose a classical spin Hamiltonian to obtain spin configuration (Monte Carlo) • Choose sd Hamiltonian with local moment Sr from previous classical spin Hamiltonian; diagonalizeHsd • Use Kubo formula for transverse conductivity

26. One can prove that any COLLINEAR spin configuration automatically gives sxy = 0, hence noncollinear spin configuration is a pre-requisite of AHE

27. Evolution of sxywith magnetic field SCh SP sxy averaged over ~ 100 MC spin configurations SS

28. sxyfor various Skyrme crystal states (T/J=0.1) • Hc > 0 Hc = 0 Hc = 0

29. sxyfor various Skyrme crystal states (T/J=0.5) • For SC1 & SC2, skyrmion number is locally nonzero, but globally zero. • External field renders a nonzero global average, hence nonzero sxy

30. Comparison to AHE seen in A-phase • Experimentally, onset of anomalous part occurs above threshold (HC > 0) Pfleiderer’s group (Beff ~ 2.5T) Ong’s group (Beff ~ 40T)

31. MnSi A phase (neutron diffraction & AHE) is consistent with condensation of SCh • Is there an analogue of SC2 in MnSi?

32. MnSi under high pressure • MnSi under large pressure has diffuse Bragg peaks along [110] and its symmetry equivalents – “Partial order” • When partial order is interpreted as multiple-spiral order, our phase diagram bears similarity to the experiment Pfleiderer et al. Nature 427, 227 (2004)

33. Theoretical proposals for “partial order” • Several theories of Ginzburg-Landau variety have been proposed Tewari et al. PRL 96, 047207 (2006) Binz, Vishwanath, Aji, PRL 96, 207202 (2006) Fischer, Shah, Rosch, PRB 77, 024415 (2008) • Our classical spin model is (in some sense) a microscopic answer to these phenomenological theories • GL models were in 3D, ours is in 2D • In our 3D simulation, we are unable to identify a 3D Skyrme crystal phase

34. Suppression of sxxin AHE regime • Numerically we find suppression of sxxwhen sxyis significant • Black- sxy • Gray - sxx • Kinks in magnetoresistancerxxis expected

35. Summary (scientific) We address the problem of electron conduction in a spiral metallic magnet We adopt a two-stage approach: Use (classical) spin Hamiltonian (J+K+A+H) to capture magnetic configurations Use sd Hamiltonian to calculate conductivities for given spin configuration The approach is ad-hoc, but appears to capture a lot of physics ofMnSi