Skyrmions and Anomalous Hall Effect in a Dzyaloshinskii-Moriya Magnet Jung Hoon Han (SungKyunKwanU, Suwon) Su Do Yi SKKU Shigeki Onoda RIKEN Naoto Nagaosa U of Tokyo arxiv:0903.3272v1
Nearly ferromagnetic metal What is MnSi? Spiral spins with a long modulation period λ 180Α below T c ~29.5K Dzyaloshinskii-Moriya (DM) interaction is found responsible for spirality 2 4 Mn/Si atoms in a unit cell Curie-Weiss fit moment ~ 2.2μ B Ordered moment ~ 0.4μ B Nakanishi et al. SSC 35, 995 (1980)
Per Bak s Model of Spiral Spins 3 Due to lack of inversion symmetry, a term with a single gradient is allowed in GL theory; a spiral spin structure with nonzero modulation vector k is stabilized Bak & Jensen J Phys C 13, 881 (1980)
Bogdanov ogy 4 Initial suggestion of a Skyrmionic magnetic spin configuration under magnetic field in a chiral magnet Facilitation of Skyrmionic state for uniaxial anisotropy Stable Skyrme crystal solution of square symmetry without magnetic field for soft spins Ref: Bogdanov & Yablonskii, JETP (1988) Bodanov & Hubert, JMMM (1994, 1999) Bogdanov & Roessler, PRL (2001) Roessler, Bogdanov & Pfleiderer, Nature (2006)
Phase Diagram of MnSi (ambient pressure) Muhlbauer et al. Science 323, 915 (2009)
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
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 One should evaluate and compare free energies of Skyrme crystal states against other possible states such as conical, or spiral (Muhlbauer s Science paper; also Bogdanov s early papers)
Hall effect of topological origin in A-phase MnSi 8 Neubauer et al. PRL 102, 186602 (2009)
Hall effect of topological origin in MnSi 9 Lee et al. PRL 102, 186601 (2009)
10 Skyrmion number and Topological Hall Effect (THE) A number of ideas relating the topological spin texture to AHE /THE appeared in the past decade 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) 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
THE experiments Ong s group finds THE under pressure + mag. field Pfleiderer s group finds THE under mag. Field alone Both groups say THE is due to nonzero spin chirality Both groups find THE for B>B c We will claim that both groups results are consistent with condensation of Skyrme crystal phase
A Two-step Strategy Choose a classical spin Hamiltonian to obtain spin configuration (Monte Carlo) on a lattice Choose sd Hamiltonian with local moment S r from previous classical spin Hamiltonian; diagonalize H sd Use Kubo formula for transverse conductivity
Hamiltonian - Classical FM exchange DM Kitaev-type anisotropy Single-ion anisotropy Zeeman
Hamiltonian - Classical FM exchange DM Compass term Single-ion anisotropy Zeeman
Disclaimer The lattice model we use reduces to Bak-Jensen free energy in the continuum Real MnSi structure is more complicated (4 Mn/cell) Some try to study MnSi using more realistic microscopic model (Kee&Hopkinson) Initial motivation was to study a toy model of metallic magnets with nontrivial spin texture (such as in multiferroics)
Hamiltonian - Classical FM exchange DM Compass term All three terms (J<0) appear in the superexchange calculation with spin-dependent (spin-orbit-mediated) hopping J~ λ 0, K~ λ 1, A 2 ~ λ 2, λ=spin-orbit energy
Phase Diagram (2D, fixed J & K, T=0) A 1 plays a minor role Small Zeeman and A 2 gives spiral spin (SS) Large compass term A 2 gives Skyrme crystal (SC 2 ) Large Zeeman gives hexagonal Skyrme crystal (SC h )
Skyrmion Textures Spin texture analyzed by FT: S k =Σ r S r e ik*r Spin texture analyzed by local Skyrmion number, or spin chirality χ r
Skymre Crystals of square symmetry found in GL 19 Roessler, Bogdanov, Pfleiderer, Nature 442, 797 (2006)
A Two-step Strategy Choose a classical spin Hamiltonian to obtain spin configuration (Monte Carlo) Choose sd Hamiltonian with local moment S r from previous classical spin Hamiltonian; diagonalize H sd Use Kubo formula for transverse conductivity
Evolution of σ xy with magnetic field σ xy averaged over ~ 100 MC spin configurations SS SC h SP
σ xy for various Skyrme crystal states (T/J=0.1) H c > 0 H c = 0 H c = 0
σ xy for various Skyrme crystal states (T/J=0.5)
σ xy consistent with anti-skyrmion lattice Onset of ΤΗΕ (σ xy <0) is concomitant with onset of nonzero χ Signs of χ (<0) and σ xy (<0) consistent with anti-skyrmion lattice χ/l 2 gives flux per particle, but estimate of σ xy is non-trivial due to lattice effects (recall TKNN) σ xy can be as high as ~10e 2 /h
Comparison to AHE seen in A-phase Experimentally, onset of anomalous part occurs above threshold (H C > 0) Ong s group (B eff ~ 40T) Pfleiderer s group (B eff ~ 2.5T)
MnSi under high pressure 26 MnSi under large pressure has diffuse Bragg peaks along [110] 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)
Theoretical proposals for partial order Tewari et al. produces a phase diagram reminiscent of the pressure evolution of MnSi phases 27 Tewari et al. PRL 96, 047207 (2006) Binz, Vishwanath, Aji, PRL 96, 207202 (2006) Fischer, Shah, Rosch, PRB 77, 024415 (2008)
Evolution of spiral states with A 2 28 SC 2 SC 3 A larger A 2 term favors spiral states with k-vector along the crystal axes [11]->[10] (2D) [111] -> [110] -> [100] (3D) An alternative explanation of stabilization of [110] over [111] spiral given by Plumer&Walker (1981) and Hopkinson&Kee (2007,2009)
Is it 2D or 3D? 29 The modulation vector re-orients itself perpendicular to the magnetic field; A-phase SC h is two-dimensional Partial order phase is 3D (our calculation is 2D)
Suppression of σ xx in THE regime 30 Numerically we find suppression of σ xx when σ xy is significant
Summary 31 Skyrmion spin configuration in a chiral magnet became very interesting with recent discoveries in MnSi (although that s what motivated us) 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