Jung Hoon Han (SKKU, Korea) 응집물리학의 스커미온 Jung Hoon Han (SKKU, Korea) 성균관대 한정훈

“Topological Numbers” (Almost) All important numbers in condensed matter physics are topological in nature

Examples of Topological Numbers Quantized circulation in superfluid helium Quantized flux in superconductor Chern Number for quantized Hall conductance Z2 number for quantum spin Hall and 3D topological insulators (11월 6일 연세대) Skyrmion numbers

Is electric charge a topological number? Common view: The charge of an elementary particle is just a constant of nature Tony Skyrme’s view: No, even the charge can be derived from the underlying theory In Skyrme’s theory, charge is a topological number of a particular field configuration

2007 Nobel Prize

r=R/2 r=R/4 r=R/10 3D Skyrmion r=R r=2R r=10R

2D (Baby) Skyrmion 2차원 스커미온 위상숫자 3차원 스커미온 위상숫자 Skyrme 왈: 위상수=전하 (양자홀 계에서 최초로 검증)

Study of Magnets in Physics Study of magnetism has taught us about Electromagnetism and gauge theory (B=xA) Molecular field (Curie-Weiss) theory Ginzburg-Landau theory Renormalization group idea Spontaneous symmetry breaking in phase transition Goldstone theorem, Mermin-Wagner theorem High-Tc puzzle, spin liquids Quantum Criticality Magnetic Storage (GMR, CMR, TMR) Spintronics, QSHE

Excitations of Magnets Topological: Vortices (p1(S1) homotopy, circle->circle) Skyrmions (p2(S2) homotopy, sphere->sphere) Non-topological: At low temperature, magnetic moments ORDER into e.g. a ferromagnet (a phenomenon of SSB). By virtue of Goldstone’s theorem, gapless excitations must exist – spin waves

2D XY Magnet: Vortex Vortices are natural defects in XY magnets T/J=0.05 T/J=0.90 Vortices proliferate at TKT, disorder the spin system T/J = 1.1

Vortices in Condensed Matter Superfluid 4He in 2D Josephson Junction Array 2D Superconductor

2D Heisenberg Magnet: Skyrmion 2D XY Magnet: Vortex 2D Heisenberg Magnet: Skyrmion Nontrivial homotopy map: p1(S1) = Z -> p2(S2) = Z

Unimportance of Skyrmions in Heisenberg Magnet Unlike vortices in 2D XY magnet, Skyrmions in 2D Heisenberg magnets are irrelevant excitations Polyakov’s work: spin-wave excitations already disorder the phase

Exception to the Rule: Quantum Hall FM Electrons confined to 2D under strong (~10T) magnetic field All electrons sit within a single LL K lB B Klaus von Klitzing (NP ’85)

lB Completely spin polarized due to Coulomb exchange Quantum Hall Ferromagnet (QHFM) K B lB B

Skyrmions as Quasiparticles “usual” quasiparticle Q=1, S=1/2, Energy=D “Skyrmionic” quasiparticle Q=1, S=many, Energy=(1/2)D Sondhi et al. PRB 47, 16419 (1993)

Skyrmions are cheapest charge excitations of QHFM NMR Knight shift data showed charge excitation accompanied by a large spin flip (S~4) Each Skyrmion carries electric charge Barrett et al. PRL 74, 5112 (1995)

Search for Skyrmion Crystal Skyrmion lattice as a model for nuclear matter Claims of Skyrme crystal by MacDonalds et al. Experimental evidence still indirect

Analogy to Vortex Matter NbSe2 Hess et al, PRL (89) Nb Tonomura group Nature (92) MgB2 Vinnikov et al PRB (03) Array of quantized magnetic fluxes (F=h/2e) Predicted to exist by Abrikosov based on analysis of GL Possible to find a matter consisting of Skyrmions?

“Sightings” of Skyrmion Crystal in Chiral Magnet Nearly ferromagnetic metal Spiral spins with a long modulation period l~180A Dzyaloshinskii-Moriya (DM) interaction Nakanishi et al. SSC 35, 995 (1980)

Recent Breakthroughs in Chiral Magnet (3D) A new phase with triple-Q Bragg spots in MnSi, Fe1-xCoxSi (3D) Muhlbauer et al. Science 323, 915 (2009) For cFM relevant objects are twisting spins called Skyrmions

Recent Breakthroughs in Chiral Magnet (2D) A new phase with triple-Q Bragg spots in Fe1-xCoxSi (2D) Perpendicular magnetic field drives Stripe->Skyrmion Lattice -> FM via two 1st order phase transitions Yu et al Nature (2010)

Skyrmion Lattice Recipe Take a helical spin state propagating along Q1-vector Q1

Take two helical spin states propagating along Q1 and Q2-vectors

Take three helical spin states propagating along Q1, Q2, Q3-vectors You get a Skyrmion lattice! Q1 Q2 Q3

Skyrmion Lattice as a Hidden Abrikosov Lattice Vortex: Quantum mechanical, U(1) Skyrmion: classical, O(3)

Han,Zang,Yang,Park,Nagaosa, PRB (2010) Abrikosov’s solution for Abrikosov lattice CP1 solution for Skyrmion lattice Han,Zang,Yang,Park,Nagaosa, PRB (2010)

스커미온의 쓸모 지적호기심 (예) 소자응용가능성 (?) 단위 소자 전자(electron) 잘하면 전자소자(electronics) 스핀(spin) 스핀소자(spintronics) 스커미온(skyrmion) 스커미온소자(skyrmionics)?

Magnetic Nanodot (포항공대이현우교수) Vortex in FM nanodisk [ Waeyenberge et al., Nature 444, 461 (2006) ] Shinjo et al., Science 289, 930 (2000)

Controlled motion of Skyrmionic object possible

Vortex Magnetic Nanodot (포항공대이현우교수) Vortex polarity as memory Magnetic-field-induced vortex switching Waeyenberge et al., Nature 444, 461 (2006)

Current-induced Vortex Reversal (포항공대이현우교수) Theory Experiment Sang-Koog Kim et al., APL 91, 082506 (2007) Yamada et al., Nature Materials 6, 269 (2007)

Vortex as Microwave Source (포항공대이현우교수) GMR and microwave data Line narrower than 300 kHz at ~1.1 GHz Magnetic dynamics of thick layer Pribag et al., Nature Physics 3, 498 (2007)

Uses of Magnetic Vortex Magnetic memory Source of microwave generation Current control of vortices and Skyrmions possible through spin transfer torque (STT) Chiral FM Ordinary FM Competition Short-range FM Short-range DM Long-range Dipolar Skyrmion size 10nm<R<100nm R>100nm Natural state Lattice Single Magnetic field Required Not required Disk size Large/small small Chirality Yes No Can we replace conventional magnetic vortex by Skyrmion?

Collaboration Park Jin-Hong, Yang Zhihua (SKKU) Naoto Nagaosa (U. Tokyo) Tokura group (U. Tokyo) Jiadong Zang (Fudan U)