Topological phases driven by skyrmions crystals Joaquín Fernández Rossier 1,2 INL, Braga, Portugal Univ. Alicante, Spain SPIN ORBIT COUPLING AND TOPOLOGY IN LOW DIMENSIONS Spetses, June 29 2016

5 minute summary Joaquin.fernandez-rossier@inl.int

Topological phases: the Quantum Hall trio W Topological phases: the Quantum Hall trio S. Oh, Science 340, 153 (2013) Gapped bulk Chiral edge states Persistent non-dissipative currents Quantized, material independent Hall response given by winding number Joaquin.fernandez-rossier@inl.int

W A skyrmion lattice Alejandro Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, 045015 (2016) Joaquin.fernandez-rossier@inl.int

1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!” Two topological phases Graphene Magnetic material With skyrmion lattice 1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!” J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B 92, 115433 (2015) Joaquin.fernandez-rossier@inl.int

1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!” 1) AHE for graphene on skyrmions Graphene Magnetic material With skyrmion lattice 1. Quantum Anomalous Hall effect in graphene with B=0 and “no SOC !!” J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B 92, 115433 (2015) Joaquin.fernandez-rossier@inl.int

2) Skyrmion crystal magnons are topological W 2) Skyrmion crystal magnons are topological Joaquin.fernandez-rossier@inl.int

People Alejandro Roldán-Molina (Chile) Álvaro S. Núñez (U. Chile) José Luis Lado (INL, Portugal) Francesca Finnochiaro (IMDEA-NANO, Madrid) Joaquin.fernandez-rossier@inl.int

What are skyrmions? Joaquin.fernandez-rossier@inl.int

W Skyrmions Non-coplanar spin texture Skyrmion lattice = ground state in some materials (MnSi) Competition of exchange and DM interaction Topological spin texture Joaquin.fernandez-rossier@inl.int

W Magnetic skyrmions N=+1 N=-1 Joaquin.fernandez-rossier@inl.int

W Skyrmions: different sizes STM Images, spontaneous skyrmion lattice Fe(ML)/Ir(111) Ta(5nm)/Co20Fe60B20(CoFeB)(1.1nm)/TaOx(3nm STM Images, spontaneous skyrmion lattice (B=0) W. Jiang et al, Science DOI: 10.1126/science.aaa1442 S. Heinze et al., Nat. Phys. 7, 713 (2011) Joaquin.fernandez-rossier@inl.int

W Graphene + Skyrmions: experiments !! Jens Brede et al, Nature Nano 9 1019 (2014) Joaquin.fernandez-rossier@inl.int

Electrons surfing non-collinear magnetic landscapes Joaquin.fernandez-rossier@inl.int

Electrons surfing a non collinear magnetic landscape W Electrons surfing a non collinear magnetic landscape Bruno, P., V. K. Dugaev, and M. Taillefumier. "Topological Hall effect and Berry phase in magnetic nanostructures." Physical Review Letters 93 096806 (2004)

Electrons surfing a non collinear magnetic landscape W Electrons surfing a non collinear magnetic landscape

Strong coupling : spinless fermions in a “magnetic” field W Strong coupling : spinless fermions in a “magnetic” field X X X X

W Total flux= skyrmion number Pseudo field flux = Skyrmion number

W Total flux= skyrmion number Pseudo field flux = Skyrmion number CONSEQUENCES Non-quantized anomalous Hall effect (aka topological Hall effect) expected for electrons interacting with skyrmions (in the strong coupling ) Yi, Onoda, Nagaosa, Han arXiv:0903.3272 (2009) Quantized Anomalous Hall effect predicted for Kagome Lattice with non-coplanar magnetization K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B 62, R6065 (2000).

Graphene electrons surfing a skyrmion lattice Joaquin.fernandez-rossier@inl.int

W The Quantum Hall trio in graphene No reports so far Haldane (88): weird B Qiao (‘10): M + Rashba SOC Kane-Mele (‘05): SOC Abanin-Lee-Levitov (‘06): B + FM order (Routinely) observed experimentally B>1T Joaquin.fernandez-rossier@inl.int No reports so far Joaquin.fernandez-rossier@inl.int

W The Quantum Hall trio in graphene Haldane (88): weird B Qiao (‘10): M + Rashba SOC Kane-Mele (‘05): SOC Abanin-Lee-Levitov (‘06): B + FM order (Routinely) observed experimentally B>1T Quantum Hall trio in graphene: B and/or Spin-orbit needed Joaquin.fernandez-rossier@inl.int

Quantum Anomalous Hall effect in graphene with B=0 and no SOC !! Graphene coupled to Skyrmions Graphene Magnetic material With skyrmion lattice J. L. Lado, and J. Fernández-Rossier , Phys. Rev. B 92, 115433 (2015) Quantum Anomalous Hall effect in graphene with B=0 and no SOC !! Joaquin.fernandez-rossier@inl.int

W Hall conductivity Hall conductivity Berry curvature Berry connection Quantized Hall Conductance in a Two-Dimensional Periodic Potential TKNN PRL 49, 405 (1982) Joaquin.fernandez-rossier@inl.int

Quantum Hall conductivity as a topological invariant W Quantum Hall conductivity as a topological invariant Chern number = Integer number Berry curvature Berry connection Quantized Hall Conductance in a Two-Dimensional Periodic Potential TKNN PRL 49, 405 (1982) Joaquin.fernandez-rossier@inl.int

W Haldane model Bulk Spinless fermions in honeycomb lattice Local magnetic flux, zero total B Broken time reversal symmetry Gapped bulk Chern number = 1 Quantized Hall conductance 1 Edge states (index theorem) Quantum Anomalous Hall effect Strip (2 edges) 1 edge F. D. M. Haldane Phys. Rev. Lett. 61, 2015 (1988) Joaquin.fernandez-rossier@inl.int

Quantized Anomalous Hall effect W Quantized Anomalous Hall effect Wanted: 2 Dimensional system Broken time reversal symmetry Insulating Joaquin.fernandez-rossier@inl.int

Turning graphene magnetic: “theory” W Turning graphene magnetic: “theory” Graphene FM material

Opening a gap in magnetic graphene W Opening a gap in magnetic graphene Spin mixing term needed: Spin orbit coupling Non-collinear magnetism

Qiao et al model for QAH in graphene W Qiao et al model for QAH in graphene Bulk Graphene + Zeeman off-plane + Rashba Gapped bulk Chern number = 2 Quantized Hall conductance 2 Edge states (index theorem) Quantum Anomalous Hall effect Strip (2 edges) 1 edge Z. Qiao, S. A. Yang, W. Feng, W. Tse, J. Ding, Y. Yao, J. Wang, Q. Niu Phys. Rev. B82, R161414 (2010) Joaquin.fernandez-rossier@inl.int

QAHE in graphene + Skyrmions W QAHE in graphene + Skyrmions J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B 92, 115433 (2015) Joaquin.fernandez-rossier@inl.int

1st neighbor tight-binding W Models and methods Classical spins 1st neighbor tight-binding Joaquin.fernandez-rossier@inl.int

1st neighbor tight-binding W Models and methods Classical spins 1st neighbor tight-binding Dirac cones Joaquin.fernandez-rossier@inl.int

W Open source code: Quantum Honeycomp (Contact José Luis Lado) Models and methods Classical spins 1st neighbor tight-binding J. L. Lado, and JFR, PRB 92, 115433 (15) Weak coupling: Two Geometries: 2D crystal: standard band calculation enlarged unit cell Edge states (semi-infinite 2D crystal): recursive Green function Open source code: Quantum Honeycomp (Contact José Luis Lado) Joaquin.fernandez-rossier@inl.int

Graphene + triangular lattice Skyrmions W Graphene + triangular lattice Skyrmions Gap opens Joaquin.fernandez-rossier@inl.int

Graphene + triangular lattice Skyrmions W Graphene + triangular lattice Skyrmions Gap opens Finite Berry curvature Chern number= 2 N Topological imprinting Quantized Anomalous Hall phase Joaquin.fernandez-rossier@inl.int

W Edge states, 1 per spin channel 2 co-propagating edge states Joaquin.fernandez-rossier@inl.int

Topological wires: 2 and 4 lanes J. L. Lado, and J. Fernández-Rossier, Phys. Rev. B92, 115433 (2015) Joaquin.fernandez-rossier@inl.int

Non-quantized Anomalous Hall effect W Non-quantized Anomalous Hall effect Triangular lattice Joaquin.fernandez-rossier@inl.int

Dirac electrons interacting with 1 skyrmion Trying to understand: Dirac electrons interacting with 1 skyrmion Joaquin.fernandez-rossier@inl.int

Dirac electrons surfing 1 skyrmion valley Dirac Hamiltonian + exchange spin sublattice Joaquin.fernandez-rossier@inl.int

Dirac electrons surfing 1 skyrmion Dirac Hamiltonian + exchange Skyrmion field: Joaquin.fernandez-rossier@inl.int

Hamiltonian in the skyrmion frame Rotated Dirac Hamiltonian + exchange Joaquin.fernandez-rossier@inl.int

Hamiltonian in the skyrmion frame Rotated Dirac Hamiltonian + exchange Spin dependent magnetic field Joaquin.fernandez-rossier@inl.int

Hamiltonian in the skyrmion frame Rotated Dirac Hamiltonian + exchange Effective Rashba-type spin-orbit coupling Joaquin.fernandez-rossier@inl.int

A hand waving argument Joaquin.fernandez-rossier@inl.int

Does it work without skyrmions? Joaquin.fernandez-rossier@inl.int

W Do we need skyrmions? “in-plane skyrmion” opens a gap Zero Berry curvature Trivial insulator Joaquin.fernandez-rossier@inl.int

W How big is the gap? Joaquin.fernandez-rossier@inl.int

W How big is J? Skyrmion J Gap [1] [2] [3] J (meV) 37 70 50 t (eV) 2.8 2.6 2.7 Material EuO BiFeO3 YIG Skyrmion Gap (mev) 0.1 0.4 0.2 J Skyrmion Gap [1] H. X. Yang et al, PRL. 110, 046603 (‘13) [2] Z. Quai et al, PRL. 112, 116404 (‘14) [3] Z. Wang, PRL 114, 01660 (‘15) Joaquin.fernandez-rossier@inl.int

Graphene on skyrmion lattice material (Fe ML/Ir(111)) [1] Wish list Graphene Hall bar on top of insulating magnetic material with skymion lattice Graphene on skyrmion lattice material (Fe ML/Ir(111)) [1] AHE in graphene/YIG: ferromagnetic proximity [2] Skyrmions in insulating material Cu2OSeO3 [3] [1] Jens Brede et al, Nature Nano 9 1019 (2014) [2 ]Z. Wang, Phys. Rev. Lett. 114, 016603 (2015) [3] S. Sekil et al., Science 336, 198 (2012) Joaquin.fernandez-rossier@inl.int

Turning graphene magnetic: experiments W Turning graphene magnetic: experiments AHE Z. Wang, Phys. Rev. Lett. 114, 016603 (2015) Joaquin.fernandez-rossier@inl.int

A new way to have a Chern insulator: Conclusions part 1 A new way to have a Chern insulator: Graphene interacting with a skyrmion lattice No B and no graphene-SOC needed Weak exchange coupling is ok Topological imprinting: Joaquin.fernandez-rossier@inl.int

Topological spin waves in the atomic-scale magnetic skyrmion crystal PART II Topological spin waves in the atomic-scale magnetic skyrmion crystal A. Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, 045015 (2016)

Skyrmion lattice: the model W Skyrmion lattice: the model Joaquin.fernandez-rossier@inl.int

Skyrmion lattice: the model W Skyrmion lattice: the model (1) (2) (3) (4) Heisenberg exchange (FM) (J=1 meV) DM coupling (D=1 meV) (breaks inversion symmetry, requires SOC) Zeeman Uniaxial anisotropy K=0.5 meV Joaquin.fernandez-rossier@inl.int

Skyrmion lattice: the model W Skyrmion lattice: the model (1) (2) (3) (4) J Promotes (FM) (J=1 meV) K defines easy axis D promotes helical phase Off plane Zeeman promotes non-coplanarity Joaquin.fernandez-rossier@inl.int

Skyrmion lattice: classical approximation W Skyrmion lattice: classical approximation A. Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, 045015 (2016) Joaquin.fernandez-rossier@inl.int

Spin waves: reminder of Holstein-Primakoff Representation of spin operators in terms of HP Bosons Spin quantization axis is Density of HP bosons = = deviation from the maximal spin projection along the quantization axis = metric of quantum fluctuations T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940). Joaquin.fernandez-rossier@inl.int

Spin waves: reminder of Holstein-Primakoff Representation of spin operators in terms of HP Bosons Linearization: small density of HP bosons T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940). Joaquin.fernandez-rossier@inl.int

Spin waves: reminder of Holstein-Primakoff Representation of spin operators in terms of HP Bosons Linearization: small density of HP bosons Quadratic Hamiltonian for bosons (Bogoliubov de Gennes type) Joaquin.fernandez-rossier@inl.int

Spin waves for magnetic crystals Bloch quadratic Hamiltonian for bosons (Bogoliubov de Gennes type) After Bogoliubov transformation, the true excitations are found Joaquin.fernandez-rossier@inl.int

For classical ground states with crystal symmetry After Bogoliubov transformation, the true excitations are foundc And their wave functions are also found: That permit to compute the density of HP bosons A. Roldán-Molina , MJ Santander, AS Núñez and J. Fernández-Rossier Phys. Rev. B 89 054403 (2015) Joaquin.fernandez-rossier@inl.int

A simple case: a FM 1D chain W A simple case: a FM 1D chain No anomalous terms: HP bosons = magnon excitation Joaquin.fernandez-rossier@inl.int

Spin waves of a single skyrmion A. Roldán-Molina , MJ Santander, AS Núñez and J. Fernández-Rossier Phys. Rev. B 89 054403 (2015) Joaquin.fernandez-rossier@inl.int

Lowest energy bands magnons skyrmion lattice A. Roldán-Molina, A. S. Núñez, J. Fernández-Rossier, New J. Phys. 18, 045015 (2016) Joaquin.fernandez-rossier@inl.int

W Density of HP bosons Joaquin.fernandez-rossier@inl.int

Berry curvature of magnons in skyrmion crystal W Berry curvature of magnons in skyrmion crystal Berry curvature for a given band: Joaquin.fernandez-rossier@inl.int

Chern number for magnons in skyrmion crystal W Chern number for magnons in skyrmion crystal Berry curvature for a given band: Chern number of a given band: Winding numbers Joaquin.fernandez-rossier@inl.int

What are the consequences of finite winding numbers? Joaquin.fernandez-rossier@inl.int

W Edge states 2 3 2 1 1 Joaquin.fernandez-rossier@inl.int

W Edge states Joaquin.fernandez-rossier@inl.int

W Edge states Joaquin.fernandez-rossier@inl.int

Other “topological magnon systems” W Other “topological magnon systems” Zhang L, Ren J, Wang J S and Li B Phys. Rev. B 87 144101 (2013 ) Mook A, Henk J and Mertig I Phys. Rev. B 90 024412 (2014 ) Mook A, Henk J and Mertig I Phys. Rev. B 91 224411 (2015 ) Joaquin.fernandez-rossier@inl.int

Other “topological magnon systems” W Other “topological magnon systems” Joaquin.fernandez-rossier@inl.int

Other “topological magnon systems” W Other “topological magnon systems” Self-assembled nanostructure Artificial nanotructure Atomic crystal Joaquin.fernandez-rossier@inl.int

W Quarks, electrons Atoms, electrons Crystals, Magnets Emergence: symmetry breaking and topology Quarks, electrons Atoms, electrons Crystals, Magnets Skyrmion crystal Topological spin waves Joaquin.fernandez-rossier@inl.int

Thanks for your attention Conclusions part 2 Magnons in skyrmion crystal have finite Chern number Chiral edge states for magnons First example in a self-assembled mesoscale structure Thanks for your attention Joaquin.fernandez-rossier@inl.int