Spin-Orbit Interaction A Path to Topological Matter in Real and - PowerPoint PPT Presentation

Spin-Orbit Interaction – A Path to Topological Matter in Real and Momentum Space Peter Grünberg Institute and Institute for Advanced Simulation Stefan Blügel PAGE 1 Trieste MaX Conference, 31. Jan. 2018

Topology of electrons in an insulator H(k) Fibre bundle theory integer Chern number – topological invariant of fibre bundles Nash and Sen, Topology and Geometry for Physicists PAGE 2 Trieste MaX Conference, 31. Jan. 2018

Topological insulators Topological matter Topology of Bloch wavefunction conduction conduction band band energy edge 2D states valence valence band band momentum Topological classification Z 2 = 1 Z 2 = 0 PAGE 3 Trieste MaX Conference, 31. Jan. 2018

Topological insulators Topological matter Dissipationless edge states conduction defect band energy edge 2D states valence band Quantum Spin Hall Effect momentum PAGE 4 Trieste MaX Conference, 31. Jan. 2018

Topological Characterization of Solids 3D Topological Insulators Response Properties Relativistic GW electronic structure causes non- Bi 2 Se 3 trivial topological invariants Aguilera et al. , PRB 88 , Goal: 045206 (2013) Energy Bi 2 Te 3 § Exploration of topological Bi 2 Se 3 Berry curvature phase space Sb 2 Te 3 ZZ Ω ˆ m k x dk x d ˆ Z ∼ m { r , p , M , t } Spin Orbit Torque Generation of Spin-Currents M (b) T odd z z even ˆ T ˆ m m QAnomalous HE , j E , j H. Zhang et al ., PRL (2012) PAGE 5 Trieste MaX Conference, 31. Jan. 2018

Chiral magnetic skyrmion Im Ma Juba Da Sa from Bertrand Dupé PAGE 6 Trieste MaX Conference, 31. Jan. 2018

Chiral magnetic skyrmion Im Ma Juba Da from Karin Everschor-Sitte from Bertrand Dupé Sa PAGE 7 Trieste MaX Conference, 31. Jan. 2018

Chiral magnetic skyrmion Skyrmion= non-trivial, hedgehog smooth mapping from S d vector field of to order parameter space magnetization (“trivial winding at infinity”) direction magnetization direction m ( x,y )= M / M m ( x,y ) Smooth mapping Here d=2, S 2 → S 2 ✓ ∂ m ◆ Q = 1 Z ∂ x × ∂ m dxdy R 2 m · 4 π ∂ y PAGE 8 Trieste MaX Conference, 31. Jan. 2018

Skyrmions: Experimental observations Layers of materials with intrinsic chirality Ultrathin films with induced chirality (cubic helimagnets FeGe, MnSi, Fe 1-x Co x Si) (Fe/Ir, Mn/W, Pd/Fe/Ir) Lorentz Transmission Electron Microscopy Spin-Polarized Scanning Tunneling Microscopy X.Z. Yu et al. Nature 465 , 90 (2010) N. Romming et al. Science 341 , 636 (2013) Magnetic Force Microscopy B app ≠ 0 P. Milde et al., Science 340, 1076 (2013) PAGE 9 Trieste MaX Conference, 31. Jan. 2018

Multiscale modeling v Micromagnetic-model: Z A | r m | 2 + D : ( r m ⇥ m ) + m · K · m � B m · ˆ ⇥ ⇤ E ( m ) = e z d r R 2 v Spin-Lattice Model: c H = 1 1 X X X X z }| { ⇥ ⇤ m i m j − ( m i ˆ e i )( m j ˆ J ij m i m j + D ij m i × m j + m i K m i + e i ) r 3 2 ij ij ij i ij v DFT-model: E DFT e rot ) = E DFT noSOC ( q ) + ∆ E DFT tot ( q , ˆ SOC ( q , ˆ e rot ) From total energy calculation to • A, D , K • J ij , D ij M. Heide, G. Bihlmayer, and S. Blügel, Physica B 404 , 2678 (2009) B. Zimmermann, M. Heide, G. Bihlmayer, and S. Blügel, PRB 90 , 115427 (2014) B. Schweflinghaus, B. Zimmermann, G. Bihlmayer and S. Blügel, PRB 94 , 024403 (2016) PAGE 10 Trieste MaX Conference, 31. Jan. 2018

Multiscale modeling v Micromagnetic-model: Z A | r m | 2 + D : ( r m ⇥ m ) + m · K · m � B m · ˆ ⇥ ⇤ E ( m ) = e z d r R 2 v Spin-Lattice Model: c H = 1 1 X X X X z }| { ⇥ ⇤ m i m j − ( m i ˆ e i )( m j ˆ J ij m i m j + D ij m i × m j + m i K m i + e i ) r 3 2 ij ij ij i ij v DFT-model: E DFT e rot ) = E DFT noSOC ( q ) + ∆ E DFT tot ( q , ˆ SOC ( q , ˆ e rot ) ˆ e rot ˆ e rot q M. Heide, G. Bihlmayer, and S. Blügel, Physica B 404 , 2678 (2009) B. Zimmermann, M. Heide, G. Bihlmayer, and S. Blügel, PRB 90 , 115427 (2014) B. Schweflinghaus, B. Zimmermann, G. Bihlmayer and S. Blügel, PRB 94 , 024403 (2016) PAGE 11 Trieste MaX Conference, 31. Jan. 2018

Ab-initio A, D, K v Micromagnetic-model: Z A | r m | 2 + D : ( r m ⇥ m ) + m · K · m � B m · ˆ ⇥ ⇤ E ( m ) = e z d r R 2 v Spin-Lattice Model: c H = 1 1 X X X X z }| { ⇥ ⇤ m i m j − ( m i ˆ e i )( m j ˆ J ij m i m j + D ij m i × m j + m i K m i + e i ) r 3 2 ij ij ij i ij v DFT-model: E DFT e rot ) = E DFT noSOC ( q ) + ∆ E DFT tot ( q , ˆ SOC ( q , ˆ e rot ) § Spin Stiffness: A = ∂ 2 X ∂ q 2 E DFT J 0 j R 2 tot ( q ) ∝ 0 j j > 0 § Spiralization (micromagnetic D) D = ∂ ∂ q E DFT X tot ( q ) ∝ D 0 j ⊗ R 0 j j > 0 PAGE 12 Trieste MaX Conference, 31. Jan. 2018

KKRnano: all-electron linear scaling for thousands of atoms i Setting up reference system Preconditioned iterative solution of sparse linear equation (Dyson-equation) Calculate charge density Calculate new potential Mixing potentials f Schematic representation of workflow in KKRnano PAGE 13 Trieste MaX Conference, 31. Jan. 2018

What happens when space inversion symmetry broken (GaAs, InSb, interfaces, surfaces, ...) Time reversal + space inversion symmetry: Time reversal only , Effective spin-orbit (“magnetic”) field Ω: Time reversal symmetry: I. Zˇuti ́c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004) . PAGE 14 Trieste MaX Conference, 31. Jan. 2018

Spin-Orbit Coupling v spin-orbit coupling has fascinating realizations and ramifications in solids Examples: o Orbital and topological orbital magnetic moment o Magnetic Anisotropy o Dzyaloshinskii-Moriya Interaction o Rashba Effect , Dresselhaus Effect o Topological Insulator, Weyl Semimetals o Spin-Relaxation (Elliot-Yafet, Dyakonov-Perel) o Anomalous Hall Effect, Spin Hall Effect o Spin-Orbit torque o Quantum Spin Hall Effect, Quantum Anomalous Hall Effect PAGE 15 Trieste MaX Conference, 31. Jan. 2018

Magnetic materials & spintronics have a market Energy Storage Memory permanent magnets hard disk drive MRAM IoT magneto-caloric materials TMR magnetic sensors PAGE 16 Trieste MaX Conference, 31. Jan. 2018

Example 1: Bandstructure of topological insulator PAGE 17 Trieste MaX Conference, 31. Jan. 2018

GW with spin-orbit coupling (SOC) MOST GW WORKS PUBLISHED a posteriori SOC: LDA (without SOC) + GW (without SOC) + SOC(LDA) GW +SOC 18 Trieste MaX Conference, 31. Jan. 2018

GW with spin-orbit coupling (SOC) MOST GW WORKS PUBLISHED a posteriori SOC: LDA (without SOC) + GW (without SOC) + SOC(LDA) GW +SOC OUR WORK full SOC: Sakuma et al. , PRB 84 085144 (2011) LDA (with SOC) + GW (with SOC) (more accurate but ~10 times more time-consuming) G SOC W SOC 19 Trieste MaX Conference, 31. Jan. 2018

GW with spin-orbit coupling (SOC) MOST GW WORKS PUBLISHED a posteriori SOC: LDA (without SOC) + GW (without SOC) + SOC(LDA) GW +SOC OUR WORK full SOC: LDA+SOC G SOC W SOC Sakuma et al. , PRB 84 085144 (2011) GW +SOC LDA (with SOC) + GW (with SOC) (more accurate but ~10 times more time-consuming) G SOC W SOC Aguilera, Friedrich, Blügel, PRB 88 , 165136 (2013) 20 Trieste MaX Conference, 31. Jan. 2018

(~100 nm) 100 QL slab of Bi 2 Se 3 contribution " GW " "LDA" of the 1 st QL max min 21 Trieste MaX Conference, 31. Jan. 2018

(~100 nm) 100 QL slab of Bi 2 Se 3 contribution " GW " "LDA" of the 1 st QL max min Dispersion of the lower Dirac cone? 22 Trieste MaX Conference, 31. Jan. 2018

Comparison with ARPES: Bi 2 Se 3 23

Comparison with ARPES: Bi 2 Se 3 " GW " "LDA" ARPES 1 eV 1 eV 0.3 Å -1 0.3 Å -1 0.3 Å -1 24

Example 2: Skyrmion design PAGE 25 Trieste MaX Conference, 31. Jan. 2018

Skyrmions for Spintronics The Fert criteria • Chiral magnetism in thin films, but not too thin (min 3 layers) • Try find small but not too small skyrmions ≈ 5-10 nm • Above room temperature and zero magnetic field • Fit to the field of spintronics: injection , transport , detection , manipulation at reasonable fields and currents • Fast & energy efficient • Also for logic operation • Metallic magnetism Albert Fert, Vincent Cross and João Sampaio, Nature Nanotechnology 8 , 152 (2013) PAGE 26 Trieste MaX Conference, 31. Jan. 2018

Multiscale modeling v Micromagnetic-model: Z A | r m | 2 + D : ( r m ⇥ m ) + m · K · m � B m · ˆ ⇥ ⇤ E ( m ) = e z d r R 2 v Spin-Lattice Model: c H = 1 1 X X X X z }| { ⇥ ⇤ m i m j − ( m i ˆ e i )( m j ˆ J ij m i m j + D ij m i × m j + m i K m i + e i ) r 3 2 ij ij ij i ij PAGE 27 Trieste MaX Conference, 31. Jan. 2018

Exchange bias stabilized skyrmions Mn/W(100) (nm) 1.3 -1.3 -2.5 2.5 20 4 ) m o 15 t 2 a / V K 100 e m 0 ( T 10 O T E -2 -0.2 0.0 0.2 -1 (nm -1 ) 5 K 110 0 -5 SR SOC -10 -0.8 -0.4 0.0 0.4 0.8 Spin-polarized STM image -1 (nm -1 ) Theory result Ferriani et al., PRL 101 027201 (2008) c H = 1 1 X X X X z }| { ⇥ ⇤ J ij m i m j + D ij m i × m j + m i K m i + m i m j − ( m i ˆ e i )( m j ˆ e i ) r 3 2 ij ij ij i ij PAGE 28 Trieste MaX Conference, 31. Jan. 2018

Interlayer Exchange Bias Skyrmions Spontaneous nucleation of individual skyrmion with finite life-time Skyrmion lattice (SkL) phase � � iSk phase SkL phase <010> SS phase <100> � � Nandy, Kiselev, Blügel, PRL. 116 , 177202 (2016) PAGE 29 Trieste MaX Conference, 31. Jan. 2018