MP - Magnetisation Processes Joo-Von Kim Centre for Nanoscience and - PowerPoint PPT Presentation

ESM 2018 Krakow MP - Magnetisation Processes Joo-Von Kim Centre for Nanoscience and Nanotechnology, Université Paris-Saclay 
 91120 Palaiseau, France joo-von.kim@c2n.upsaclay.fr


 � 2 Outline European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV MP1 
 Quasi-static processes, domain states, nontrivial spin textures 
 MP2 
 Precessional dynamics, dissipation processes, elementary and soliton excitations 
 MP3 
 Spin-transfer and spin-orbit torques, current topics in magnetization dynamics 


� 3 MP1: Quasi-static processes European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Overarching theme: Hysteresis loop 
 Energy landscapes 
 Which magnetisation configurations are possible, favourable? 
 Reversal mechanisms 
 How do we navigate this energy landscape? 
 Time-dependent and thermal e ff ects 
 “Slow” dynamics and the limits of what we mean by “quasi-static”

� 4 Length scales European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Magnetism Spin transport nm Magnetic 0.1 Local moments nanostructures 1 Fermi wavelength 10 Exchange correlation Electron mean free path 100 Spin diffusion length Macroscopic domains 10 3

� 5 Time scales European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Ultrafast laser-induced thermalisation Domains in nanoparticle arrays Data storage Spin waves Y h s µs ns ps fs x U L c Domain wall creep Spin glass relaxation Conduction spin relaxation

� 6 The hysteresis loop European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Common characterisation of a magnetic material Captures physics across many length and time scales M Saturation Remanence Coercivity

� 7 “Quasi-statics”: Navigating the energy landscape European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV As field is varied, magnetic system may move through a variety of metastable energy states “Quasi-static” processes dominated by energy considerations, rather than torques (i.e., precessional dynamics) “Slow” dynamics, compared with ns-scale of fs-scale processes M Energy Saturation Remanence Coercivity

� 8 What contributes to the Energy terms - Brief overview energy landscape? European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Exchange Atomistic Micromagnetic E ex = A ( r m ) 2 E ex = − J ij S i · S j J > 0: ferromagnetic J < 0: antiferromagnetic m = M /M s k m k = 1 J 1 , J 2 , J 3 , … Anisotropy E 1 e ) 2 E K = − K ( m · ˆ E 1 ≠ E 2 Uniaxial form shown, higher orders are possible E 2

� 9 What contributes to the Energy terms - Brief overview energy landscape? European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Dipolar " # − 3 ( µ i · r ij ) ( µ j · r ij ) µ i · µ j E d = µ 0 r 3 r 5 ij ij r ij E d = − 1 H d = �r Φ m 2 µ 0 M · H d Demagnetising Magnetostatic µ = gµ B S field potential Zeeman H 0 E Z = − µ 0 M · H 0 Applied field

� 10 What contributes to the Energy terms - Brief overview energy landscape? European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Dzyaloshinskii-Moriya E 1 E DMI = D ij · S i × S j E 1 ≠ E 2 E 2 Example of a chiral interaction lemons oranges

��� � 11 What contributes to the Energy terms - Brief overview energy landscape? European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Dzyaloshinskii-Moriya E 1 E DMI = D ij · S i × S j E 1 ≠ E 2 E 2 Non-centrosymmetric crystals (e.g., B20) Interface-induced e f Ge Co, Fe … S 2 D 12 S 1 Pt, Ir … Large SOC Substrate with large Mn Si spin-orbit coupling Mn (SOC) E DM = D m · ( r ⇥ m ) E DM = D [ m z ( r · m ) � ( m · r ) m z ] ^

� 12 Energy terms - Interlayer coupling European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Néel “Orange Peel” coupling Dipolar coupling due to induced magnetic charges at rough interfaces fringing fields σ = ⃗ Stray magnetic M · ⃗ n charges appear at surface of rough film surface magnetic charge M density Upper interface Similar phenomenon D for rough interfaces in multilayer Lower interface σ U ( ⃗ R ) σ L ( ⃗ R ′ ) � � d 2 R d 2 R ′ E = µ 0 � D 2 + ( ⃗ R − ⃗ R ′ ) 2 Upper Lower interface interface

� 13 Energy terms - Interlayer coupling European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV RKKY Coupling Indirect exchange coupling mediated by conduction electrons in spacer layer Related to Ruderman-Kittel-Kasuya-Yosida interaction between two magnetic impurities in an electron gas E RKKY = − J ( d ) m i · m j GMR signal Fe/Cr, Co/Cr, Co/Ru, 
 Co/Cu/, Fe/Cu, ... Coupling oscillates with spacer layer thickness

� 14 Magnetic states Domains European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV On length scales of ~100 nm and above, magnetic order can be subdivided into di ff erent domains 2 µm Compromise between the short-range ferromagnetic exchange interaction and the long-range antiferromagnetic dipolar interaction + + + + + – – – – – – – – – – + + + + +

� 15 Magnetic states Domain walls European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV The boundary between two magnetic domains is called a domain wall . Wall structure mainly determined by competition between the ferromagnetic exchange interaction (favours parallel alignment with neighbouring spins) and the uniaxial anisotropy (favours alignment along easy axis). Di ff erent wall types exist: Bloch, Néel, Vortex, Transverse ... 
 Each minimises part of the dipolar energy Transverse wall Vortex wall


 
 � 16 Magnetic states Bloch walls European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Profile obtained by minimising the energy functional for the exchange and uniaxial anisotropy energies 
 Suppose m varies along x axis, with anisotropy axis along z 
 We seek to minimise � ∂θ � � � 2 θ � + K u sin 2 θ dx A E = ∂ x x Using variational calculus, obtain Euler-Lagrange equation for function that minimises integral 2 A ∂ 2 θ ∂ x 2 − K u sin 2 θ = 0 p A/K u ∆ = Solution is example of a topological soliton domain wall width θ ( x ) = 2 tan − 1 [exp ( x/ ∆ )] p AK u σ = 4 domain wall energy density

� 17 Magnetic states Vortices European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV p, C In thin circular submicron magnetic elements (“ dots ”), dipolar energy can be minimised by forming vortex states. L Magnetisation curls in the film plane and culminates perpendicular to the film plane at the vortex centre. Region with perpendicular component is called the vortex core. R Another example of topological solitons. Phase diagram for existence of vortex state MFM images of vortex cores 6 4 R/L E 2 0 0 1 2 3 L/L E s 2 A L E ~ 10-20 nm L E = (exchange length) µ 0 M 2 s

� 18 Magnetic states Vortices p, C European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Suitable ansatz for vortex profile: θ = θ ( r ) φ ( r ) = ϕ ± π sign determines chirality m = (cos φ sin θ , sin φ sin θ , cos θ ) 2 r = ( r, ϕ ) Profile minimises volume charges and surface charges at edges �r · m = 0 m · ˆ r = 0 Volume charges vanish Edge surface charges vanish Simulated core profile Energy costs: Vortex core leads to surface magnetic charges at the top and bottom surfaces, curling configuration costs exchange energy. The core profile results from a minimisation of these two energies.

� 19 Magnetic states Skyrmions European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Skyrmions are like vortices but with m z varying from +1 to -1 Vortex Result from competition between exchange, anisotropy, and the chiral Dzyaloshinskii-Moriya interaction Another example of topological solitons Pt/Co/Ox M Schott et al, Nano Lett (2017) [Ir/Co (0.6 nm)/Pt] n C Moreau-Luchaire et al, Skyrmion Nat Nanotechnol (2016) c d Pt/Co (1 nm)/MgO O Boulle et al, Nat Mater (2016) 0.15 mT 𝑞 1 0.15 mT −5 V +10 V. 𝑢 = 0.468 nm , 𝜈 0 𝐼 = 0.15 mT , 𝑁 𝑡 = 0.92 +/- 0.05 MA/m 𝜏 𝑥 = 1.33 +/- 0.16 mJ/m² 0 V, +5 V −5 V 𝐹 𝑜 𝐹 𝑏 +5 V 𝑢, 𝐼, 𝑁 𝑡 𝜏 𝑥

� 20 Magnetic states Skyrmions European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Inexact but useful ansatz for skyrmion profile: 4 cosh 2 c cos θ ( r ) = cosh 2 c + cosh(2 r/ ∆ ) − 1 Double wall N Romming et al, Phys Rev Lett 117 , 177203 (2015) m = (cos φ sin θ , sin φ sin θ , cos θ ) r = ( r, ϕ ) φ ( r ) = ϕ + ψ Helicity Néel Bloch Néel

� 21 Reversal mechanisms European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV Magnetization reversal involves navigating through an energy landscape May involve intermediate states with nontrivial magnetization configurations Intermediate states are metastable energy states Minimizing energies allow us to guess/predict/describe intermediate states Energy