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

MP - Magnetisation Processes

Joo-Von Kim

Centre for Nanoscience and Nanotechnology, Université Paris-Saclay
 91120 Palaiseau, France joo-von.kim@c2n.upsaclay.fr

ESM 2018 Krakow

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Outline

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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
 


European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

MP1: Quasi-static processes

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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 effects
 “Slow” dynamics and the limits of what we mean by “quasi-static”

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Length scales

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Magnetic nanostructures

Local moments Exchange correlation

0.1 nm 1 10 100 103

Macroscopic domains

Magnetism Spin transport

Fermi wavelength Electron mean free path Spin diffusion length

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Time scales

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Y h s µs ns ps fs

Lc U x Domain wall creep Spin glass relaxation Domains in nanoparticle arrays Data storage Spin waves Conduction spin relaxation Ultrafast laser-induced thermalisation

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

The hysteresis loop

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Common characterisation of a magnetic material Captures physics across many length and time scales M

Remanence Coercivity Saturation

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

“Quasi-statics”: Navigating the energy landscape

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M

Remanence Coercivity Saturation

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 Energy

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Energy terms - Brief overview

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Exchange Anisotropy

Eex = A (rm)2 Eex = −JijSi · Sj EK = −K (m · ˆ e)2

Atomistic Micromagnetic

Uniaxial form shown, higher orders are possible J > 0: ferromagnetic J < 0: antiferromagnetic

What contributes to the energy landscape?

J1, J2, J3, …

m = M/Ms kmk = 1

E1 E2 E1 ≠ E2

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Ed = −1 2µ0M · Hd

Energy terms - Brief overview

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Dipolar Zeeman

Ed = µ0 " µi · µj r3

ij

− 3 (µi · rij) (µj · rij) r5

ij

# Hd = rΦm

Magnetostatic potential Demagnetising field

EZ = −µ0M · H0 H0

Applied field

What contributes to the energy landscape?

rij

µ = gµBS

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Energy terms - Brief overview

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Dzyaloshinskii-Moriya

EDMI = Dij · Si × Sj

What contributes to the energy landscape? E1 E2 E1 ≠ E2 Example of a chiral interaction

lemons

• ranges

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Energy terms - Brief overview

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Dzyaloshinskii-Moriya

Non-centrosymmetric crystals (e.g., B20) Interface-induced

^

• Large SOC

D12 S2 S1

e f

Co, Fe … Pt, Ir …

EDM = D [mz (r · m) (m · r) mz] EDM = Dm · (r ⇥ m)

What contributes to the energy landscape?

Si Mn Mn Ge Substrate with large spin-orbit coupling (SOC)

EDMI = Dij · Si × Sj

E1 E2 E1 ≠ E2

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Energy terms - Interlayer coupling

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σ = ⃗ M · ⃗ n

surface magnetic charge density fringing fields

M

Stray magnetic charges appear at surface of rough film

Upper interface Lower interface

D

Similar phenomenon for rough interfaces in multilayer

E = µ0

• d2R
• d2R′

σU(⃗ R)σL(⃗ R′)

• D2 + (⃗

R − ⃗ R′)2

Upper interface Lower interface

Néel “Orange Peel” coupling

Dipolar coupling due to induced magnetic charges at rough interfaces

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Energy terms - Interlayer coupling

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Fe/Cr, Co/Cr, Co/Ru, 
 Co/Cu/, Fe/Cu, ... Coupling oscillates with spacer layer thickness

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

GMR signal

ERKKY = −J(d)mi · mj

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic states

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Domains

On length scales of ~100 nm and above, magnetic order can be subdivided into different domains Compromise between the short-range ferromagnetic exchange interaction and the long-range antiferromagnetic dipolar interaction

2 µm + + + + + – – – – – – – – – – + + + + +

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic states

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Domain walls

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). Different wall types exist: Bloch, Néel, Vortex, Transverse ...
 Each minimises part of the dipolar energy

Transverse wall Vortex wall

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic states

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Bloch walls

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

E =

• dx
• A

∂θ ∂x 2 + Ku sin2 θ

• Using variational calculus, obtain Euler-Lagrange equation for

function that minimises integral

2A ∂2θ ∂x2 − Ku sin 2θ = 0

Solution is example of a topological soliton

x θ

domain wall width domain wall energy density

σ = 4 p AKu ∆ = p A/Ku θ(x) = 2 tan−1 [exp (x/∆)]

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic states

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Vortices

In thin circular submicron magnetic elements (“dots”), dipolar energy can be minimised by forming vortex states. 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. Another example of topological solitons.

3 2 1

L/LE R/LE

6 4 2

Phase diagram for existence of vortex state LE ~ 10-20 nm (exchange length) MFM images of vortex cores

p, C

L R

LE = s 2A µ0M 2

s

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic states

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Vortices

Suitable ansatz for vortex profile:

p, C

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. Profile minimises volume charges and surface charges at edges

sign determines chirality Volume charges vanish Edge surface charges vanish Simulated core profile

r · m = 0 m · ˆ r = 0

m = (cos φ sin θ, sin φ sin θ, cos θ)

r = (r, ϕ)

θ = θ(r) φ(r) = ϕ ± π 2

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic states

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Skyrmions

Skyrmions are like vortices but with mz varying from +1 to -1 Result from competition between exchange, anisotropy, and the chiral Dzyaloshinskii-Moriya interaction Another example of topological solitons

Vortex Skyrmion [Ir/Co (0.6 nm)/Pt]n

C Moreau-Luchaire et al, Nat Nanotechnol (2016)

Pt/Co/Ox

M Schott et al, Nano Lett (2017)

0.15 mT 𝑞1 0.15 mT −5 V +10 V. 𝑢 = 0.468 nm, 𝜈0𝐼 = 0.15 mT, 𝑁𝑡 = 0.92+/-0.05 MA/m 𝜏𝑥 = 1.33+/-0.16 mJ/m² 0 V, +5 V −5 V 𝐹𝑜 𝐹𝑏 +5 V 𝑢, 𝐼, 𝑁𝑡 𝜏𝑥

c d

Pt/Co (1 nm)/MgO

O Boulle et al, Nat Mater (2016)

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic states

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Skyrmions

Inexact but useful ansatz for skyrmion profile: m = (cos φ sin θ, sin φ sin θ, cos θ)

r = (r, ϕ)

cos θ(r) = 4 cosh2 c cosh 2c + cosh(2r/∆) − 1

Double wall

φ(r) = ϕ + ψ

Helicity Néel Bloch Néel

N Romming et al, Phys Rev Lett 117, 177203 (2015)

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

Reversal mechanisms

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Energy

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

E = −µ0HMs cos θ − Ku cos2 θ

Coherent reversal

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The magnetic configuration at a given applied field represents the local energy minimum under that applied field. Simplest example that contains essential physics: magnetic nanoparticle with uniaxial anisotropy

H M E θ θ = 0 θ = π

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

E = −µ0HMs cos (θ − θH) − Ku cos2 θ

Coherent reversal: Stoner-Wohlfarth astroid

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Metastable states: Minimize Zeeman and anisotropy energy (in macrospin approximation) for arbitrary field angles

Hy Hx

Astroid
 (critical curve) Inside astroid, system is bistable Outside astroid, M aligned along field

HK = 2Ku µ0Ms

Critical field

Hsw HK = h sin2/3 θH + cos2/3 θH i−3/2

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Stoner-Wohlfarth Astroid

Hysteresis loops for various angles Switching behavior Hysteresis loops for various angles For arbitrary angles one obtains a mixture of both cases. The coercive fields (HC) for switching into the stable minimum for arbitrary directions show a

• minimum. The switching fields for coherent rotation can be summarized in

the switching asteroid.

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Stoner-Wohlfarth: Hysteresis loops

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0º 10º 45º 70º 90º

Astroid
 (critical curve)

30º

Hsw HK = h sin2/3 θH + cos2/3 θH i−3/2

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Examples of astroids

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Hx (mT) Hy (mT)

W Wernsdorfer et al, Phys Rev Lett 78, 1791 (1997)

First experimental observation (2D system) 25 nm Co cluster

T Devolder et al, Appl Phys Lett 98, 162502 (2011)

Experimental astroid for magnetic nanopillar (200 x 100 x 2 nm) Magnetic tunnel junction, typical MRAM device

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall nucleation and propagation

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In some circumstances it is more favourable to nucleate a domain wall, rather than rotate all moments coherently across sample

10 μm

H0

Kerr microscopy image

• f domain growth

Need to balance energy cost in creating a domain wall with energy gain from Zeeman interaction

E(r) = (2πrd)σ − (πr2d)(2µ0MsH0)

Zeeman energy Wall energy

σ = 4 p AKu

r E r ΔE

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall nucleation and propagation

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Reversal through domain walls generally leads to lower coercivities than coherent reversal

• 0.4
• 0.2

0.0 0.2 0.4

• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.0 0° 20° 40° 60°

(b) (a)

• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0° 20° 40° 60°

down→up up→down

RAHE (Ω) RAHE (Ω) μ0Hex (T) μ0Hex (T)

θ Hex z

S Kim et al, Phys Rev B 95, 220402 (2017) Courtesy of J Ferré, Laboratoire de Physique des Solides, Orsay

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Reversal in dots

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In circular dots where vortices are metastable states, magnetisation reversal occurs through the nucleation and annihilation of vortices

p, C

K Guslienko et al, Phys Rev B 65, 024414 (2001) A Fernández & C Cerjan,
 J Appl Phys 87, 1395 (2000) Reversal in Co dots

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Reversal in dots

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1 µm diameter dots K Zeissler et al, Sci Rep 7, 15125 (2017)

In perpendicularly-magnetised dots with DMI, reversal can take place through skrymion nucleation and annihilation

| 7: | DOI:10.1038/s41598-017-15262-3

Micromagnetic Simulations.

• 10 mT
• 20 mT
• 52 mT

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Reversal in dots

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Simulations show that intermediate states can depend strongly on magnetic parameters

A G Kolesnikov, J Magn Magn Mater 429, 221 (2017) Lower Ms, Higher Ku Higher Ku

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Hysteresis: sweep rate

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M

Remanence Coercivity Saturation

Does it matter how fast we sweep the field? What does “quasi-statics” mean in this context? Slow dynamics … but slow compared to what? Fluctuations and energy barriers are the key

Energy

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Sweep rates matter

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How fast you navigate the energy landscape matters

Hysteresis loop for a Pt/Co/Pt thin film Courtesy of J Vogel, Institut Néel, Grenoble

H = 0 H < 0 H > 0

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Thermal fluctuations

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Brownian motion Magnetisation precession

Particle (red) experiences random collisions (forces) due to thermal environment (blue) Precessing magnetic moment experiences random fields due to thermal environment

W F Brown, Phys Rev 130, 1677 (1963) R Brown, Phil Mag 4, 161 (1828)

Heff

T ≠ 0

(Animation) (Animation)

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Thermal activation

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Energy Probability function Transition rate out of well

P = 1 τ exp ✓ − t τ ◆

A S Ea B

Reaction coordinate

Courtesy of Louise Desplat, C2N, Palaiseau

Arrhenius law

Attempt frequency
 (Prefactor)

Thermal fluctuations give you a finite probability of escaping a metastable state. How patient are you?

1 τ = 1 τ0 exp ✓ − Ea kBT ◆

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Activation pathways

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H A S B A S B

?

Current research topic, subject of debate

Coherent reversal Skyrmion annihilation

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Magnetic aftereffect

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Thermal effects necessarily introduce the notion of time into a measurement What happens when a field is suddenly applied? Thermal fluctuations eventually drive system into lower energy state

J-P Jamet et al, Phys Rev B 57, 14320 (1998) Magnetic aftereffect in ultrathin Co film

Energy

H = 0 H ≠ 0 Ea

m(t) = 2 exp ✓ − t τ ◆ − 1 1 τ = 1 τ0 exp ✓ − Ea kBT ◆

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall hopping

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J-P Tetienne et al, Science 334, 1366 (2014)

Thermally-activated domain wall hopping between two metastable states can be revealed using scanning probe techniques
 
 Example: 
 Nitrogen-vacancy centre magnetometry

• n 1-nm thick CoFeB films

Stray field measurement Domain wall AFM

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall hopping

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500 1000 1500

y (nm)

• 500
• 400
• 300
• 200

x (nm)

q1 q2

~140 nm

• 70

70 140 210

q (nm)

• 0.5

0.5 1

U (eV)

• 70

70 140 210

q (nm)

• 0.5

0.5 1

U (eV)

(b) (c)

a = 0 a = 0.02 a = 0.04

(a)

J-P Tetienne et al, Science 334, 1366 (2014)

Laser-induced heating can control hopping rates between two pinning sites

(a) (b) (c)

Γ ≡ 1 τ = 1 τ0 exp ✓ − Ea kBT ◆

Modelling with two-state system accounts for experimental results

Laser on site 1 Laser on site 2

Laser power

Measured domain wall position

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Transition state theory

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Connection to (higher-frequency) modes through the Arrhenius prefactor (attempt frequency) Example: Langer’s theory of transition rates

Γ ≡ 1 τ = λ+ 2π Ω0 exp ✓ − Ea kBT ◆

J S Langer, Ann Phys 54, 258 (1969)

dM/dt M H

(a)

dM/dt M H M× dM/dt

(b)

Linearised dynamics at S, 
 rate of growth of unstable mode

Ratio of curvatures Dynamical prefactor

H = ⇢ ∂2E ∂ηi∂ηj

• Ω0 =

s det HA | det HS| = s ΠiλA

i

Πj|λS

j |

Hessian matrix Ratio of products of eigenvalues of H

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Hysteresis loop as theme for quasi-statics
 
 
 Energy landscapes 
 (Meta)stable magnetisation configurations
 Domain walls, vortices, skyrmions, …
 
 Reversal mechanisms
 Navigate energy landscape through nontrivial states
 Rotation; nucleation, propagation, annihilation
 
 Time-dependent and thermal effects
 Measurement times matter
 Fluctuations drive transitions out of metastable states

Summary

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10 μm

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall depinning

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C Burrowes et al, Nat Phys 6, 17 (2010)

Fluctuations can drive domain walls out of a local potential well Probability distribution of residence (depinning) times used to determine energy barriers

Depinning time (s) Probability density 0.1 0.2 0.3 0.4 0.5 20 40 60 80

P = 1 τ exp ✓ − t τ ◆

1 τ = 1 τ0 exp ✓ − Ea kBT ◆

Distribution of depinning times

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall creep

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In disordered films, motion is more complicated under low fields Competition between domain wall energy and disorder potential Creep motion occurs, involving thermally-activated avalanches Useful analogy: Elastic band moving across rough surface

V Jeudy et al, Phys Rev Lett 117, 057201 (2016) S Ferroro et al, Phys Rev Lett 118, 147208 (2016) Colours indicate times

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall creep: Energetics

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Defect (attractive potential) Flat wall Deformed wall

u L + δL = L r 1 + 4u2 L2 L +

Eel = 1 2σ Z L dx ✓∂u ∂x ◆2 ' σu2 L Epin = −fpin p ξLn

S Lemerle et al, Phys Rev Lett 80, 849 (1998)

Balance between increase in elastic energy and decrease in pinning energy

Elastic energy Disorder (pinning) energy

σ = 4 p AKu

Characteristic length of disorder Defect density Characteristic strength of disorder Domain wall energy

European School on Magnetism 2018, Krakow – Magnetisation Processes (MP1) – Kim,JV

Domain wall creep: barriers and motion

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v = v0 exp ✓ − ∆E kBT ◆ ∆E = kBTd "✓ H Hd ◆−µ − 1 #

S Lemerle et al, Phys Rev Lett 80, 849 (1998) V Jeudy et al, Phys Rev Lett 117, 057201 (2016)

Energy barrier has power law dependence (µ = 1/4 for 2D systems) Avalanches – critical phenomena