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Modelling & SimulationThursday 14 August14:00-15:3016:00-17:30
Robert Stamps received BS and MS degrees from the University of Colorado, and a PhD in Physics from Colorado State University. He was with the University of Western Australia until 2010, and is currently Professor of Solid State Physics at the University of Glasgow in Scotland. He was an IEEE Magnetics Society Distinguished Lecturer in 2008 (including visits to CBPF and elsewhere in Brazil), and he was the IEEE/IOP Wohlfarth Lecturer in 2004. He is chair of the IRUK IEEE Magnetics Society Chapter, was chair of the 2007 MML Symposium, and will co-chair the Joint European Magnetics Symposia in 2016. This is the fourth time he has lectured at an IEEE Magnetics School.
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To provide an introduction to the philosophy and art of modelling of the essential physics at play in magnetic systems.
Examples will be given of how simple models can be constructed and applied to understand and interpret observable phenomena, ranging from magnetisation processes to high frequency spin wave dynamics.
Along the way, an introduction to some general tools will be provided, including Monte Carlo models and micromagnetics.
Aim of lectures:
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Modelling and Simulation
Robert Stamps
IEEE Magnetics Society School 2014
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Outline• Modelling: where to start?
– Starting points
– Phenomenology
• Some generic tools:
– Micromagnetics
– Mean field theory & Monte Carlo • Spin dynamics
– Torque equations
– Spinwaves & resonances• Domains and domain walls
– Stoner-Wohlfarth models
– Magnetic domains and domain walls
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Modelling: where to start?
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Models for research & development: magnetic ordering,
dynamics, transport ...
Some starting points for model makers
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1) Simulations do not by themselves provide interpretations or insights
2) Analytic/conceptutal models often go where simulations cannot
Tools
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The dark arts of simplification:
phenomenology
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EnergiesRelevant energy scales (P. W. Anderson, 1953):
1 – 10 eV Atomic Coulomb integralsHund's rule exchange energyElectronic band widthsEnergy/state at e
f
0.1 – 1.0 eV Crystal field splitting
10-2 – 10-1 eV Spin-orbit couplingk
BT
C or k
BT
N
10-4 eV Magnetic spin-spin couplingInteraction of a spin with 10 kG field
10-6 – 10-5 eV Hyperfine electron-neuclear coupling
mag
non
reg
ion
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Concept: Exchange EnergyPauli exclusion separates like spins:
Can be energetically favourable: suppose alignment determines average separation. Then if:
⟨ra⟩∼0.3 nm
rp
ra
e2
r a
∼4.8 eV
⟨r p⟩∼0.31nm e2
r p
∼4.75eV
E↑↑−E↑↓=0.05 eV (580 K )
… equivalent field: E↑↑−E↑↓μB
=870T
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Exchange Interactions
Hamiltonian as spin functions: (Dirac & Heisenberg)
H̃=−J 1,2σ1⋅σ2
Pauli spin matrices
Generalised for multi-electron orbitals (van Vleck):
H ex=−∑a ,bJ (ra−rb)S (ra)⋅S (rb)
total spin at sites r
Exchange: electrostatic repulsion + quantum mechanics.
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Using Symmetry: ExchangeMeasurable moment density (not an operator):
Exchange still in Heisenberg form:
m(r )=Tr (ρ M̂ (r))density matrix
E ex=∑ j ,δ J m(r j)⋅m(r j+ δ)
neighbours
Atomic to continuum: Expand m field about rj
m(r j+ δ)=m(r j)+ [(δ⋅∇ )m(r j+ δ)] j=δ+12 [(δ⋅∇ )2m(r j+ δ)] j=δ+ ...
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Example: isotropic medium
Using Symmetry: Exchange
When lattice symmetry allows:
Exchange energy must be compatible with symmetry of the crystal
E ex=∑α k l C kl∂mα(r)∂ r k
∂mα(r)∂ r l
Eex=mx(∇2 mx)+m y(∇
2 m y)+mz(∇2 mz)
δx∂m∂ x+ (−δx)
∂m∂ x=0
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Using Symmetry: AnisotropyLocal atomic environment affects spin orientation:
[Kittel, Introduction to Solid State]
Anisotropies & symmetries: (u = m/Ms)
E ani=−K u(1)u z
2−K u
(2)u z4+...• Uniaxial:
• Cubic:
E ani(u z)=E ani(−uz)
Spin orbit interaction and crystal field effects
E ani(u x , u y , uz)=E ani (−u x , u y , u z), etc.
E ani=K 4(ux2 u y
2+u x
2 u z2+u y
2 u z2)+...
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Using Symmetry: Anisotropy
wondermagnet.com
All moments interact throughout sample. Sample shape creates an anisotropy:
θM
E ani=M 2 V2μo
(N x sin2θ cos2ϕ+N y sin2θsin2ϕ+N z cos2θ)
ϕ
Easy direction Hard direction
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Dzyaloshinski-Moriya Interaction
H=∑i , j [ J S i⋅S j−D⋅(S i×S j)]
Asymmetric interaction possible when inversion symmetry absent:
Describes weak ferromagnetism of canted antiferromagnets:
D = 0 D 0
M
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It's Only Angular Momentum
Bohr and Pauli Study Angular Momentum
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Some generic tools
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Magnetostatics and DomainsDipolar fields compete with other local fields:
S
N
S
N
S
N
S
N
S
N
S
N
S
N
S
NHigh magnetostatic
energyLow magnetostatic
energy
Competition between exchange, anisotropy and magnetostatic energies
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Domain Patterns
ifw.dresden.de
ww
w.z
eis
s.d
e
http://www.nanotec.es/FeBsquares_mag.jpg
http://www.phy.cam.ac.uk/
Pattern detail depends on magnetization, exchange, anisotropy...
… and geometry!
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The Problem of Dipolar InteractionsMagnetic fields decrease slowly with distance-- sample shape matters
Magnetisation is generally not uniform:
wikipedia.org
Nmag.soton.ac.uk
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Micromagnetics
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Minimising the Energy
Goal: find stable (and metastable) configurations that define minima of the total energy E
u⃗=M⃗M s
E(u⃗)=∫[A (∇ u⃗)2−K n ( n̂⋅⃗u )2−μo M s(u⃗⋅H⃗ a+ u⃗⋅h⃗d)]dV
exchange
anisotropy
applied fieldmagnetostatic
reduced MδE=0
Minimisation = vanishing torques:
u⃗×(−∂E∂ u⃗ )=0
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Finding Zero Torque Solutions
Strategy: Numerically integrate torque equations
∂ u⃗∂ t∝ u⃗×(−∂E
∂ u⃗ )+Γdamping→0
Gilbert damping:
(1+α2
|γ| )∂ M⃗∂ t=−M⃗×H⃗ eff−
αM s
M⃗×M⃗×H⃗ eff
H⃗ eff=−∂E(M⃗ )
∂ M⃗
damping parameter
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Finite Differences
Convert differential equations to difference equations:
uβ(x+Δ x)=uβ(x)+Δ x ∂∂ x
uβ(x)+12(Δ)
2 ∂2
∂ x2uβ(x)
uβ(x−Δ x)=uβ(x)−Δ x ∂∂ x
uβ(x)+12(Δ)
2 ∂2
∂ x2uβ(x)
Divide magnetisation into blocks,replace differentials, construct torque equations for each block
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Magnetostatic Terms
The magnetostatic terms link all blocks throughout the sample
Maxwell equations define a magnetostatic potential
Blocks are sources of H field
B⃗=μo(H⃗+M⃗ )
∇⋅B⃗=0
∇×H⃗=∂ D⃗∂ t≈0
H⃗=−∇Φ
∇2Φ=−∇ M⃗
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Micromagnetics and GPU'sThe magnetostatic calculation involves convolution over all blocks:
Accelerate calculations using Graphical Processing Units
H⃗ (i)=K̂ (i , j)∗M⃗ ( j)
block i
blocks j≠i
[Vansteenkiste, et al. arXiv:1406.7635]
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Example: Mumax3// Standard Problem #4
SetGridsize(256, 64, 1)SetCellsize(500e-9/256, 125e-9/64, 3e-9)
Msat = 800e3Aex = 13e-12alpha = 0.02
m = uniform(1, .1, 0)relax()save(m) // relaxed state
autosave(m, 200e-12)tableautosave(10e-12)
B_ext = vector(-24.6E-3, 4.3E-3, 0)run(1e-9)
define grid and sizes (m)
parameters (SI)
initialse Mfind zero torque configurationsave configuration
save configurations every 0.2 nscreate table of m(t)
apply magnetic fieldtime evolution
Information & Download: http://mumax.github.io/index.html
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Run Standard Problem 4
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Approaches (with example codes)
VAMPIRE http://www-users.york.ac.uk/~rfle500/research/vampire/
http://nmag.soton.ac.uk
Nmag http://nmag.soton.ac.uk/nmag/
Finite element: useful for complex geometries
Atomistic: model atomic lattice scale variations
Finite difference: mumax3, OOMMF
MAGPARhttp://magnet.atp.tuwien.ac.at/
… and many more !
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Limitations!
Lengthscales are limited
Shapes are approximate
Timescales are limited
Classical limits: dynamics & thermodynamics
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Questions?
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Mean field approximation
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Thermal FluctuationsReduction in magnetisation:
M (T )∼⟨S ⟩
Replace local site field with averaged effective field:
Dynamic correlations are replaced by a static field:
H=−2∑i , jJ ex S i⋅S j≈−2∑i , j
S i⋅Bex Bex=2 Z J ex
N gμB
⟨S ⟩
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Heisenberg Model and Mean FieldHeisenberg exchange energy:
H=−∑i , jJ ij S⃗ i⋅S⃗ j
Thermal averaged magnetisation (N moments):
M⃗=N gμB ⟨ S⃗ ⟩
Fluctuations:s⃗i= S⃗ i−⟨ S⃗ ⟩
H=−∑i , jJ ij ( s⃗i+ ⟨ S⃗ ⟩)⋅( s⃗ j+ ⟨ S⃗ ⟩)
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Heisenberg Model and Mean FieldZ near neighbours:
H=−J∑i , js⃗i⋅s⃗ j−2 Z J∑i
S⃗ i⋅⟨ S⃗ ⟩+ Z N∣⟨ S⃗ ⟩∣2
Second term is the mean field:
B⃗ex=−2 Z J ⟨ S⃗ ⟩
Mean field approximation: neglect first term (correlations)
H fluctuations=−J∑i , js⃗i⋅s⃗ j
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Reminder: ParamagnetismProbabilities to be antiparallel (down) and parallel (up):
n↑N∝exp(
μB Bk B T
)n↓N∝exp(
−μB B
k B T)
Magnetisation = difference:
⟨S ⟩=(N↑−N↓
N )=tanh(μB B
k B T )
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Generalised Paramagnetism
Brillouin function for any J:
Average magnetisation from: M ∝⟨S ⟩=BJ (x)
Angular momentum states ( J = 1/2, 3/2, 5/2, … ):
B
B J (x)=2J+ 1
2Jcoth(2J+ 1
2Jx)− 1
2Jcoth ( 1
2Jx)
x=g J μB B
k B T
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Exchange: Replace B by Bex
⟨ S⃗ ⟩=BJ( gμB Z J ex ⟨ S⃗ ⟩
k B T )Average M with mean field B
ex:
Plot left and right hand sides to see graphical solution:
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Example: MultiferroicsCoupled order parameters: M & P
(M = sum of canted antiferromagnetic sublattices)
Challenges:
- correlations between spin and charge distributions- how to describe dynamics?- how to describe effects of thermal fluctuations?
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Example: MultiferroicsCoupled order parameters: M & P
F=F FE (P)− P⃗⋅E⃗−λ m⃗a⋅m⃗b−K (maz2 +mbz
2 )−m⃗⋅H⃗ +F ME
polarization part magnetization part
Mean field approximation for free energy:
magneto-electric
coupling
Approach:(Vincinsius Gunawan PhD 2012)
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Example: Multiferroics
dd
F=0
[Gunawan et al., JPCM (2011)]
Landau-Ginzburg meanfield theory for P:
Minimise free energy for P and q:
dd P
F=0
ms ,α=gμB J BJ (m⃗s⋅B⃗s)Brillouin function for components of m:
F FE(P)=αo(T−T c)P2+βP4
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Monte Carlo methods
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Ising model and Monte CarloSuppose two possible states: 'up' and 'down'
Boltzman probability for individual flips:
P (−S i)∼exp(−J (∑i
S i)
k B T)
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Sampling Random Fluctuations Thermal fluctuations and 2 dimensional Ising model:
Low T Near Tc
Above Tc
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Constructing AveragesFluctuations drive the system towards thermal equilibrium.
Sample a distribution for averages:
P↑↓∼exp(−Δ E (↑⇒↓)k B T )
P↓↑∼exp(−Δ E (↓⇒↑)k B T )
⟨A ⟩=∑σA (σ)ρ(σ) ρ(σ)=
1Z
exp(−E(σ)kB T )
Key idea: s is a configuration from the ensemble of equilibrium spin configurations
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The Metropolis AlgorithmSample from {s}: Start with some x, generate a s' with a single spin flip.
Rules: Calculate DE = E(x) - E(s') 1) If DE < 0, accept s' as an equilibrium fluctuation2) If DE > 0, accept s' if P(DE) < 1
For equilibrium fluctuations, P(DE) must satisfy detailed balance:
P(σ ')W (↑⇒↓)=P(ξ)W (↓⇒↑)
W (↑⇒↓)W (↓⇒↑)
=P(ξ)
P(σ ')=P(ΔE)=exp(−E(ξ)−E(σ ' )
kB T )
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Run Monte Carlo
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Example: Interacting magnetic particles
Challenges: - large arrays of submicron elements - super-paramagnetic - long range interactions
Approach: (Zoe Budrikis, PhD 2012)Combine Mean Field & Monte Carlo
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Example: Artificial Antiferromagnet(artificial square spin ice)
Shape anisotropy: Ising spins
Dipolar interactions
6 interactions but can only minimise 4
Wa
ng
et
al.,
Na
ture
(2
00
6)
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Thermal fluctuations on 2 timescales: - small volumes (reversal)
- thermal reduction of element M
Thermal evolution of domains
Configuration dependent local M
KVk BT
∼1
enhancement
suppression
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Mean field model: thermal dynamics
Mean field model for element magnetisations:
⟨m j⟩=B1 /2[βm j⋅(hc+∑k≠ jJ j , k ⟨mk ⟩)]
Algorithm: - self consistent iteration for <m
j>
- stochastic reversal (Monte Carlo)
Disorder: uniform distribution for K centred on Ko
K=K o(1+ r2) r∈[− ,]
hc=K ⟨m j⟩
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Configurations
Type I
(ground)
Type II
(wall)
Type III
(defect)
Local spin configurations:
Nisoli et al., Nature Physics (2010)
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T
Monte Carlodynamics
Approach to Ground StateInitial T=0 Type I state (~ “FC”): thermal decay
Initial T=0 quenched random state (~ “ZFC”):
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Growth of Domains and Wall MotionType I domains separated by Type II walls:
Type III 'charge' production during wall motion
• Type I
• Type II
• Type III
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Thermal fluctuations at walls
Thermal fluctuations largest on domain walls
• M = 1
• M = 0.1
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Challenge: modelling kinetics in real time with Monte Carlo
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Continuous Time Monte CarloProbability for acceptance of a single flip (out of N spins):
Q=1N∑Δ E
n(ΔE)P(ΔE)
number of spins with DE
Probability that a spin will flip in time Dt: P flip(Δ t)=exp(−Δ t
τ Q)
Rejection free algorithm: 1) track all possible transitions2) accept one according to random R 3) update time according to Δ t=− τ
Qln R
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Example: Exchange Bias
Stamps, PRB 2000
M Kirschner http://magnet.atp.tuwien.ac.at
Time dependent coercivity: Field sweep rates
Thermal setting of bias:
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Note on phase transitions: Scaling near critical points
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Schematic of the Transition (2nd order)
[Weiss & Forrer]
Linear spinwaves
Large amplitude fluctuations
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Scaling
Mean field theory: M (T )∼(T−T C)1/2
Reality includes correlations: M (T )∼(T−T C)β
β≈0.34
Note on dimensionality: • Ultra thin films ~ two dimensional systems • fluctuations destroy long range order • nano-thermodynamics for small elements (~ 0 D!)
Remember this for later when we talk about domain wall creep
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Break!
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Spin Dynamics
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Low Temperature Fluctuations
Energy to reverse one spin: 2 J
Superposition of ways to flip one spin:
Spinwave excitation
n 1
H J iji j
S i S j
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Torque equations
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Excitations: Spin Waves
Excitations: Precessional dynamics
slide courtesy J-V Kim
Ground state magnetic orderings:
Note: The excitations are bosons!
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Spin Waves and MicromagneticsProcedure:
1) Relax to steady state2) Use broadband pulse to excite spin waves3) Record time evolution (for spectral analysis)
Example: exciting precession in mumax3 script
defregion(1,rect(10e-9,125e-9))save(regions)
driv := 0.001 // amplitude driving fieldf := 1.0e9 // frequency unitsfdel := 20.*f*2.*pi // frequency windowtime := 1000./fdel // evolve timetoff := 3./f // offset
B_ext = vector(-24.6E-3, 4.3E-3,driv*sin( (t-toff)*fdel )/(2*pi*(t-toff)*fdel))run(time)
define antenna region
sinc function pulse
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Results
Ground state:
Antenna:
h(w)
Mz(w)
Spectra:
Note: Spectral analysis performed separately on mumax generated data.
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Geometry Example: Antidot Array
// array of holes set in repeating frame
SetPBC(3, 3, 0) // for periodic bc
ndots := 9 // number of dots in each framer := 100e-9 // period of latticeb := ndots * rc := b * 1.3 // frame size
h := 55e-9hole := cylinder(h, h) EdgeSmooth = 8 // use small blocks to smooth dots
setgeom( rect(c,c).sub(rect(b,b)).add( rect(b,b).sub(hole.repeat(r,r,0))) )
surround by copies, depth 3
combine basic elements to construct geometry
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Results (Francisco Trinidade PhD ~2015)
HDC
Fre
quen
cy
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Example: (Pablo Boyrs, PhD ~2015)
Dyzaloshinskii-Moriya Interaction (DMI) and Spin Waves
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Spin Waves and DMILow symmetry allowed exchange terms
E DM=D m⃗⋅∇×m⃗
Phenomena:• weak ferromagnetism and multiferroics• helicoidal and skyrmionic spin textures• exchange bias (Dong et al. PRL 2009, Yanes et al. ArXiv 2013)• metal films (Fert & Levy, PRL 1980; Bodanov et al. PRL 2001)• domain wall structures (Thiaville et al. EPL 2012; K-J Lee)
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Asymmetry of Spin Wave Profiles
Cou
rtes
y J-
V K
im
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Spin Waves & DMI: Domain WallsProfile determined by exchange + anisotropy
E=∫[[A(∂θ∂ x )2
+ sin2θ(∂ϕ∂ x )
2
]+ D sin2θ∂ϕ
∂ x−k u cos2
ϕsin2θ+ k p cos2
θ]dx
Heisenberg DM exchange
Anisotropies
Example: Bloch-Neél wall
[Thiaville, et al ELP 2013]
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DMI Edge States & Nonreciprocity
PRB 2014
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Note: Spin Wave DispersionsSpin wave w(k) from micromagnetics:
Venkat, Fangohr, et al., IEEE Trans. Magn. 49 (2013)
h(x , y , t)=sinc ((x−x ')k x) sinc(( y− y ')k y)sinc ((t−t ')ω)
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Don't forget analytic models!
Example: (Rhet Magaraggia, PhD 2011)Microwave spectroscopy of thin films
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Microwave Spectroscopy
nanotechweb.org
• Resonant absorption and standing spinwaves • Energies (~ µ eV): fine scale electronic states and processes• Buried interfaces and surfaces
magnetbackscattering optics
sample
physics.colostate.edu
• 1-20 (40) GHz range• Sensitivity to ~ 3 nm thick Py• Parameter extraction: vary field magnitude and orientation
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FMR Spectra
Magaraggia, et al., PRB 83, 054405 (2011)
Broadband FMR with 60 nm FeNi:
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Exchange Anisotropy
y
T⃗ surf=−M⃗×∇M E SA=−M⃗× p⃗
Require: p(FMR) = p(FEX)
'Kittel' formula for FMR:
Pinning changes ky:
Simplify: assume single uniform thin film
p(θ)=( 2AM s)[
−k y(θ)
cot (k y(θ)t eff ) ]
(ωγ )2
=[H f (θ)+D k y2(θ)]
×[H f (θ)+D k y2(θ)+μo M S]
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Pinning & Effective Thickness
p(θ)=( 2AM s)[
−k y(θ)
cot (k y(θ)t eff ) ]Angular dependence:
Pinning parameter:
Magaraggia, et al., PRB 83, 054405 (2011)
fit parameter
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Don't forget analytic models!
Example: (Karen Livesey, PhD 2009)Nonlinear spin waves
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Spinwave InteractionsBeyond linearisation: spin wave interactions
M z=M s[1−mx2m y
2
M s2 ]
1/2
=M s[1−m+2m-
2
M s2 ]
1 /2
m'±'=mx±i m y
Expand in spin wave amplitudes:
m+=1
√N∑k
e- i k⋅r ck+ m-=
1
√N∑k
e ik⋅r ck
Energy now includes interactions:
Example three wave processes: g k ck c-k c0
+g*
k ck+ c- k
+ c0+
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Reversal & Spin Waves
Instability for growth of mode amplitudes:
Apparent reduction of M with onset of instability:
[Livesey et al, Phys. Rev. B (2007)]
small angle precession large angle precession
mz(t)
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Reversal & Spin Waves
Onset of 4-wave processes:• threshold angle• depends on spin wave (k)• (k) depends on bias direction
Apparent reduction of M because of spinwave
turbulence
c00
Nembach et al., Phys. Rev. B 84, 184413 (2011)
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Domains and Domain Walls
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Switching of Single Domain Particles
Dynamics: Precessional reversal
Stability: Thermal activation
€
H ≥ HcHc
€
H < HcH
H
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Challenge: fluctuations over long time scales
Approach: Stoner-Wohlfarth models
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Independent Particles: H<Hc
Climbing to the top: fluctuations
Energy transfer between spin system and heat bath
Torque equation of motion: thermal fluctuation ‘field’
random thermal ‘driving torque’
ddtm=− m× d E
d m h f
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Single Domain Rotation
Approximate reversal as pure relaxation:
E=V (−B M cosθ+K sin2θ)
ϵ=V K [1+(B M2K )
2
]
Rate depends on activation energy and attempt frequency
Stoner-Wohlfarth Model
=1= f o exp −/ k BT
H
B
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Reversal of a Particle Ensemble
Ensemble of particles: B=0, thermal fluctuations reduce M
Approach to equilibrium: Chemical rate problem
dndt=W n−W n
dndt=W n−W n
mt =n−n=A e− t
B = 0B
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Reversal of a Particle Ensemble
Ensemble of particles: H=0, dipolar fields drive m to 0
m =1 m =0
Approach to equilibrium: Distribution of rates
Can one measure the distribution of rates P( )? (Rebecca Fuller, PhD 2010)
mt =A∫P e− t d
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Relaxation: DistributionsDistribution of energy barriers:
Magnetic viscosity: ln(t) for broad distributions
m(t )=C−S (H ) ln (t Γo)
= f oexp −/k B T
m(t )=m(∞)+A∫0
∞
P (ϵ)e−tΓ(ϵ)d ϵ
[Fuller, et al., JPCM 2009]
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Relaxation: Energy BarriersUseful measure: study dm/dt at different T
P (ϵ=k B T ln (t / τo))≈1
Ak B T (−td md t )=
SAk B T
Viscosity at different temperatures and
fields provides estimates for energy barrier distribution
[Fuller, et al., JPCM (2009)]
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Questions?
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Magnetic domains and domain walls
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Nucleation of domains and domain walls
[Slaughter, 2000]
Routes to Reversal
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Challenge: fluctuations over long times and large lengthscales
Approach: a Stoner-Wohlfarth model for fluctuating lines
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Domain & Wall Dynamics
Co0.5 nm
Ag
Ag
• Example: MOKE study• Perpendicular M in Co• Method: - saturate - apply field pulse - image & repeat
H
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Magnetization Processes & Domains
H
Nucleation processes:
Growth of a critical domain volume
E Zeeman=−M V H EDW= A
VAc=
M H
Surface energy
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Magnetisation Processes & Domains
Growth stops at local field gradients
(pinning 'pressure')
Hpressure∼−∇ E local
Stroboscopic 'movie' of domain growth
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Magnetization Processes & DW's
Wall structure:– Topological excitation
– Surface tension
– Characteristic width
Dynamics:– Translation & fluctuations
– Pinning & 'creep'
– Internal modes
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Magnetization Processes & DW's
Domain walls define spin reorientation:
Energy: exchange + anisotropy
E=∫ A
M s2 [∇mx
2∇my2∇mz
2]− K
M s2mz
2d r
Allowed spin configurations minimise E
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Domain WallsSuppose wall along y direction:
mz=M s cos y my=M s sin y
A ∂2
∂ y2K sin2=0
cos=tanh[ y− yo
A/K ]
Minimum energy requires
Solution:
=4 A K
width = l energy
q(y)
y
E=∫[ A
M s2(∂mz ( y)
∂ y )2
−K
M s2 mz
2]dx
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DW Mobility: High Field FlowViscous Flow: High field driven dynamics
H
vDW ~ H
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DW Mobility: High Field FlowViscous Flow: High field driven dynamics
H
V(x) x
vDW ~ H
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DW Mobility Theory: Flow
Time averaged velocity (in Flow regime):
v∝∫∣m×H∣2d 3 x
precessional torqueGilbert damping
Effective local field: (exchange, anisotropy, dipolar) H=H applied−
∂ E∂m
∂m∂ t=−m×H
M s
m×∂m∂ t
Torque equations of motion:
[X Wiang, P Yan, J Lu]
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DW Mobility: Low Field CreepCreep: low field thermally activated dynamics
H t = t1
t = t1+
t = t1+2
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DW Mobility: Low Field CreepCreep: low field thermally activated dynamics
H t = t1
t = t1+
V(x) x
V(x) x
t = t1+2
V(x) x
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DW Mobility Theory: Creep
Pinning sites oppose wall motion:
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DW Mobility Theory: Creep
Pinning sites oppose wall motion:
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DW Mobility Theory: CreepNumber of pinning sites:
L
E p=√ f pin2 N pξ
pin force
Macroscopic wall motion through avalanche:
[Eelastic−E Zeeman]=E B≈U C( H dep
H applied)
2 ζ−2+ D2−ζ
Scaling: critical field for avalanche onset
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DW Mobility Theory: CreepDepinning rate:
Multiply by distance travelled to give velocity:
1τ(L)
=1τ0
exp[−E B(L)
k BT ]
v=w (L)τ (L)
≈ξτ0
exp[−U C
k B T ( H dep
H applied)μ
]Expect µ = ¼ for ultra thin films.
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DW Motion: TransitionThreshold: transition from creep to viscous flow
vDW
Hc
H
Zero temperature
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DW Motion: TransitionThreshold: transition from creep to viscous flow
vDW
Hc
H
Zero temperature
H
H
Tk
Uvv p
B
co exp
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DW Motion: TransitionObserved transition from creep to viscous flow:
(Peter Metaxas, PhD 2009)
[Metaxas, et al., Phys. Rev. Lett., 2007]
H
H
Tk
Uvv p
B
co exp
Hv
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Challenge: What are magnetisation processes in chiral spin systems?
Approach: (Pablo Boyrs, PhD ~2015)Visualising 1 D spin textures
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Experimental System
Cr Nb3S
6: P6
322
Tc = 130 K
Conductor
L~
48 nm
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soliton (~domain wall)
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Useful Visualisation
x
1 2 3
1
2
3
Along the axis view
Down the axis view
In order to follow rotations, map thespin tips along thechiral axis (x):
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Soliton FormationH = 0 H = 0.2 H = 0.4
H = 0.6 H = 0.7 H = 0.8
note: curl at end is amplified by this method of plotting
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Confined Solitons
R LL R
Pinning & confinement by chiral
boundaries
R L
Lorentz TEM (Y Togawa & S McVitie)
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Confined Solitons
H = 0.2
Nucleation occurs at chiral boundaries-- deformation of kinks produce solitons
H =0.2267
Chiral boundaries define twist direction reversal: creates soliton pinning sites
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Summary
•Approximations: Heisenberg exchange, anisotropy, mean field theory
•Simulations: Micromagnetic, Monte Carlo
•Analytic models: spin waves, domain walls, thermal activation
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The End