Advanced micromagnetics and atomistic simulations of magnets Richard - - PowerPoint PPT Presentation

Advanced micromagnetics and atomistic simulations of magnets

Richard F L Evans ESM 2018

Overview

• Landau-Lifshitz-Bloch micromagnetics
• Applications of atomistic spin dynamics
• Simulations of ultrafast magnetisation processes

! −

Landau Lifshitz Bloch micromagnetics

Next generation micromagnetics: Landau Lifshitz Bloch equation

• Conventional micromagnetics ubiquitous but does a poor

job of thermodynamics of magnetic materials

• Atomistic models in principle resolve this but

horrendously computationally expensive

• Landau Lifshitz-Bloch micromagnetics is an advanced

micromagnetic approach which attempts to correctly simulate the intrinsic thermodynamic properties of magnets

• Still only a partial solution - crystal structure, interfaces,

surfaces, local defects, finite size effects all still not really accessible to a micromagnetic model

Landau Lifshitz Bloch (LLB) equation

• An additional dynamic term compared to the LLG equation
• Derived from the thermodynamic behaviour of a collection of classical

spins by D. Garanin [1]

• Longitudinal fluctuations (and damping) of the magnetization are now

included in the dynamics, enabling simulations up to and above the Curie temperature

• Also quantum flavours of the LLB

[1] D. A. Garanin, Phys. Rev. B 55, 3050 (1997) ˙ m = γ [m × Heff] + |γ |α|| m2 (m · Heff)m − |γ |α⊥ m2 [m × [m × (Heff + η⊥)]] + η||

Longitudinal term in the Landau Lifshitz Bloch (LLB) equation

• Longitudinal fluctuations of the

magnetization have their own dynamics

• Different effects below and above the

Curie temperature, Tc

• The effective magnetic field that

constrains the magnetization length is given by

+ |γ |α|| m2 (m · Heff)m

Heff = H + HA + ⎧ ⎨ ⎩

1 2 ˜ χ∥

• 1 − m2

m2

e

• m,

T Tc, − 1

˜ χ∥

• 1 + 3

5 Tc T −Tc m2

m, T Tc

Energy terms in the Landau Lifshitz Bloch (LLB) equation

• Conventional energy terms used in micromagnetics cause numerical

issues for the LLB, as any “applied” magnetic field will cause the moment length to grow

• Therefore need to treat internal fields in a special way so that in thermal

equilibrium, the net magnetic field is zero

Evans et al, Phys. Rev. B 85, 014433 (2012)

+ |γ |α|| m2 (m · Heff)m

F M0

s V =

⎧ ⎪ ⎨ ⎪ ⎩

m2

x+m2 y

2 ˜ χ⊥

+ (m2−m2

e) 2

8 ˜ χ∥m2

e

, T Tc,

m2

x+m2 y

2 ˜ χ⊥

+

3 20 ˜ χ∥ Tc T −Tc

• m2 + 5

3 T −Tc Tc

2 , T > Tc.

Parameters for the LLB equation can be derived from mean field or atomistic/multiscale simulations

Kazantseva et al, Phys. Rev. B 77, 184428 ︎(2008)

Tc T [K] me 900 600 300 1.0 0.8 0.6 0.4 0.2 0.0

me(T)

˜ χ ˜ χ⊥ T [K] ˜ χ [1/T] 900 600 300 0.2 0.15 0.1 0.05

χ(T)

Tc T [K] A [J/m] 900 600 300 5 10−12 1 10−11 1.5 10−11 2 10−11 2.5 10−11

x x x x x

A(T)

Comparative dynamics for LLB and atomistic simulations

T = 800 K T = 650 K T = 500 K T = 300 K t [ps] mz 5 4 3 2 1 1 0.8 0.6 0.4 0.2

Kazantseva et al, Phys. Rev. B 77, 184428 ︎(2008)

Applications of atomistic spin dynamics

Atomistic spin dynamics and temperature dependent properties of Nd2Fe14B

Permanent magnetic materials

Source: siemens.com

Structure at atomic and granular length scales determines overall material performance

Acta Materialia 77, 111-124 (2014)

Atomistic spin Hamiltonian for Nd2Fe14B

H = HNd +HFe HNd = −∑

i,δ

JNdFeSi ·Sδ −∑

i

Ek,Nd

i

− µNd∑

i

Happ ·Si HFe = −∑

ν,δ

JFe(r)Sν ·Sδ −∑

ν, j

JNdFeSν ·S j −∑

ν

Ek,Fe

ν

− µFe∑

ν

Happ ·Sν

Fe-Fe Exchange interactions

JFe(r) = J0 +Jr exp(−r/r0)

−0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 2 2.5 3 3.5 4 4.5 5 Exchange energy (× 10−21 J) Interatomic spacing (Å) ab−initio data Fitted function Scaled data

ab-initio data for BCC Fe from

• M. Pajda et al,
• Phys. Rev. B 64,

174402 (2001)

Magnetic anisotropy energy

• J. F. Herbst, Rev. Mod. Phys. 63, 819 (1991)

Fe Nd

1.0 1.5 2.0 2.5 3.0 k2

Fe(T) / k2 Fe(0)

a

10 20 30 100 200 300 400 500 600 Hk

Fe (kOe)

Temperature (K)

b

kFe

2 (e

T) = f(σ(e T))

f(e σ) = 1+ κca r tanh(re σ)

Ek,Nd

i

= −κNd

2

e P

2 −κNd 4

e P

4

e e e P

2 = − 1 3(3S2 z −1)

e P

4 = − 1 12(35S4 z −30S2 z +3)

Simulated temperature dependent magnetization

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 400 500 600 Magnetisation (m/ms) Temperature (K)

a

Simulation Fit

The temperature-dependent xpression m(T ) = (1 T/Tc)β (sho T = 631 82 K and exponent

Quantitative modelling of temperature dependent properties in ferromagnets

Richard F L Evans, Unai Atxitia and Roy W Chantrell

Evans et al, Phys. Rev. B 91, 144425 (2015)

Classical spin model m(T) simulation

Real ferromagnets: Kuz’min equation

m 1 s3=2 1 sp1=3;

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 m τ (b) Fe 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 m τ (c) Co v |

Real ferromagnets very different from classical model → problem!

• M. D. Kuz’min, Phys. Rev. Lett. 94, 107204 (2005)

Classical model Assume m(T) well fitted by Curie-Bloch equation Classical model: α = 1 Real ferromagnets: α ≠ 1 Simplest rescaling:

Phenomenological temperature rescaling

• τ = τ

1 α

m(τ) = (1−τα)β

xpression m(T ) = (1 T/Tc)β fitted T = 631 82 K and exponent

Texp = 300 K Simulation Tsim = 50 K Universe msim = 0.9 mexp = 0.9

Spin temperature rescaling (STR) method

Tsim Tc = Texp Tc α

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Experimental Temperature τ ~ Simulation Temperature τ α = 1

3/2 5/2

Spin temperature rescaling (STR) method

Classical spin model with Heisenberg exchange

Hexc =

X

i6= j

Ji jSi · S j

• vampire.york.ac.uk

VAM PIR E

R F L Evans et al, J. Phys.: Condens. Matter 26 103202 (2014)

20 nm 20 nm 20 nm

0.0 0.2 0.4 0.6 0.8 1.0 200 400 600 800 1000 1200 1400 1600 1800 Magnetization (m/ms) Temperature (K)

Co

Fit Calculated Kuzmin Corrected 0.0 0.2 0.4 0.6 0.8 1.0 200 400 600 800 1000 1200 1400 1600 Magnetization (m/ms) Temperature (K)

Fe

Fit Calculated Kuzmin Corrected 0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 400 500 600 700 800 900 Magnetization (m/ms) Temperature (K)

Ni

Fit Calculated Kuzmin Corrected 0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 400 500 Magnetization (m/ms) Temperature (K)

Gd

Fit Calculated Kuzmin Corrected

a! b! c! d!

• 2
• 1

1 0.2 0.4 0.6 0.8 1 Relative error (%) τ

• 1

1 0.2 0.4 0.6 0.8 1 Relative error (%) τ

• 1

1 0.2 0.4 0.6 0.8 1 Relative error (%) τ

• 1

1 2 3 0.2 0.4 0.6 0.8 1 Relative error (%) τ

Quantum

S

Classical

S S

Rescaled

Physical picture of temperature rescaling

Back to NdFeB

Apply temperature rescaling to achieve exact agreement with experimental Ms(T)

0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 400 500 600 Magnetisation (m/ms) Temperature (K)

b

Corrected simulation Classical fit Kuz’min fit

• R. F. L. Evans et al,
• Phys. Rev. B 91, 144425 (2015)

Tsim Tc = Texp Tc α

m(τ) = (1−τα)β

Temperature rescaling

5 10 15 20 25 30 35 40 100 200 300 400 500 600 Magnetisation (µB / f.u.) Temperature (K) Nd2Fe14B Fe fit Nd

Temperature dependent magnetization with temperature rescaling

Spin-reorientation transition

5 10 15 20 25 30 20 40 60 80 100 120 140 Angle from c axis (°) Temperature (K)

b

Model Experiment

𝜄

Ek,Nd

i

= −κNd

2

e P

2 −κNd 4

e P

4

Anisotropy field calculation

5 10 15 20 25 30 35 40 5 10 15 20 25 Magnetic moment (µB / f.u.) Applied field strength (T) Simulation [001] [110] Experiment [001] [110]

Domain wall structures in Nd2Fe14B

10 nm

Domain wall profile

1 nm

−1.0 −0.5 0.0 0.5 1.0 −4 −2 2 4 δ001 = 2.182 nm δ100 = 2.471 nm Reduced magnetization Distance (nm) δ001 δ100

0.

! −

Temperature dependent properties and dynamics of IrMn3 antiferromagnets

Sarah Jenkins and Richard F L Evans

Simple antiferromagnets

• ‘Simple’ antiferromagnets consist of two

magnetic sublattices

• Total magnetic moment is zero

(macroscopically)

• Can consider two antiparallel contributions

from each ‘colour’ of spin

• This is called the sublattice magnetization
• The Néel vector n is the equivalent order

parameter for antiferromagnets

ma = ∑

a

Sa mb = ∑

b

Sb n = ma − mb

Motivation: exchange bias and antiferromagnetic spintronics

http://nabis.fisi.polimi.it/research-areas/antiferromagnet-spintronics/

I r M n 1 M n 2 M n 3 M n 4

Ordered L12 IrMn3 Disordered ɣIrMn3

Crystallographic structure

[111]

Atomistic spin model

• Ir

Mn Ir JNN-JNNN model Néel pair anisotropy

IrMn IrMn3 Rij (˚ A) Jij (meV) 10 9 8 7 6 5 4 3 2 20 10

• 10
• 20
• 30
• 40
• 50

AFM FM

H = ∑

i<j

JijSi ·Sj kN 2

z

∑

i6=j

(Si ·ei j)2

! −

a

−

[001] [010] [100]

L12 - IrMn3 ɣ - IrMn3 Triangular (T1) Tetrahedral (3Q)

Simulated ground state structures

Ordered L12 IrMn3 Disordered ɣIrMn3 TN = 1000 K TN = 680 K

0.2 0.4 0.6 0.8 1 200 400 600 800 1000 1200 Sublattice magnetisation (n(T)) Temperature (K) L12-IrMn3 γ-IrMn3

b

Simulated Néel temperatures

Use spin dynamics to calculate switching rate for small system near blocking temperature

𝜐0 = 3.84 ⨉ 1011

Much higher attempt frequency than equivalent ferromagnets (~1010)

Thermodynamics of ultrafast magnetization processes

Ultrafast demagnetization in Ni

• E. Beaurepaire et al, Phys. Rev. Lett. 76 4250 (1996)

Laser excitation M

Origin of thermal fluctuations in the atomistic model

• Lets go back to the thermal fluctuations in the atomic

model

• Physically caused by spin scattering phenomena
• electron-spin, spin-phonon, spin-photon
• Laser interaction causes heating of the electrons and

more scattering events -> fast increase in the effective temperature in the material

Equilibrium properties of Ni

• Use spin temperature rescaling to accurately reproduce temperature

dependent magnetization

Temperature (K) 0.0 0.2 0.4 0.6 0.8 1.0 100 200 300 400 500 600 700 800 900 Magnetization (m/ms) Temperature (K)

Ni

Fit Calculated Kuzmin Corrected

c! d!

• 1

1 0.2 0.4 0.6 0.8 1 Relative error (%) τ

Evans et al, Phys. Rev. B 91, 144425 (2015)

Simulating a laser pulse: two temperature model Free electron approximation

400 600 800 1000 1200 1400 1600 0.5 1 1.5 2 Temperature (K) Time (ps) Tp Te

Ce ∂Te ∂t = − G(Te − Tp) + S(t) Cp ∂Tp ∂t = − G(Tp − Te)

• S. Anisimov, B. Kapeliovich, and T. Perelman, Sov. Phys. JETP 39, 375 (1974).

Ce = C0Te

Te Tp

0.5 0.6 0.7 0.8 0.9 1.0 5 10 15 Normalized magnetization Time (ps) Classical Rescaled Experiment

Ni

• R. F. L. Evans et al, Phys. Rev. B 91, 144425 (2015)
• E. Beaurepaire et al, Phys. Rev. Lett. 76, 4250 (1996)

Ultrafast demagnetization in Ni

damping-constant = 0.001

What about magnetic alloys?

• Ni shows ultrafast response to a

laser excitation?

• What about alloys? How do

different magnetic moments inside a material respond to ultrafast laser excitation?

• Consider permalloy - alloy of 80%

Ni, 20% Fe

• With XMCD can measure response
• f each sublattice separately

Demagnetization dynamics in bulk Ni80Fe20 Permalloy

0.0 0.2 0.5 0.8 1.0 0.2 0.4 0.6 0.8 1 1.2 Normalized magnetization ( M / Ms ) Time (ps) Ni Fe

• I. Radu et al, SPIN 5, 1550004 (2015)

𝝊 ∝𝜈/𝛽

𝜈Fe = 2.30 𝜈B 𝜈Ni = 0.98 𝜈B 𝛽 = 0.0065

Artificial frustration - a route to tuneable dynamics?

𝜈eff ~ 0.13 𝜈s 𝝊 ∝𝜈eff/𝛽 ?

Permalloy nanodot simulation Ni80Fe20

70 nm 20 nm

𝜈Fe = 2.30 𝜈B 𝜈Ni = 0.98 𝜈B

8,815,413 spins

Include spin temperature rescaling to get correct dynamics

Parallel scaling of VAMPIRE code

100 nm x 100 nm x 20 nm (18M spins)

100 1000 10000 24 48 96 192 384 768 1536 3072 6144 Runtime (s) Number of CPUs

ARCHER Group Cluster 10K, 100K and 1M spins

Demagnetization process in a Ni80Fe20 nanodot

Short time demagnetization dynamics in Ni80Fe20 comparing bulk and vortex samples

Vortex structure has no effect on dynamics!

25 50 75 100 0.2 0.4 0.6 0.8 1 1.2 Normalized magnetization (%) Time (ps) Ni (vortex) Fe (vortex) Ni (bulk) Fe (bulk)

Longer timescale remagnetisation dynamics are different - topology?

25 50 75 100 2 4 6 8 10 12 14 16 18 20 Normalized magnetization (%) Time (ps) Ni (vortex) Fe (vortex) Ni (bulk) Fe (bulk) To be continued…

25 50 75 100 200 400 600 800 1000 Normalized magnetization (%) Time (ps) Ni (vortex) Fe (vortex)

After thermal “kick”, oscillatory dynamics are long lived

Ultrafast heat-induced switching of GdFeCo

GdFe ferrimagnet Gd Fe

Ferrimagnetic nature of GdFe(Co) and spin models

JTM−RE=1.00Jmax JTM−RE=0.75Jmax JTM−RE=0.50Jmax JTM−RE=0.25Jmax RE TM T [K] Reduced Magnetisation 900 750 600 450 300 150 1.00 0.75 0.50 0.25 0.00

• 0.25
• 0.50
• 0.75
• 1.00

H = −1 2

• ⟨i,j⟩

JijSi · Sj −

N

• i=1

Di(Si · ni)2 −

N

• i=1

µiB · Si,

T Ostler et al, Phys. Rev. B 84, 024407 (2011)

Ultrafast magnetization dynamics measured with XMCD

Complex reversal mechanism owing to different sub lattice magnetization dynamics

• I. Radu et al, Nature 472, 205–208 (2011)

Ultrafast magnetization dynamics simulated with atomistic spin model

• I. Radu et al, Nature 472, 205–208 (2011)
• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 Normalized magnetization Time (ps) Fe Gd

Atomistic prediction of heat induced switching

• l
• en

e , en h ld

• ).

in ic ues h l

• s
• le
• n

e

a b c

400 800 1,200 Te (K) –1.0 –0.5 0.0 0.5 1.0 Mz/M0 –0.6 –0.3 0.0 0.3 0.6 1 2 M (µ B per atom) 250 500 750 Time (ps)

• T. Ostler et al, Nat. Commun. 3, 666 (2012)

Experimental confirmation of heat-induced switching

• T. Ostler et al, Nat. Commun. 3, 666 (2012)

Difference in scale for magnetisation dynamics

What about the role of inhomogeneity in the sample?

±

±

± ±

Graves et al, Nature Materials (2013)

Different dynamics based on Gd and Fe concentrations

Total Concentration resolved

• E. Iococca et al, arXiv:1809.02076

Large scale simulation 1 µm x 1 µm x 10 nm

• E. Iococca et al, arXiv:1809.02076

Summary

• Introduced the basic background of

Landau-Lifshitz-Bloch micromagnetics

• Presented simulations of the static and

dynamic properties of more complex magnets

• Thermodynamics is a significant and

important contribution to ultrafast magnetic processes

! −