[PPT] - COMPUTER SIMULATION STUDIES OF SKYRMIONIC TEXTURES IN HELIMAGNETIC PowerPoint Presentation

SLIDE 1

COMPUTER SIMULATION STUDIES OF SKYRMIONIC TEXTURES IN HELIMAGNETIC NANOSTRUCTURES

Marijan Beg*, Hans Fangohr

Faculty of Engineering and the Environment, University of Southampton, Southampton, United Kingdom

*email: mb4e10@soton.ac.uk

SLIDE 2

OVERVIEW

1. Initial states (analytic model)
2. Equlibrium states in a nano disk
3. Ground state phase diagram
4. Robustness
5. Hysteretic behaviour (DMI anisotropy)
6. Reversal mechanism
7. Summary

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SLIDE 3

SKYRMIONIC TEXTURES IN CONFINED HELIMAGNETIC NANOSTRUCTURES

3

SLIDE 4

MOTIVATION

Magnetic skyrmions possess interesting

properties promising for the development of future data-storage and information processing devices.

One of the main problems, obstructing the

development of skyrmion-based devices using helimagnetic materials, is their magnetic and thermal stability.

In infinitely large thin film or bulk B20

helimagnetic samples, skyrmion phase is stabilised in presence of an external field.

The motivation for this work is to explore

the skyrmionic textures in finite size B20 helimagnetic nanostructures.

4

b a

Yu et. al., Nature 465, 901-4 (2011)

Schematic Lorentz TEM Thin film phase diagram

SLIDE 5

SYSTEM UNDER STUDY

Sample geometry is 10 nm thin film

disk with varying diameter

Cubic B20 helimagnetic FeGe
MS = 3.84 x 105 Am-1
A = 8.78 x 10-12 Jm-1
D = 1.58 x 10-3 Jm-2.
Helical period 4πA/D = 70 nm
Finite elements mesh maximum

neighbouring node spacing smaller than 3 nm.

External field applied uniformly and

perpendicular to the film in +z direction.

zero temperature micromagnetic

model

5

Sample geometry and sample skyrmion ground state

Finite size effects, stability, hysteretic behaviour, and reversal mechanism of skyrmionic textures in nanostructures, Marijan Beg, Dmitri Chernyshenko, Marc- Antonio Bisotti, Weiwei Wang, Maximilian Albert, Robert L. Stamps, Hans Fangohr, arxiv:1312.7665 http://arxiv.org/abs/1312.7665 (2014)

SLIDE 6

MICROMAGNETIC MODEL

HAMILTONIAN AND DYNAMICS -
FINMAG
Finite elements based simulator.
successor of Nmag, http://nmag.soton.ac.uk
HAMILTONIAN:
No anisotropy (isotropic helimagnetic B20 material).
Full 3D model - no assumption about translational invariance of

magnetisation in out-of-film direction which radically changes the skyrmion energetics [Rybakov et al., PRB 87, 094424 (2013)].

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W = Z ⇥ A(rm)2 + Dm · (r ⇥ m) µ0m · H + wd ⇤ d3r

SLIDE 7

MAGNETISATION DYNAMICS

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∂m ∂t = γ∗m × Heff + αm × ∂m ∂t

Magnetisation dynamics is governed by the LLG EQUATION.

precession damping

Heff Heff Heff Heff Heff Heff

+ + = =

SLIDE 8

ENERGY LANDSCAPE

8

initial state

relaxed state

SLIDE 9

SIMULATION METHOD

d and H are varied in steps:

Gilbert damping  
System is relaxed from multiple initial

states by computing the magnetisation’s time development

The relaxed state with the lowest energy is chosen as the ground state for

the phase space point (d, H).

The scalar parameter Sa is computed as:
Phase diagram: Sa = f(d, H)

9

∆d = 2 nm

µ0∆H = 2 mT

α = 1 Sa = 1 4π

m ·

✓∂m ∂x × ∂m ∂x ◆

d3r

SLIDE 10

INITIAL CONFIGURATIONS

10

SLIDE 11

DEFINING SKYRMIONIC INITIAL STATES – ANALYTIC MODEL

The effective field due to

symmetric exchange and DMI (no external field, isotropic B20 material):

11

~ Heff = − 1 µ0Ms w ~ m ~ Heff = 2 µ0MS ⇥ Ar2 ~ m D(r ⇥ ~ m) ⇤

mr = 0 mθ = sin(kr) mz = − cos(kr)

b a

Yu et. al., Nature 465, 901-4 (2011)

Schematic Lorentz TEM

The chiral skyrmion profile is approximated in cylindrical coordinates:

1

1

x

y=0

mx my mz

SLIDE 12

ANALYTIC MODEL

ZERO TORQUE EQUATION -
In equilibrium state, the torque is zero:
Computing the zero radial torque at r=R for assumed chiral skyrmion

profile results in condition:

This equation has solution if:
Two scalar parameters are computed:

12

g(kR) ≡ − D kA sin2(kR) − sin(2kR) 2kR + 1 = 0

P = D kA > 2 3 ⇒ D > 2 3kA

m × Heff = 0

S = Z m · ✓∂m ∂x × ∂m ∂y ◆ dxdy

Sa = Z

m ·

✓∂m ∂x × ∂m ∂y ◆

dxdy

SLIDE 13

13

SOLUTION A

mr = 0 mθ = sin(kr) mz = − cos(kr)

3.5

kR (π) g(kR)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 P=2.0 A B C D E F 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

kR (π) S(kR), Sa(kR)

z 2R R H=0 A B C D E F A B C D E F Sa S

1

1

x

y=0

mz(x)

SLIDE 14

14

SOLUTION B

mr = 0 mθ = sin(kr) mz = − cos(kr)

3.5

kR (π) g(kR)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 P=2.0 A B C D E F 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

kR (π) S(kR), Sa(kR)

z 2R R H=0 A B C D E F A B C D E F Sa S

1

1

x

y=0

mz(x)

SLIDE 15

15

SOLUTION C

mr = 0 mθ = sin(kr) mz = − cos(kr)

3.5

kR (π) g(kR)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 P=2.0 A B C D E F 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

kR (π) S(kR), Sa(kR)

z 2R R H=0 A B C D E F A B C D E F Sa S

1

1

x

y=0

mz(x)

SLIDE 16

16

SOLUTION D

mr = 0 mθ = sin(kr) mz = − cos(kr)

3.5

kR (π) g(kR)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 P=2.0 A B C D E F 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

kR (π) S(kR), Sa(kR)

z 2R R H=0 A B C D E F A B C D E F Sa S

1

1

x

y=0

mz(x)

SLIDE 17

17

SOLUTION E

mr = 0 mθ = sin(kr) mz = − cos(kr)

3.5

kR (π) g(kR)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 P=2.0 A B C D E F 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

kR (π) S(kR), Sa(kR)

z 2R R H=0 A B C D E F A B C D E F Sa S

1

1

x

y=0

mz(x)

SLIDE 18

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SOLUTION F

mr = 0 mθ = sin(kr) mz = − cos(kr)

3.5

kR (π) g(kR)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 P=2.0 A B C D E F 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

kR (π) S(kR), Sa(kR)

z 2R R H=0 A B C D E F A B C D E F Sa S

1

1

x

y=0

mz(x)

SLIDE 19

19

ANALYTIC MODEL RESULTS

P = D kA

3.5

kR (π) g(kR)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.0
1.5
1.0
0.5

0.0 0.5 1.0 1.5 P=0 P=2.5 P=2.0 P=1.5 P=0.5

a

P=2/3 P=1.0 A B C D E F 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1.0
0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

kR (π) S(kR), Sa(kR)

z 2R R H=0 A B C D E F A B C D E F Sa S 1

1

mz(x)

y=0

c A

1

1

x

y=0

B C mz(x) D x E F b

Du, H., Ning, W., Tian, M., & Zhang,

Y. (2013), Physical Review B, 87, 014401.

Du, H., Ning, W., Tian, M., & Zhang,

Y. (2013). EPL, 101(3), 37001.

SLIDE 20

SIMULATION RESULTS

20

SLIDE 21

EQUILIBRIUM CONFIGURATIONS

21

SLIDE 22

GROUND STATE PHASE DIAGRAM

We select the state with the lowest energy
Two different ground states.

22

FeGe thin film disk phase diagram

SLIDE 23

23

INCOMPLETE SKYRMION (ISK)

23

No complete spin rotation.
|S| < 1
In literature also called “quasi-

ferromagnetic” or “vortex” state.

mz(x)

1

1

x

d/2

d/2

d=80 nm μ0H=0.2 T

SLIDE 24

24

ISOLATED SKYRMION (SK)

Complete spin rotation

present.

Significant tilt of magnetisation

at the edge which reduces |S|.

mz(x)

1

1

x

d/2

d/2

d=160 nm μ0H=0.3 T

SLIDE 25

25

ENERGIES OF METASTABLE STATES

40 60 80 100 120 140 160 180

d (nm)

0.0 0.2 1.2 0.4 1.0 0.6 0.8

μ0H (T)

z d 10 nm d/2 H

iSk Sk

Sk

2Sk 3Sk H2 H3 H

SLIDE 26

26

ROBUSTNESS

Skyrmionic textures able to adapt

their size to accommodate the size of a hosting nanostructure.

This provides the robustness of

technology built on skyrmions.

iSk and Sk have different core
rientation.
90
60
30

30 60 90

1.0
0.5

0.0 0.5 1.0

mz(x) x (nm)

d=180 nm d=160 nm d=140 nm

d/2

d/2 z 10 nm x d/2 H=0 d=80 nm d=100 nm d=120 nm

40 60 80 100 120 140 160 180 80 100 120 140 160 180 200 220

s=2π/k (nm)

x d/2

d/2

z s

cos(kx)

mz z d 10 nm d/2 H

d (nm)

Sk iSk

Du, H., Ning, W., Tian, M., & Zhang,

Y. (2013), Physical Review B, 87, 014401.

SLIDE 27

POSSIBLE STABILISING MECHANISM

27

no demagnetisation

Rybakov et al., PRB 87, 094424 (2013)

2D

SLIDE 28

POSSIBLE STABILISING MECHANISM

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SLIDE 29

29

HYSTERETIC BEHAVIOUR

0.5
0.4
0.3
0.2
0.1

0.0 0.1 0.2 0.3 0.4 0.5

1.0
0.5

0.0

μ0H (T)

0.5 1.0

mz

< <

z d=80 nm 10 nm d/2 H

a

with dipolar interactions no dipolar interactions iSk iSk iSk iSk

0.5
0.4
0.3
0.2
0.1

0.0 0.1 0.2 0.3 0.4 0.5

1.0
0.5

0.0

μ0H (T)

0.5 1.0

mz

< <

b

z d=150 nm 10 nm d/2 H with dipolar interactions no dipolar interactions Sk Sk Sk Sk

c

rientation

mz(x)

1

1

x

d/2

d/2

mz(x)

1

1

x

d/2

d/2

mz(x)

1

1

x

d/2

d/2

mz(x)

1

1

x

d/2

d/2

mz(x)

1

1

x

d/2

d/2

mz(x)

1

1

x

d/2

d/2
rientation
rientation
rientation

with dipolar interactions no dipolar interactions

incomplete Skyrmion (iSk) Skyrmion (Sk) mz(x)

1

1

x

d/2

d/2

mz(x)

1

1

x

d/2

d/2

iSk iSk iSk iSk Sk Sk Sk Sk

SLIDE 30

30

HYSTERETIC BEHAVIOUR (ISK)

Hysteretic behaviour remains in absence of magnetocrysralline anisotropy and dipolar-based shape anisotropy, suggesting the existence of Dzyaloshinskii-Moriya based shape anisotropy.

0.5
0.4
0.3
0.2
0.1

0.0 0.1 0.2 0.3 0.4 0.5

1.0
0.5

0.0

μ0H (T)

0.5 1.0

mz

< <

z d=80 nm 10 nm d/2 H

a

with dipolar interactions no dipolar interactions iSk iSk iSk iSk

SLIDE 31

31

HYSTERETIC BEHAVIOUR (SK)

Hysteretic behaviour remains in absence of magnetocrysralline anisotropy and dipolar-based shape anisotropy, suggesting the existence of Dzyaloshinskii-Moriya based shape anisotropy.

0.5
0.4
0.3
0.2
0.1

0.0 0.1 0.2 0.3 0.4 0.5

1.0
0.5

0.0

μ0H (T)

0.5 1.0

mz

< <

b

z d=150 nm 10 nm d/2 H with dipolar interactions no dipolar interactions Sk Sk Sk Sk

SLIDE 32

REVERSAL MECHANISM

32

Skyrmionic texture core

reverses via Bloch point

ccurrence and

propagation.

Reversal from core down to

core up.

External field reduced abruptly

from -210 mT to -260 mT.

Magnetisation dynamics

recorded for 1 ns. Liu, Y., Du, H., Jia, M., & Du, H. (2015), Physical Review B, 91, 094425.

SLIDE 33

REVERSAL MECHANISM

1. Core shrinks
2. Profile lowers
3. Core reverses
4. Size increases to adapt to the size of

nanostructure.

33

SLIDE 34

CORE REVERSAL MECHANISM

Bloch Point (BP):

1. enters the sample at the top boundary,
2. propagates to the bottom boundary, and
3. leaves the sample.

34

SLIDE 35

DIFFERENT BLOCH POINT PROPAGATION

One may ask whether different BP propagation direction is allowed.
This process is stochastic and depends on simulation parameters.
We choose 0.35 for Gilbert damping.
Bloch point structure changes.

35

SLIDE 36

In finite size B20 helimagnetic samples skyrmionic textures are ground state at zero field and in

absence of magnetocrystalline anisotropy.

Skyrmionic textures occur in a much wider range of external field values.
Demagnetisation energy and magnetisation variance in out-of-plane direction crucial for the stability.
Bistability: Two energetically equivalent configurations with different core orientation identified .
Writability: Skyrmionic textures undergo hysteretic behaviour when the core orientation is changed

using an external field.

Large hysteresis identified in absence of magnetocrystalline and dipolar-based shape anisotropies,

suggesting the existence of DMI-based shape anisotropy.

Skyrmionic textures reverse via Bloch-point occurrence and propagation.
Results fully scalable for helimagnetic materials with different helical period.
Presented stable skyrmionic textures could be used for hosting the information bit.
Paper: Beg et al. Scientific Reports 5, 17137 (2015)
Acknowledge the financial support from the EPSRC’s DTC grant EP/G03690X/1
Acknowledge the help of Marijan Beg, Weiwei Wang, David Cortes, Rebecca Carey, Maximilian Albert,

Dmitri Chernyshenko, Marc-Antonio Bisotti, Mark Vousden and Robert L. Stamps.

email: mb4e10@soton.ac.uk

36

SUMMARY

SLIDE 37

MICROMAGNETIC MODEL

37

w(m) = A(rm)2 + Dm · (r ⇥ m) µ0Msm · H + wd Heff(m) = − 1 µoMs δw(m) δm Heff(m) = 2A µ0Ms r2m 2D µ0Ms (r ⇥ m) + H + Hd ∂m ∂t = γ∗m × Heff + αm × ∂m ∂t + u(|m| − 1)V (m)

exchange DMI Zeeman demagnetisation precession damping norm correction exchange DMI Zeeman demagnetisation

Energy density
LLG equation
Effective field

SLIDE 38

DEMAGNETISATION

volume term surface term

Magnetic scalar potential
Effective (demagnetisation) field

φ(r0) = Ms 4π ✓

Z

Ω

r · m(r0) ||r r0|| dV + Z

∂Ω

m(r0) · n ||r r0|| dS ◆ Hd(r) = rφ(r)

Demagnetisation energy density

wd = −1 2Hd · m