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May 05, 2023 627 likes •1.09k views

Topology in Magnetism a phenomenological account Wednesday: vortices Friday: skyrmions Henrik Moodysson Rnnow Laboratory for Quantum Magnetism (LQM), Institute of Physics, EPFL Switzerland Thanks to Jiadong Zang and Shinichiro Seki for

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Ronnow – ESM Cargese 2017 Slide 1

Topology in Magnetism – a phenomenological account Wednesday: vortices Friday: skyrmions

Henrik Moodysson Rønnow

Laboratory for Quantum Magnetism (LQM), Institute of Physics, EPFL Switzerland

Thanks to Jiadong Zang and Shinichiro Seki for slides, many figures copied from internet

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Ronnow – ESM Cargese 2017 Slide 2

Skyrmions in Magnetism

  • Skyrmions

– Topological solitons – 3-Q magnetic structure – Models

  • Skyrmion measurements

– SANS, LTEM, STXM, MFM, SPSTM

  • Skyrmion materials

– Bulk materials: Chiral, Polar, Frustrated – Interface systems

  • Skyrmion fundamentals

– Skyrmion types, Lattice effects, dynamics, …

  • Skyrmion control
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SLIDE 3

Ronnow – ESM Cargese 2017 Slide 3

The hairy ball theorem

  • "you can't comb a hairy ball flat

without creating a cowlick“

  • Topology concern non-local properties !
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Ronnow – ESM Cargese 2017 Slide 4

Stereographic projection

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Ronnow – ESM Cargese 2017 Slide 5

Topological charge

Q=0 Q=1 Q=2

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Ronnow – ESM Cargese 2017 Slide 6

Magnetic order - Against all odds

  • Bohr – van Leeuwen theorem:

(cf Kenzelmann yesterday)

– No FM from classical electrons

  • <M>=0 in equilibrium

(cf Canals yesterday)

  • Mermin – Wagner theorem:

– No order at T>0 from continuous symmetry in D2

  • No order even at T=0 in 1D
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SLIDE 7

Ronnow – ESM Cargese 2017 Slide 7

Derrick’s Scaling Argument: No stable local texture

𝐹[𝐧] = න 𝛼𝐧 2 + 𝑔 𝐧 𝑒3𝑠 ≡ 𝐽1 + 𝐽2 𝐧𝟏(𝐬) Assume existence of stable Local Texture Scale size of texture 𝐹[𝐧0(𝜇𝐬)] is minimized at 𝐹[𝐧0(𝜇𝐬)] = ඲ 1 𝜇 ෨ 𝛼 𝐧

2

+ 1 𝜇3 𝑔 𝐧 𝑒3 ǁ 𝑠 𝜇 = 1 Τ = 𝐽1 𝜇 + Τ 𝐽2 𝜇3 ǁ 𝑠 = 𝜇𝑠 𝑒𝐹 𝑒𝜇 ቚ

𝜇=1 = 0

𝑒2𝐹 𝑒𝜇2 ቚ

𝜇=1 > 0

𝐽1 = ∫ 𝛼𝐧 2 < 0 Ways out:

Dzyaloshinskii-Moriya Interaction Finite Size

7 𝐧𝟏(𝜇𝐬)

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SLIDE 8

Ronnow – ESM Cargese 2017 Slide 8

“spin vortices” as local solitonic solution to continuum model Skyrmion lattice of individual skyrmions  3-Q magnetic structure

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Ronnow – ESM Cargese 2017 Slide 9

Physical Consequence: Dzyaloshinskii-Moriya Interaction (DMI) 9

 

  

ij j i

J H S S ) (

j i ij

S S D  

Broken Inversion Symmetry

Inversion

+

θ

θ ~ D/2J

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SLIDE 10

Ronnow – ESM Cargese 2017 Slide 10

Model

  • Microscopic
  • Coarse grained

simple cube

  • Continuum version

 

     

ij j i ij j i

J H ) ( S S D S S

 

     

ij j i ij j i

J H ) ( S S D S S

Sum over all bonds Sum over neighboring unit cells

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Ronnow – ESM Cargese 2017 Slide 11

Dzyaloshinskii-Moriya helices

H = -Ʃ JijSi·Sj + Dij·(Si×Sj) J favors parallel spins J>0 Ferromagnet J<0 Antiferromagnet D favor perpendicular spins J & D: twist spins by angle tanƟ = D/J Helix with period Q = 2a J/D

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Ronnow – ESM Cargese 2017 Slide 13

3Q structure

  • Superpose 3 helices:

Looks like “spin vortices”

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Ronnow – ESM Cargese 2017 Slide 14

Helical, conical and “A-phase”

1993

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Ronnow – ESM Cargese 2017 Slide 18

“for the theoretical prediction, the experimental discovery and the theoretical analysis of a magnetic skyrmion phase in MnSi, a new state of matter.”

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Ronnow – ESM Cargese 2017 Slide 19

Many networks developing:

Interdisciplinary network with 12 project partners from EPFL, University of Basel and Paul Scherrer Institut (PSI) Funded by SNSF via grant CRSII5_171003

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Ronnow – ESM Cargese 2017 Slide 20

Skyrmions in Magnetism

  • Skyrmions

– Topological solitons – 3-Q magnetic structure – Models

  • Skyrmion measurements

– SANS, LTEM, STXM, MFM, SPSTM

  • Skyrmion materials

– Bulk materials: Chiral, Polar, Frustrated – Interface systems

  • Skyrmion fundamentals

– Skyrmion types, Lattice effects, dynamics, …

  • Skyrmion control
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SLIDE 17

Ronnow – ESM Cargese 2017 Slide 21

Skyrmion measurements

  • SANS Small Angle

Neutron Scattering

SASXRS Small angle Soft X-ray Resonant Scattering

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Ronnow – ESM Cargese 2017 Slide 22

Skyrmion measurements

  • Magnetic contrast transmission electron microscopy (LTEM)
  • Sensitive to in-plane magnetization components
  • TIE Transfer of intensity: recover phase
  • Electron holography: towards 3D imaging of magnetic textures
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SLIDE 19

Ronnow – ESM Cargese 2017 Slide 23

Skyrmion measurements

  • Scanning Tunneling X-ray Microscopy
  • Magnetic Force Microscopy
  • Spin Polarized STM
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Ronnow – ESM Cargese 2017 Slide 24

Skyrmion measurements Transport effects

Hall effect  B Anomalous Hall  M Topological Hall  Q

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Ronnow – ESM Cargese 2017 Slide 25

Skyrmion spectroscopy

  • Seki et al.
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Ronnow – ESM Cargese 2017 Slide 26

Skyrmion hosts

  • Interface or “ABCABC”

multilayer of FM and high-Z compounds

  • Albert Fert
  • Stuart S Parkin
  • Manny others…
  • Chiral and Polar bulk materials
  • Pfleiderer
  • Tokura, Seki
  • Keszmarki
  • Many others…
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Ronnow – ESM Cargese 2017 Slide 27

Skyrmions in fabricated interfaces/multilayers

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Ronnow – ESM Cargese 2017 Slide 28

  • In plate-geometry bubbles are stabilized

by dipole fields

  • Hot topic in 1980’s
  • Reached commercial products,

but not competitive

Skyrmions and magnetic bubbles

Intel 1Mbit bubble memory

MnSi

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Ronnow – ESM Cargese 2017 Slide 29

Current Driven Skyrmions (movies Eric Fullerton)

50 µm

  • 1.2 Oe
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Ronnow – ESM Cargese 2017 Slide 30

Moving Skyrmions

Racetrack Memory Logic Gates

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Ronnow – ESM Cargese 2017 Slide 31

  • Skyrmions move in small currents
  • Race-track memory…
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Ronnow – ESM Cargese 2017 Slide 32

Skyrmion materials: Bulk materials

Material SG Ordering Temp Helimag. Period Transport property Skyrmion motion SkL Dimensi

  • n

References MnSi P213 30 K 18 nm Metallic jc~106A.m-2 DT 2D

  • S. Mühlbauer et al., Science 323, 915 (2009)
  • F. Jonietz et al., Science 330, 1648 (2010)
  • M. Mochizuki et al., Nat. Mater. 13, 241

(2014)

FeGe P213 280 K 70 nm Metallic jc<106A.m-2 2D

X.Z. Yu et al., Nat. Mater. 10, 106 (2010) X.Z. Yu et al., Nat. Comm. 3, 988 (2012)

Fe1-xCoxSi P213 11 – 36 K 40-230 nm Metal / semi- conductor 2D

  • W. Münzer et al., PRB 81, 041203(R) (2010)

X.Z. Yu et al., Nature 465, 901 (2010)

Mn1-xFexSi P213 7-16.5 K 10-12 nm Metallic 2D

S.V. Grigoriev et al., PRB 79, 144417 (2009)

Mn1-xFexGe P213 150-220 K 5 - 220 nm Metallic 2D

  • K. Shibata et al., Nature Nano. 8, 723 (2013)

CoxZnyMnz P4132 140-480K 110-190nm Metallic 2D

  • Y. Tokunaga et al., Nat. Com. 6, 7638 (2015)

GaV4S8 C3v 13 K 22nm Semi- conductor 2D anisotrop Cu2OSeO3 P213 58 K 50 nm Insulating Magneto- electric DT E<105 V/m 2D

  • S. Seki et al., Science 336, 198 (2012)
  • T. Adams et al., PRL 108, 237204 (2012)
  • M. Mochizuki et al., Nat Mat 13, 241 (2014)

MnGe P213 170 K 3 nm Metallic 3D?

  • N. Kanazawa et al., PRL 106, 156603 (2011)
  • N. Kanazawa et al., PRB 86, 134425 (2012)
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Ronnow – ESM Cargese 2017 Slide 33

  • Large sample  100000 skyrmions resolved (Cu2OSeO3)
  • Allows quantitative analyses, such as delauney triangulation

Bulk systems have more stable skyrmions

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Ronnow – ESM Cargese 2017 Slide 34

Defects and angles

  • Defects classifiable – eg a 5-7 or a 5-8-5 defect

– “loss” of row along 2 directions

Map SkL angle:

  • r peak in “local Fourrier”

– Defects creates far-stretching rotations – Model system for understanding lattice defects

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Ronnow – ESM Cargese 2017 Slide 35

Skyrmions as arena for real-space imaging of phase transitions

c b a

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Ronnow – ESM Cargese 2017 Slide 36

The 3rd dimension – how protected?

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Ronnow – ESM Cargese 2017 Slide 37

White Karube Tokunaga

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Ronnow – ESM Cargese 2017 Slide 38

Metastability could come from topological protection ?

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Ronnow – ESM Cargese 2017 Slide 39

Square lattice ?

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Ronnow – ESM Cargese 2017 Slide 40

Topological protection + D/J(T) => long skyrmions

  • Consequence:

– Relationship helical domains / elongated skyrmions – Edges of helical domains carry half-skyrmions = merons – Crossing phase transition can pump skyrmions

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Ronnow – ESM Cargese 2017 Slide 41

Skyrmion hosting insulator Cu2OSeO3

  • S. Seki et al., Science 336, 198 (2012)
  • S. Seki et al., Phys. Rev. B 86, 060403(R) (2012)

‘Generic’ magnetic phase diagram + SkL phase

Crystal structure, P213, no inversion symmetry Cubic unit cell contain 16 Cu2+ S=1/2 4 tetrahedral forming “3-up-1-down” S=1 Combined to single S=4 in skyrmion simulations

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Ronnow – ESM Cargese 2017 Slide 42

Can create skyrmions with electric field

  • A. Kruchkov, arxiv 1702.08863 & 1703.06081, to be submitted soon…
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Ronnow – ESM Cargese 2017 Slide 43

pre-treatments

  • aligning
  • Area

selection

  • filtering

43

Identification of skyrmions in image data – easy if complete skyrmion phase

dynamical box algorithm

  • scanning by a box
  • finding local minima
  • overlapping minima with

centers manual revisions

  • adding and/or

removing

  • Delaunay triangulation
  • interactive programs

raw data LoG filtered identification triangulation

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Ronnow – ESM Cargese 2017 Slide 44

Counting skyrmions in mixed phase

Previous algorithm gets confused Use orientational map Inspired by finger-print algorithms Rau & Schunck 1989

𝑊

𝑦 𝑣, 𝑤 =

෍

𝑗=𝑣−𝑥 2 𝑣+𝑥 2

෍

𝑘=𝑤−𝑥 2 𝑤+𝑥 2

2𝜖𝑦 𝑗, 𝑘 𝜖𝑧 𝑗, 𝑘 𝑊

𝑧 𝑣, 𝑤 =

෍

𝑗=𝑣−𝑥 2 𝑣+𝑥 2

෍

𝑘=𝑤−𝑥 2 𝑤+𝑥 2

𝜖𝑦

2 𝑗, 𝑘 𝜖𝑧 2 𝑗, 𝑘

𝜄 𝑣, 𝑤 = 1 2 tan−1 𝑊

𝑧 𝑣, 𝑤

𝑊

𝑦 𝑣, 𝑤

Inspect frame by hand (worst case) Skyrmion counts: Hand inspection 90 Algorithm: 132 Missed: 37 Extra: 79 So we count skyrmions with an offset

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Ronnow – ESM Cargese 2017 Slide 45

Reproducible writing and erasing

Sample configuration

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Ronnow – ESM Cargese 2017 Slide 46

D> D<

Nano-structured bulk materials: Ultra-narrow MnSi Nanowires

46

B

  • Wire width (40 nm) comparable to

skyrmion diameter (18 nm)

  • High quality MnSi nanowires
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Ronnow – ESM Cargese 2017 Slide 47

Magnetoresistance

47

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Ronnow – ESM Cargese 2017 Slide 48

Cascading Transitions of Skyrmion Cluster

48

F: Ferromagnet C: Conical S: Skyrmion H: Helical

  • H. Du, JZ, M. Tian, S. Jin et. al, Nature Commun. (2015)
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Ronnow – ESM Cargese 2017 Slide 49

Skyrmions in Magnetism

  • Skyrmions

– Topological solitons – 3-Q magnetic structure – Models

  • Skyrmion measurements

– SANS, LTEM, STXM, MFM, SPSTM

  • Skyrmion materials

– Bulk materials: Chiral, Polar, Frustrated – Interface systems

  • Skyrmion fundamentals

– Skyrmion types, Lattice effects, dynamics, …

  • Skyrmion control