Physik-DepartmentTechnische Universität München

Christian Pfleiderer

TOPFIT

Magnetic Textures II

plots from https://en.wikipedia.org/wiki/Liquid_crystal https://www.aps.org/about/physics-images/archive/umbilic.cfm

On the Characterisation of Condensed Phases

from

Wik

iped

ia

minimize total energy:reduction of dipolar stray field

Pierre-Ernest Weiss

Lev Landau

On the Nature of the Magnetisation Process

competition:K: anisotropyA: exchange

from Hubert & Schäfer, Springer

minimize total energy:reduction of dipolar stray field

width

energy density

On the Nature of the Magnetisation Process

from Hubert & Schäfer, Springer

On the Hierarchy of Magnetic Phenomena

Outline

Quantum Phase Transitions● Thermal vs quantum melting● Transverse field Ising transition

Fractionalisation in Spin Ice● From water ice to spin ice● Monopole condensation

Emergent Electrodynamics in Chiral Magnets● A new form of magnetic order● Consequences of topology● Towards Skyrmionics

Strong Global Anistropy:Quantum Phase Transition

Freon

T=Tc

T>Tc

T<Tc

Textbook Example of a Phase Diagram

Freon

classical thermal fluctuations

7

s = −kB

V

!

pσ

[npσ ln(npσ) + (1 − npσ) ln(1 − npσ)] (94)

s = −kB

V

!

pσ

[npσ ln(npσ) (95)

+(1 − npσ) ln(1 − npσ)] (96)

npσ =1

e(ϵ−µ)/kBT + 1(97)

hΓq ≪ kBT (98)

hΓq ≫ kBT (99)

d → deff = d + z (100)

deff ≥ 4 (101)

Tc → 0 (102)

Q

r(103)

Q

re−κr (104)

2D Ising model

Tc

T/Tc=1.15 T/Tc=1.06

T/Tc=1.01 T/Tc=1.00

T/Tc=0.99 T/Tc=0.97

Critical Slowing Down

for T→Tc: fluctuations slow down

correlation length correlation time

consider free energy

change of

7

s = −kB

V

!

pσ

[npσ ln(npσ) + (1 − npσ) ln(1 − npσ)] (94)

s = −kB

V

!

pσ

[npσ ln(npσ) (95)

+(1 − npσ) ln(1 − npσ)] (96)

npσ =1

e(ϵ−µ)/kBT + 1(97)

hΓq ≪ kBT (98)

hΓq ≫ kBT (99)

d → deff = d + z (100)

deff ≥ 4 (101)

Tc → 0 (102)

Q

r(103)

Q

re−κr (104)

7

s = −kB

V

!

pσ

[npσ ln(npσ) + (1 − npσ) ln(1 − npσ)] (94)

s = −kB

V

!

pσ

[npσ ln(npσ) (95)

+(1 − npσ) ln(1 − npσ)] (96)

npσ =1

e(ϵ−µ)/kBT + 1(97)

hΓq ≪ kBT (98)

hΓq ≫ kBT (99)

d → deff = d + z (100)

deff ≥ 4 (101)

Tc → 0 (102)

Q

r(103)

Q

re−κr (104)

for

quantum fluctuations

classical fluct.

for

7

s = −kB

V

!

pσ

[npσ ln(npσ) + (1 − npσ) ln(1 − npσ)] (94)

s = −kB

V

!

pσ

[npσ ln(npσ) (95)

+(1 − npσ) ln(1 − npσ)] (96)

npσ =1

e(ϵ−µ)/kBT + 1(97)

hΓq ≪ kBT (98)

hΓq ≫ kBT (99)

d → deff = d + z (100)

deff ≥ 4 (101)

Tc → 0 (102)

Q

r(103)

Q

re−κr (104)

reduction of many systems:

system in quantum-mechanical ground state

Thermal versus Quantum Melting

Thermal versus Quantum Meltingconsider free energy

change of

7

s = −kB

V

!

pσ

[npσ ln(npσ) + (1 − npσ) ln(1 − npσ)] (94)

s = −kB

V

!

pσ

[npσ ln(npσ) (95)

+(1 − npσ) ln(1 − npσ)] (96)

npσ =1

e(ϵ−µ)/kBT + 1(97)

hΓq ≪ kBT (98)

hΓq ≫ kBT (99)

d → deff = d + z (100)

deff ≥ 4 (101)

Tc → 0 (102)

Q

r(103)

Q

re−κr (104)

7

s = −kB

V

!

pσ

[npσ ln(npσ) + (1 − npσ) ln(1 − npσ)] (94)

s = −kB

V

!

pσ

[npσ ln(npσ) (95)

+(1 − npσ) ln(1 − npσ)] (96)

npσ =1

e(ϵ−µ)/kBT + 1(97)

hΓq ≪ kBT (98)

hΓq ≫ kBT (99)

d → deff = d + z (100)

deff ≥ 4 (101)

Tc → 0 (102)

Q

r(103)

Q

re−κr (104)

for

quantum fluctuations

classical fluct.

for

7

s = −kB

V

!

pσ

[npσ ln(npσ) + (1 − npσ) ln(1 − npσ)] (94)

s = −kB

V

!

pσ

[npσ ln(npσ) (95)

+(1 − npσ) ln(1 − npσ)] (96)

npσ =1

e(ϵ−µ)/kBT + 1(97)

hΓq ≪ kBT (98)

hΓq ≫ kBT (99)

d → deff = d + z (100)

deff ≥ 4 (101)

Tc → 0 (102)

Q

r(103)

Q

re−κr (104)

reduction of many systems:

system in quantum-mechanical ground state

Quantum Phase Transition:(-) Phase transition driven by quantum fluctuations.

(-) Phase transition between quantum phases.(-) Phase transition at zero temperature.

(-) Phase transition characterized by change of quantum entanglement.

Quantum Criticality in an Ising Chain

Quantum Phase Transition(weak residual interactions)

Coldea et al., Science 327 177 (2010)

Quantum Criticality in CoNb2O6

spin flip scattering

Quantum Criticality in CoNb2O6

Coldea et al., Science 327 177 (2010)

Quantum Criticality in CoNb2O6

Coldea et al., Science 327 177 (2010)

Quantum Criticality in CoNb2O6

Coldea et al., Science 327 177 (2010)

Coldea et al., Science 327 177 (2010)

Quantum Criticality in CoNb2O6

semi-base : height : apotherm E8 Lie group

Emergent E8 Symmetry

Affleck, Science (2010)

Quantum Phase Transition(coupling to nuclear spins)

Quantum Criticality in LiHoF4

LiHoF4

must include hyperfine coupling Bitko et al., PRL 77, 940 (1996)

LiHoF4

Quantum Criticality in LiHoF4

must include hyperfine couplingspectrum gappedRonnow, et al., Science 308, 389 (2005)

Quantum Criticality in LiHoF4

must include hyperfine coupling

Quantum Criticality in LiHoF4

easy axis

must include hyperfine coupling

Quantum Criticality in LiHoF4

hard axis

Unresolved Issues in LiHoF4

Ronnow et al., Science 308 389 (2005) Ronnow et al, PRB 75 054426 (2007)

no account of B=0 transitioncf Chakravarthy arXiv/0402051

B=Bc transitionno quantitative account of gap

From Global to Local Ising Anisotropy:Fractionalisation in Spin Ice

On the Analogy of Spin-Ice with Water-Ice

„2 in 2 out“

Linus PaulingNP: 1954 & 1963

http

://w

ww.

nobe

lpriz

e.or

g/

huge degeneracy:

Hun

klin

ger:

Fest

körp

erph

ysik

, Spr

inge

r

Pauling. J. Am. Chem. Soc., 57, 2680 (1935)Bernal, Fowler, J. Chem. Phys., 1, 515 (1933)

typically:res. entropy:

Entropy of Spin Icepyrochlore structure

Ramirez et al. Nature 399, 333 (1999)

Emergent Magnetic Monopoles in Spin Ice

2/2 3/1

2/2

2/2

2/2

1/3

2/2

2/2

1/3

2/2

2/2

Castelnovo, Sondhi, Moessner, Nature 451, 42 (2008)Ryzkhin, JETP 101, 481 (2004)

pyrochlore structure

Coulombic interaction

Ramirez et al. Nature 399, 333 (1999)

cf ice entropy for T ➞ 0

fractionalisation

Emergent Magnetic Monopoles in Spin Icepyrochlore structure

● quantised (magnetic) charge● 1/r (Coulomb) potential

Paul Dirac

cf. prediction of magnetic monopoles to explain quantized electric charge

Dirac PRS A133, 60 (1931)

Emergent Magnetic Monopoles in Spin Ice

pinch points:Dirac strings

Morris et al. Science 326, 411 (2009)

pyrochlore structure Dy2Ti2O7

cf neutron scatteringMorris et al. Science 326, 411 (2009)

Sakakibara et al. PRL, 90 207205 (2003)

Emergent Magnetic Monopoles in Spin IceDy2Ti2O7pyrochlore structure

cf neutron scatteringMorris et al. Science 326, 411 (2009)

Sakakibara et al. PRL, 90 207205 (2003)

Emergent Magnetic Monopoles in Spin IceDy2Ti2O7

Morris et al. Science 326, 411 (2009)

pyrochlore structure Ho2Ti2O7

Krey et al. PRL, 108 257204 (2012)

cf neutron scatteringSakakibara et al. PRL, 90 207205 (2003)

Emergent Magnetic Monopoles in Spin Ice

Sakakibara et al. PRL, 90 207205 (2003)

Dy2Ti2O7

Ho2Ti2O7Dy2Ti2O7 Tb2Ti2O7

Role of hyperfine cplg.?

Prediction of quantum spin iceMolavin, Gingras J PCM, 21, 172201 (2009)

Crystal Electric Field Levels

Glassyness & Metamagnetism in Spin IceDy2Ti2O7

Sakakibara et al. PRL, 90 207205 (2003)

Ho2Ti2O7

Krey et al. PRL, 108 257204 (2012)

Glassyness & Metamagnetism in Spin IceHo2Ti2O7Dy2Ti2O7 Tb2Ti2O7

No QSI(quantum spin ice)

Krey et al. PRL, 108 257204 (2012) Legl et al. PRL, 109, 047201 (2012)Sakakibara et al. PRL, 90 207205 (2003)

Towards Very Weak Magnetic Anisotropies:Emergent Electrodynamics of Chiral Magnets

competition:K: anisotropyA: exchange

from Hubert & Schäfer, Springer

minimize total energy:reduction of dipolar stray field

width

energy density

What happens for negative domain wall energy?

Type 2 Superconductivity Revisited

A. Abrikosov

superconducting rigidityvs field penetration depth

http://www.mn.uio.no/fysikk/english/research/groups/amks/superconductivity/sv/index.html

Fluxlines in Type 2 Superconductors

Magneto-Optical Imaging in NbSe2 Neutron Scattering in Nb

Mühlbauer et al., PRL 102 136408 (2009)

Magnetic Order akin the „Shubnikov-Phase“?

FM AFM

spin spirals

Blochwall

Neelwall

anisotropy versus exchangeadd chiral „twisting“

Bogdanov, Yablonskii, JETP 68 101 (1989)

Text Book Forms of Magnetic Order Text Book Forms of Domainsfrom Huber & Schäfer, Springer

B20: no inversion center

TMSi,Ge

left-handed right-handed

B20: no inversion center

TMSi,Ge

Hierarchical Energy Scales in B20 CompoundsLandau-Lifshitz vol. 8, §52

(2) Dzyaloshinsky-Moriya (3) crystal field (P213): locked to <111> or <100>

(1) ferromagnetism

l

TN (K) l (Å)

180 to 120> 300700

30 to 60Mn1-xFexSiFe1-xCoxSiFeGe

MnGe< 28< 45280

170

Cu2OSeO3 54 620

Skyrmion Lattice in Chiral Magnets

TEM Imaging in FeGe Neutron Scattering in MnSi

B=0 ➞ 300mT, 6mT/s

TEM data by Xiuzhen Yu (RIKEN)

Mühlbauer, et al. Science 323, 915 (2009)Yu et al., Nature Materials 10, 106 (2010)

=

Magnetic Phase Diagram of MnSi

Bauer, Garst, CP, PRL 110, 177207 (2013)Mühlbauer et al, Science 323, 915 (2009)

B

field-polarized conical

skyrmion lattice

helical

paramagnet

Monte-Carlo simulation(includes thermal fluctuations)

triple-q + uniform MBinz, Vishwanath, Aji PRL (2006)

Mühlbauer, et al. Science 323, 915 (2009) Buhrandt, Fritz PRB 88, 195137 (2013)

Fluctuation-Stabilized Multi-q Structure

Remarks on Topology & Skyrmions

from

Wik

iped

ia

Topological Winding in One- & Two Dimensions

skyrmion

n=+1

(trivial) vortex

n=0

plots by M. Rahn, K. Everschor

Werner Heisenberg (1901-1976)

Towards a Unified Field Theory

Can bosons exist as non-linear excitations of fermion fields?

RMP 29 296 (1957)

Derrick-Hobart theorem:Localized states (solitons) in nonlinear field models in 2D and 3D are in general unstable and collapse spontaneously into topological singularities.

R. H. Hobart, Proc. Phys. Soc. 82 201 (1963); G. H. Derrick, J. Math. Phys. 5, 1252 (1964)

Towards a Unified Field Theory

Can fermions arise as non-linear excitations of boson fields?

Tony Skyrme (1922-1987)

Nuclear Physics 31 556 (1962) Proc. Royal Society London, Series A 260, 130 (1961)Proc. Royal Society London, Series A 262, 237 (1961) (cf. Fadeev-Skyrme model & Hopfions)

Towards a Unified Field Theory

„Discovery“ of the Skyrme Model

0

220

cita

tions

per

yea

r

20161962 1983

PRSL - Ser. A 260, 130 (1961)PRSL - Ser. A 262, 237 (1961)Nuclear Physics 31 556 (1962)

ISI: >3200 citations(170 before ’83)

Ed WittenNucl. Phys. B 228 552 (1983)

Towards a Unified Field Theory

Contents 1994 / 2010Hadrons and Nuclear MatterString TheoryCondensed Matter

Quantum Hall FerromagnetsQuantum Phase Transitions

Phys. Rev. B 47, 16419 (1993)

Skyrmions in Quantum Hall Systems

triangular lattice

Skyrmions in Different Areas

Contents 1994 / 2010Hadrons and Nuclear MatterString TheoryCondensed Matter

Quantum Hall FerromagnetsQuantum Phase Transitions

Bogdanov & Yablonskii JETP 68 101 (1989); Bogdanov & Hubert JMMM 138, 255 (1994)

Dzyaloshinsky-Moriya:

animation S. Maekawa

What about Spin Transfer Torques in Helimagnets?A. Rosch, R. Duine et al. (November 2006)

typical current density1012 A/m2

cf. Goto, et al. condmat/0807.2907; Wessely et al., PRL 96, 256601 (2006); both consider j ≈ 1012 Am-2

Bloch wallhelimagnetism

tem

pera

ture

B

curr

ent

Jonietz et al, Science 330, 1648 (2010)

Antisymmetric Rotation of the Magnetic Diffraction Pattern

Mühlbauer et al, Science 323, 915 (2009)

expe

rimen

ts @

MIR

A, F

RM

II

From Topological Winding to Berry‘s Phase

Michael Berry

cf. Pancharatnam, PIAS, A 44 247 (1956)Berry, PRSL A392 45 (1984)

in magnetic field: Aharonov-Bohm Phase

slowly changing quantum system(remains in ground state)

wave function changes phase:

Tracking Topology by Berry‘s Phase

Berry Phase Contributions to the Hall Effect

electron-like

hole-like

B

ρxy

anomalous HEnormal HE

Berry phase (real space)

cf Nagaosa et al.,RMP 82, 1539 (2010)

ρxy = R0B

cf Ritz, et al. PRB 87, 134424 (2013); Freimuth et al. PRB 88, 214409 (2013)

topological HE

exptl. unexplored: mixed Berry phases

decreases forT→0 increases for T→0

dkx

dky

Berry phase (momentum space)

Neubauer, et al. PRL 102 186602 (2009)

Pfleiderer, Rosch Nature (N&V) 465 880 (2010)

Emergent Magnetic Field of Skyrmions

after subtracting anomalous and normal Hall contributions

Binz, Vishwanath Physica B 403 1336 (2008)

collect Berry phaseconduction electron tracks spin structure:

trivial topology:

1

Φ = 0 (1)

∆C

T∝ − ln(T ) (2)

χ ∝ χ0 − χ1T3/4 (3)

ρ − ρ0 ∝ T 5/3 (4)

ωL =2µB

!(5)

mexp ≈ 0.012µB (6)

mcalc ≈ 0.015(5)µB (7)

∆a

a= 4.3 · 10−4 (8)

∆c

c= 2.1 · 10−4 (9)

∆η

η

!

!

!

c= 5 · 10−4 (10)

f"∆η

η

#

FWHM= 3.8 · 10−4 (11)

t (12)

B/B0 (13)

$

" U

JQ2

#3B (14)

|B| = 150 mT (15)

non-trivial topology:

1

Beff ≈ −2.5 T (1)

Φ = 0 (2)

∆C

T∝ − ln(T ) (3)

χ ∝ χ0 − χ1T3/4 (4)

ρ − ρ0 ∝ T 5/3 (5)

ωL =2µB

!(6)

mexp ≈ 0.012µB (7)

mcalc ≈ 0.015(5)µB (8)

∆a

a= 4.3 · 10−4 (9)

∆c

c= 2.1 · 10−4 (10)

∆η

η

!

!

!

c= 5 · 10−4 (11)

f"∆η

η

#

FWHM= 3.8 · 10−4 (12)

t (13)

B/B0 (14)

$

" U

JQ2

#3B (15)

express as Aharonov-Bohm phase represents effective field

-13

(1) details of FS(2) carrier lifetimes(3) adiabatic approximation

beware:

Franz, et al. PRL 112 186601 (2014)

Origin of Ultralow Current Densities in STT

very efficient gyro-coupling via Berry phase (entire domain)very weak pinning forces+ low defect concentration (RRR>100)+ very smooth magnetic texture (200Å)+ very stiff magnetic order (cf. collective pinning in SC)

Rotating Skyrmion Lattices

restoring force: higher-order spin-orbit cplg.

Everschor et al., PRB 86 054432 (2012)see also Yu et al. Nature Comm. 3 988 (2012) Mochizuki et al., Nat. Mat. 13 241 (2014)

Neutron Scattering in MnSi Lorentz-TEM in Cu2OSeO3

animation A. Rosch

How about Faraday’s Law of Induction?

current density: 106 A/m2 !!!

Emergent Electric Field due to Skyrmion Motion

Schulz et al. Nature Physics 8 301 (2012)

Emergent Electric Field due to Skyrmion Motion

Schulz et al. Nature Physics 8 301 (2012) see also Iwasaki et al. Nature Comm. 4 1463 (2013)

...

http://4.bp.blogspot.com/-23eJwg9ESF0/VI3feW6ysGI/AAAAAAAAAQY/5a5zU4nOMyw/s1600/cartoons+animals.png

Zoology of Magnetic-Skyrmion Materials

P213 insulator: Cu2OSeO3

B20 metals & semiconductor

P4132 & P4332: CoxZnyMnz

SrFeO3, EuO, Spinels

Sc-doped Ba-FerriteHeusler compoundsFe or PdFe-layer on Ir (111)Heterostructures....

Magnetic Phase Diagram of B20 Compounds

P213 insulator: Cu2OSeO3

Seki al., Science 336 198 (2012)Adams et al., PRL, 108 237204 (2012)

Tokura@KITP

Fe1-xCoxSix=20%

Fe1-xCoxSi x=20%

B20 semiconductor: Fe1-xCoxSi

Münzer, et al. PRB(R) 81 041203 (2010)Yu et al., Nature 465, 901 (2010)

P4132 & P4332: CoxZnyMnz

Tokunaga et al., arXiv/1503.05651

Ishi

wat

a et

al.,

PR

B 84

054

427

(201

1)

SrFeO3

Yu e

t al.,

PN

AS 1

09 8

856

(201

2)Sc-doped Ba-Ferrite

GaV4S8

Bloch-type Neel-typeKezsmarki et al., Nat. Mater 14 1116 (2015) helicity-reversal

Further Classes of Bulk Materials

Monopole-Antimonopole Lattice in MnGe

Kanazawa, et al. PRL (2011)Kanazawa et al., Nat. Comm. (2016)

Dzyaloshinsky-Moriya interaction

1

Exc = SA · J · SB (1)

= SA ·

⎛

⎝

J Dy −Dx

−Dy J Dx

Dz −Dx J

⎞

⎠ · SB (2)

= J SA · SB + D · SA × SB (3)

(4)

EDM = (SA · s)(l · s)(s · SB) (5)

= D · (SA × SB) (6)

EDM = D · (SA × SB) (7)

L(k)ij = (mi∂kmj − mj∂kmi) (8)

fDM = D(L(x)xz − L(y)

yz ) (9)

fDM = D m · (∇× m) (10)

Cp = T∂S

∂T

∣

∣

∣

p(11)

Cp = T∂S

∂T

∣

∣

∣

p(12)

α =1

V

dV

dT(13)

λ =π!

m∗αR(Ez)(14)

H =p2∥

2m∗+

αR

!σ(p × E) (15)

=1

2m2c2

1

r

dV (r)

drS · L (16)

=1

2

gsµB

!S ·B (17)

∆U = −µs · B (18)

Smith, J. Mag. Mag. Mater. 1, 214 (1976)

1

Exc = SA · J · SB (1)

= SA ·

⎛

⎝

J Dy −Dx

−Dy J Dx

Dz −Dx J

⎞

⎠ · SB (2)

= J SA · SB + D · SA × SB (3)

(4)

Easym = (SA · s)(l · s)(s · SB) (5)

→ D · (SA × SB) (6)

EDM = D · (SA × SB) (7)

L(k)ij = (mi∂kmj − mj∂kmi) (8)

fDM = D(L(x)xz − L(y)

yz ) (9)

fDM = D m · (∇× m) (10)

Cp = T∂S

∂T

∣

∣

∣

p(11)

Cp = T∂S

∂T

∣

∣

∣

p(12)

α =1

V

dV

dT(13)

λ =π!

m∗αR(Ez)(14)

H =p2∥

2m∗+

αR

!σ(p × E) (15)

=1

2m2c2

1

r

dV (r)

drS · L (16)

=1

2

gsµB

!S ·B (17)

∆U = −µs · B (18)

PdFe-layer on Ir (111)

Romming et al. Science 341, 636 (2013)CP, Physik Journal 12, 20 (2013)

(stochastic) reading & writing

Surface-Driven DM-Interactions & Skyrmions

cf. Bode et al. Nature 447,190 (2007)Heinze et al.; Nature Physics 7, 713 (2011)

Skyrmions in Fe-monolayer on Ir (111)

Moreau-Luchaire et al., arXiv/1502.07853Nature Nano doi:10.1038/nnano.2015.313 Jiang et al. Science 349, 283 (2015)

Towards Skyrmionics

Skyrmion „Bubbles“ in TrilayerTa(5nm)/Co20Fe60B20(CoFeB)(1.1nm)/TaOx(3 nm)

Skyrmion & DM EngineeringIr/Co/Pt multilayer

Controlled Creation and Manipulation of Skyrmions

Hoffmann et al., Argonne & UCLA, Science 2015

j j

Towards Skyrmion Logic Gates

Zhang, Ezawa, Zhou arXiv/1410.3086

skyrmion bubbles (µm size) @ room temperature magneto-optical Kerr imaging (MOKE)

Epilogue

http://www.mairdumont.com/de/dumont.html

(You only ‚„appreciate“ what you know…)

Outline Revisited

Quantum Phase Transitions● Thermal vs quantum melting● Transverse field Ising transition

Fractionalisation in Spin Ice● From water ice to spin ice● Monopole condensation

Emergent Electrodynamics in Chiral Magnets● A new form of magnetic order● Consequences of topology● Towards Skyrmionics

From Reductionism to Emergence

complex many-body systems

emergent properties● classical order● rigidity & symmetry breaking● elementary excitations● quantum order● Higgs mechanism● magnetic monopoles● Majorana fermions● string theory●

From Reductionism to Emergence

R. B. LaughlinP. W. AndersonS. Kauffman

J. HuxleyAristoteles J. S. Mill

Science 177, 393 (1972)

Phil AndersonNP 1977