A. de Saint-Exupéry (1943)

Christian Pfleiderer (TU Munich)

Emergent Electrodynamics of Skyrmions in Chiral Magnets

KITP Fragnets12 Conf 11 Oct 2012

A. de Saint-Exupéry (1943)

0

20

40

60

0 5 10 15 20

T (K

)

FM1

FM2

TC

Tx

Ts

* 10

p (kbar)

UGe2

0

2

4

6

8

10

0 10 20 30

T (K

)

p (kbar)

2Ts

TN

CePd2Si

2

AFM

BaCuSi2O6

C. Pfleiderer RMP 81, 1551 (2009)

0

10

20

30

40

1975 1980 1985 1990 1995 2000 2005 2010

tota

l num

ber

year

number of f-electron SC

year

num

ber

0

10

20

30

40T

c (ρ, χ)

T0 (ENS)

Tc,L

(Larmor)T

TE (Larmor)

0 5 10 15 20 25

T (K

)

NFLFL

p (kbar)

Science 323, 915 (2009)PRL 102, 186602 (2009)Science 330, 1648 (2010)Nature Physics 8 301 (2012)

0

10

20

30

40T

c (ρ, χ)

T0 (ENS)

Tc,L

(Larmor)T

TE (Larmor)

0 5 10 15 20 25

T (K

)

NFLFL

p (kbar)

Nature 414, 427 (2001)

PRB 55, 8330 (1997)Nature 427, 227 (2004)

Nature Physics 3, 29 (2007)

Science 316, 1871 (2007)

PRL 99, 156406 (2007)

Why we find MnSi interesting...Nature 442, 797 (2006)

p>pc

p=0

PRB 81, 214436 (2010)

MnSi = Manganese Silicon

Manganese SuicideXManganese SilicideX

Physik-Department E21 - Technische Universität München

Christian Pfleiderer

Emergent Electrodynamics of Skyrmions in Chiral Magnets

CollaborationsMünchen

K. EverschorM. GarstB. BinzA. Rosch

R. RitzT. SchulzM. HalderC. FranzM. WagnerA. ChaconM. Hirschberger1

F. Birkelbach

S. Mühlbauer F. JonietzT. Adams

R. GeorgiiT. Keller (MPI)W. HäußlerP. LinkB. PedersenP. BöniR. Hackl (WMI)

M. Janoschek2 F. BernlochnerJ. KindervaterM. TischendorfG. BrandlS. Dunsiger

A. BauerA. NeubauerW. MünzerS. GottliebP. Niklowitz3

UtrechtR. Duine

KölnJ. KoralekD. MaierJ. OrensteinA. Vishwanath

Berkeley

2Los Alamos1Princeton

TOPFIT3London

LausanneH. Berger

BraunschweigP. Lemmens

animation S. Maekawa

Challenges in Spintronics

typical current density1012 A/m2

Berry-phasespin torque

precession ineffective field

damping

example for a Landau-Lifshitz Gilbert equation

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 wall helimagnetism

animation S. Maekawa

animation A. Rosch

typical current density1012 A/m2

current density: 106 A/m2 !!!

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

emergent electrodynamics:

Outline● A Very Short Introduction to B20 Compounds● Exploring Spin Torques with Neutrons● Rôle of Emergent Magnetic Field● Emergent Electric Field● Perspectives & Summary

A Very Short Introduction to B20 Compounds

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

TN (K) (Å)

180 to 120> 300700

30 to 60Mn1-xFexSiFe1-xCoxSiFeGe

MnGe< 28< 45280

170

Cu2OSeO3 54 620

Fluctuation-Induced First Order Transitiona helimagnetic Brazovskii transition

S. Grigoriev et al., PRB 81, 144413 (2010)

cf. claim of a Bak-Jensen1st order transitionM. Janoschek, M. Garst et al.

arXiv/1205.4780

FM exchange J a = 11meV

DM-interactionD a2 = J Q a2 = 1.7meV

cubic anisotropy J1 a = 0.37meV

(a: lattice constant)

experiments @ MIRA, FRM II

Helimagnon-Bands as Universal Excitations

one parameter (intensity) simultaneously fits all data

Janoschek, Bernlochner, Dunsiger, CP, Böni, Roessli, Link, Rosch, PRB 81, 214436 (2010)

MnSi

cf. pump-probe measurements in Fe1-xCoxSi: Koralek et al., arXiv/1208.1462

conical phase

A phase

helical phase

Magnetic Phase Diagram of MnSiand similar B20 TM compounds

(spin-flop phase)previously unknown

Mühlbauer et al, Science 323, 915 (2009)

cf. Lebech, Bernhoeft (1993) Grigoriev, et al. (2006)

previous work

new work

Neutron Scattering Pattern in the A-Phase

=angular variation

measurements @ MIRA, FRM II

MnSi

perp. single-q: metastabletriple-q: metastable

!6 !5 !4 !3 !2 !1 0

0.2 0.4 0.6

0.01

0.02

B/Bc2

! G

mean field

fluct.

A!crystal

B paramagnet

conical

tMühlbauer et al, Science 323, 915 (2009)

thermal Gaussianfluctuations

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

Fluctuation-Stabilized Multi-q Structure

winding number per unit cell:

projection from above

!2.

!1.5

!1.

!0.5

0

0.5

φ

lattice of topological knots

Topological Properties of the Rigorous Solution

phase btw fundamental modes

skyrmions

center: M antiparallel B

Adams et al., PRL 107, 217206 (2011)

Werner HeisenbergRMP 29 296 (1957)

Tony SkyrmeNucl. Phys. 31 556 (1962)

From „Hairy Balls“ to Skyrmion Lattices

Exploring Spin Torques with Neutrons

temperature

temperature

for neutron scatteringuse special trick:

Experimental Setup

thermometers

Al window

heat sink

temperature

tem

pera

ture

B

curr

ent

tem

pera

ture

B

curr

ent

Current Dependence of the Rotation Angle

antisymmetric rotationthreshold: 106 Am-2

temperature at surface kept constant

Jonietz et al, Science 330, 1648 (2010)

Summary of Experimental Observations

● rotation for current perpendicular to field● no effect for current parallel to applied field● no effect in helical state● no effect in conical state

sense of rotation changes under:● inversion of current direction● inversion of field direction● inversion small T gradient

● low threshold for rotation: 106 Am-2

● same behavior for different sample shapesand orientations

Rôle of Emergent Magnetic Field

Neubauer, et al. PRL 102 186602 (2009) Binz, Vishwanath Physica B 403 1336 (2008)

collect Berry phaseconduction electron tracks spin structure:

trivial topology:

1

! = 0 (1)

"C

T! " ln(T ) (2)

! ! !0 " !1T3/4 (3)

" " "0 ! T 5/3 (4)

#L =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)

"$

$

!

!

!

c= 5 · 10!4 (10)

f""$

$

#

FWHM= 3.8 · 10!4 (11)

t (12)

B/B0 (13)

$

" U

JQ2

#3B (14)

|B| = 150 mT (15)

non-trivial topology:

1

!Be! ! "2.5 T (1)

! = 0 (2)

"C

T# " ln(T ) (3)

" # "0 " "1T3/4 (4)

# " #0 # T 5/3 (5)

$L =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)

"%

%

!

!

!

c= 5 · 10!4 (11)

f""%

%

#

FWHM= 3.8 · 10!4 (12)

t (13)

B/B0 (14)

$

" U

JQ2

#3B (15)

express as Aharonov-Bohm phase represents effective field

BUT: complex band structure!

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

Emergent Magnetic Field of Skyrmions

Rôle of the Temperature GradientPfleiderer, Rosch Nature (N&V) 465 880 (2010)

T gradient ⇒ gradient of spin current

counter-force to „effective Lorentz force“

resulting torque on skyrmion lattice

(assumes rigid skyrmion lattice)

Jonietz et al, Science 330, 1648 (2010)

Proposal of Rotating Skyrmion Latticesin Temperature or Field Gradients

restoring force: higher-order spin-orbit cplg.

Everschor et al., PRB 86 054432 (2012)

very efficent gyro-coupling via Berry phase+ coupling over entire magnetic domains

very weak pinning forces+ low defect concentration (RRR>100)+ very smooth magnetic texture (200Å)+ very stiff magnetic order cf. collective pinning in superconductors

Why ultra-low current densities?

consider Landau-Lifshitz-Gilbert equation

Berry-phasespin torque

precession ineffective field

damping

cf Zhang & Lee PRL (2004)addt‘l damping, cf Zang et al PRL (2010)

two types spin currents:(1) difference of spin polarisation of charge currents

(2) gradients in spin orientation: here super-spin current

cf charge supercurrents due to phase gradients

consider Landau-Lifshitz-Gilbert equation

Berry-phasespin torque

precession ineffective field

damping

cf Zhang & Lee PRL (2004)addt‘l damping, cf Zang et al PRL (2010)

two types spin currents:(1) difference of spin polarisation of charge currents

(2) gradients in spin orientation: here super-spin current

cf charge supercurrents due to phase gradients

„spin current Magnus force“

consider Landau-Lifshitz-Gilbert equation

generalized Thiele approach (project LLG on zero modes)

topological winding number

gyrocoupling vector dissipative tensor

consider Landau-Lifshitz-Gilbert equation

generalized Thiele approach (project LLG on zero modes)

topological winding number

for Fpin≈0 drift and spin-Magnus force

gyrocoupling vector dissipative tensor1

~FMagnus / ~B ⇥ (~vs � ~vd) (1)

~FMagnus +

~Ffric = 0 (2)

~Ffric ⇠ �~vd (3)

~vd ⇡ ��1 ~B ⇥ ~vs (4)

js

= j" � j# (5)

(@t

+ ~vs

r)

ˆM = � ˆM ⇥ ~He↵ +

ˆM ⇥ (↵@t

+ �~vs

r)

ˆM + . . . (6)

F [

~M ] = �F (FM)+ �F (DM)

+ �F (aniso)+ �F (new)

(7)

F [

~M ] =

Zd3r

⇣r0

~M2+ J(r ~M)

2+ U ~M4 � ~B · ~M

⌘(8)

+�F (DM)=

Zd3r

⇣2D ~M · (r⇥ ~M)

⌘(9)

+�F (4)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (10)

+�2

Rd3r M4

x

+ M4y

+ M4z

(11)

+�F (aniso)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (12)

+�2

Rd3r M4

x

+ M4y

+ M4z

(13)

+�3

Pq

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (14)

+�F (6)= �3

X

q

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (15)

⇢xy

= R0B (16)

�b

xy

= p eµ2

b

B

1 + (µb

B)

2(17)

�xy

= �b

xy

+

Gxy

t(18)

1

~FMagnus / ~B ⇥ (~vs � ~vd) (1)

~FMagnus +

~Ffric = 0 (2)

~Ffric ⇠ �~vd (3)

~vd ⇡ ��1 ~B ⇥ ~vs (4)

js

= j" � j# (5)

(@t

+ ~vs

r)

ˆM = � ˆM ⇥ ~He↵ +

ˆM ⇥ (↵@t

+ �~vs

r)

ˆM + . . . (6)

F [

~M ] = �F (FM)+ �F (DM)

+ �F (aniso)+ �F (new)

(7)

F [

~M ] =

Zd3r

⇣r0

~M2+ J(r ~M)

2+ U ~M4 � ~B · ~M

⌘(8)

+�F (DM)=

Zd3r

⇣2D ~M · (r⇥ ~M)

⌘(9)

+�F (4)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (10)

+�2

Rd3r M4

x

+ M4y

+ M4z

(11)

+�F (aniso)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (12)

+�2

Rd3r M4

x

+ M4y

+ M4z

(13)

+�3

Pq

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (14)

+�F (6)= �3

X

q

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (15)

⇢xy

= R0B (16)

�b

xy

= p eµ2

b

B

1 + (µb

B)

2(17)

�xy

= �b

xy

+

Gxy

t(18)

no friction:

with friction:

1

~FMagnus / ~B ⇥ (~vs � ~vd) (1)

~FMagnus +

~Ffric = 0 (2)

~Ffric ⇠ �~vd (3)

~vd ⇡ ��1 ~B ⇥ ~vs (4)

js

= j" � j# (5)

(@t

+ ~vs

r)

ˆM = � ˆM ⇥ ~He↵ +

ˆM ⇥ (↵@t

+ �~vs

r)

ˆM + . . . (6)

F [

~M ] = �F (FM)+ �F (DM)

+ �F (aniso)+ �F (new)

(7)

F [

~M ] =

Zd3r

⇣r0

~M2+ J(r ~M)

2+ U ~M4 � ~B · ~M

⌘(8)

+�F (DM)=

Zd3r

⇣2D ~M · (r⇥ ~M)

⌘(9)

+�F (4)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (10)

+�2

Rd3r M4

x

+ M4y

+ M4z

(11)

+�F (aniso)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (12)

+�2

Rd3r M4

x

+ M4y

+ M4z

(13)

+�3

Pq

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (14)

+�F (6)= �3

X

q

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (15)

⇢xy

= R0B (16)

�b

xy

= p eµ2

b

B

1 + (µb

B)

2(17)

�xy

= �b

xy

+

Gxy

t(18)

1

~FMagnus / ~B ⇥ (~vs � ~vd) (1)

~FMagnus +

~Ffric = 0 (2)

~Ffric ⇠ �~vd (3)

~vd ⇡ ��1 ~B ⇥ ~vs (4)

js

= j" � j# (5)

(@t

+ ~vs

r)

ˆM = � ˆM ⇥ ~He↵ +

ˆM ⇥ (↵@t

+ �~vs

r)

ˆM + . . . (6)

F [

~M ] = �F (FM)+ �F (DM)

+ �F (aniso)+ �F (new)

(7)

F [

~M ] =

Zd3r

⇣r0

~M2+ J(r ~M)

2+ U ~M4 � ~B · ~M

⌘(8)

+�F (DM)=

Zd3r

⇣2D ~M · (r⇥ ~M)

⌘(9)

+�F (4)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (10)

+�2

Rd3r M4

x

+ M4y

+ M4z

(11)

+�F (aniso)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (12)

+�2

Rd3r M4

x

+ M4y

+ M4z

(13)

+�3

Pq

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (14)

+�F (6)= �3

X

q

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (15)

⇢xy

= R0B (16)

�b

xy

= p eµ2

b

B

1 + (µb

B)

2(17)

�xy

= �b

xy

+

Gxy

t(18)

1

~FMagnus / ~B ⇥ (~vs � ~vd) (1)

~vd = ~vs (2)

~FMagnus +

~Ffric = 0 (3)

~Ffric ⇠ �~vd (4)

~vd ⇡ ��1 ~B ⇥ ~vs (5)

js

= j" � j# (6)

(@t

+ ~vs

r)

ˆM = � ˆM ⇥ ~He↵ +

ˆM ⇥ (↵@t

+ �~vs

r)

ˆM + . . . (7)

F [

~M ] = �F (FM)+ �F (DM)

+ �F (aniso)+ �F (new)

(8)

F [

~M ] =

Zd3r

⇣r0

~M2+ J(r ~M)

2+ U ~M4 � ~B · ~M

⌘(9)

+�F (DM)=

Zd3r

⇣2D ~M · (r⇥ ~M)

⌘(10)

+�F (4)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (11)

+�2

Rd3r M4

x

+ M4y

+ M4z

(12)

+�F (aniso)= �1

Pq

(q̂4x

+ q̂4y

+ q̂4z

)|~m~q

|2 (13)

+�2

Rd3r M4

x

+ M4y

+ M4z

(14)

+�3

Pq

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (15)

+�F (6)= �3

X

q

q̂2x

q̂2y

q̂2z

|~m~q

|2 + . . . (16)

⇢xy

= R0B (17)

�b

xy

= p eµ2

b

B

1 + (µb

B)

2(18)

Emergent Electric Field

standard 6-terminal,no‘ T-gradient!

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)

Emergent Electric Field due to Skyrmion Motion

Schulz et al. Nature Physics 8 301 (2012)

Perspectives & Summary

Magnetic Phase Diagram of B20 Compounds

Mühlbauer, et al. Science 323, 915 (2009)Neubauer, et al. PRL 102 186602 (2009)Janoschek, et al. J. Phys. Conf. Series (2010)

Münzer, et al. PRB(R) 81 041203 (2010)Adams, et al. J. Phys. Conf. Series (2010)Bauer, et al. PRB(R) 82 064404 (2010)

Man

yala

et a

l, N

atur

e Ph

ysic

s (2

004)

Fe1-xCoxSix=20%

Fe1-xCoxSi x=20%

P213 insulatorsCu2OSeO3

illustration from Bos et al. (2008)

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

Yu et al., Nature 465 901 (2010)

individual skyrmions

Real Space Observation with Lorentz Force Microscopy

near room temperature in multiferroics

Yu et al., Nat. Mater. 10 106 (2010)CP & Rosch, Nature (N&V) 465 880 (2010)

Seki al., Science 336 198 (2012)

Fe0.5Co0.5Si FeGe Cu2OSeO3

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

Summary

Skyrmion Lattices in Chiral Magnets» basic phenomonology in MnSi» theoretical account» proof of topology

Emergent Electrodynamics» spin torque effects (SANS)» emergent magnetic field» topological Hall effect» emergent electric field