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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