Theory of spin transport in semi-y p pconductors, ferromagnets, and

ld t tcold-atom systems

Rembert DuineSlides at:

www.phys.uu.nl/~duine/spintut.pdf

Institute for Theoretical PhysicsUtrecht UniversityUt echt Unive sity

C ll b tCollaboratorsA k l dPh. D. Students: Ties Lucassen, Aaron Swaving, Hedwig

van Driel, Erik van der Bijl, Martijn Mink, postdoc: Clement Wong

Acknowledgements:Gerrit Bauer (Delft)

Hiroshi Kohno (Osaka) M i M t (G i )Profs. Cristiane de Morais and

Henk Stoof(Utrecht University)

Maxim Mostovoy (Groningen)

( y)

Allan MacDonald (UT Austin) Paul Haney (NIST)Paul Haney (NIST)

Alvaro Núñez (Valparaiso, Chile) Jairo Sinova (Texas A&M)

Achim Rosch’s group (Cologne)Achim Rosch s group (Cologne)Christian Pfleiderers group (TU Munich)

R. Lavrijsen.+TU/e teamGi i Vi l (Mi i)Giovanni Vignale (Missouri)

Marco Polini (SNS Pisa)

O i (I)

Thanks to Dries van Oosten for picture

Overview (I)

Time-dependent magnetization induces voltage/current

Current induces magnetization dynamics

O i (II)Other experiments: Ono, Klaui, Erskine, Tsoi, Parkin, Fert, Meier, Marrows…

Overview (II)j=4.3 105 A/cm2 DW pinned by inhomogeneity

moving domain wall

current-pulsedurationduration

M. Yamanouchi, et al. (2006), expts. on magnetic semiconductor GaMnAs

O i (III)Overview (III)

O i (IV) Th bl ( )Overview (IV): The problem(s)Electric (spin) current

x t Time- and position-dependent ,x t p p

magnetization texture

Understand influence of (spin) current on dynamics magnetization texture and vice versadynamics magnetization texture, and vice versa

Potential headaches: time-dependence, non-collinearity, spin-orbit coupling, e-e interactions

B d t tBroader context:

Interplay spin current <->magnetization dynamicsmagnetization dynamics important for building magnetic memoriesmagnetic memories (spintronics)

Magnetic race track memoryMagnetic race-track memoryS. Parkin et al., IBM Almaden

O tliOutline

Semi-classical approach to: Spin transfer Spin transfer Spin motive forces + topological Hall effect Anomalous Hall effect C ld t t i d Cold-atom systems: spin drag

O tliOutline

Semi-classical approach to: Spin transfer Spin transfer Spin motive forces + topological Hall effect Anomalous Hall effect C ld t t i d Cold-atom systems: spin drag

Th bl ( )The problem(s)Electric (spin) current

x t Time- and position-dependent ,x t p p

magnetization texture

Understand influence of (spin) current on dynamics magnetization texture and vice versadynamics magnetization texture, and vice versa

Potential headaches: time-dependence, non-collinearity, spin-orbit coupling, e-e interactions

S i l i l h (I)Semi-classical approach (I) Exchange coupling is strongest: keep lowest order in ,x t

Fermi wavelength << scale of magnetic texture variation > t t l t i l i ll N t + B lt-> treat electrons semi-classically: Newton+ Boltzmann

Motivation: this approach gets band structure contribution Motivation: this approach gets band-structure contribution to anomalous Hall effect right [Jairo Sinova’s talk]

Motivation: this approach recovers correct phenomenology, even with dissipation [Disclaimer]

S i l i l h (II)Semi-classical approach (II)Conduction electron

( , , )x p s unit vector

,x t

Time- and position-dependent ,x t tdp

p pmagnetization texture (unit vector)

2

,2

dp s tdt xds t

t t t

2

,2 2pH x t sm

3

,2

,

s t x t tdt

x tx t s t a x x t

,2

x t s t a x x tt

Newton’s equations

of motion

S i l i l h (III)Semi-classical approach (III)Conduction electron

( , , )x p s

x t Time- and position-dependent ,x t p p

magnetization textureSolve equation for spin adiabatically:Solve equation for spin adiabatically:

,, d x td x t ts t x t t x t t

t

, , sss t x t t x t tdt dt

S i l i l h (IV)Semi-classical approach (IV)

Equation for electron motion, spin + magnetizationq f p g

Eliminate equation for spin precession

Equation for electron motion+extra terms: spinmotive forces and topological Hall effectmotive forces and topological Hall effect

Equation for magnetization+extra terms: spin transfer torques

O tliOutline

Semi-classical approach to: Spin transfer Spin transfer Spin motive forces + topological Hall effect Anomalous Hall effect C ld t t i d Cold-atom systems: spin drag

S i t fZhang/Li,Barnes/Meakawa, Tserkovnyak/Bauer/Brataas, Waintal/Viret, Thiaville, ...

Spin transferEquation for magnetization

in presence of current

eff s sH v vt t

fft t

Spin transfer torquesGilbertdamping

Effectivefield

,current

ss

v

currentsv

C t i d d d i ll tiSee e.g. Thiaville et al.

Current-induced domain wall motion

dwKv

2

2.5

1

1.5

0.5

j/jc

0.25 0.5 0.75 1 1.25 1.5 1.75 2

Conclusion:current

Conclusion:ratio of very important

(but hard to measure directly(but hard to measure directly, and hard to calculate)

O tliOutline

Semi-classical approach to: Spin transfer Spin transfer Spin motive forces + topological Hall effect Anomalous Hall effect C ld t t i d Cold-atom systems: spin drag

S i ti f (I)Spin motive forces (I)Conduction electron

( , , )x p s

x t

Time- and position-dependent ,x t p pmagnetization texture

This gives:This gives:

( )dp e E x B

( )s se E x B

dt

S i ti f (II)Spin motive forces (II)

( )F e E x B

Magnetization leads to effective force on electrons:

, ( )s sF e E x B

, ,x t x t

, ,

,2s

x t x tE x t

e t x

Spin motive force

, ,x t x tt x

B x x x x Effective magnetic , 2sB x x x x

e Effective magnetic field

C l l ti t lt t

Berger (1983), Barnes/Maekawa (2007), Saslow (2007), Tserkovnyak/Mecklenburg/Wong (2007)

Calculating currents, voltages, etc..I i l l i f i i Insert single-electron equation of motion in boltzmann equation to get currents, etc.:

collisions (disorder, e-e int's, ...)f f fx pt x p

Charge and spin currents:j E E j EE

“Spin drag”s sj E E s sj EE

j j j Espin current: charge current:

c sj j j Pj s sj j j E

P

current polarization

Voltage/current due to field-d i d i lldriven domain wall:

R lt f di i l ll1

Results for one-dimensional wall

0 6

0.8

1

dwKr

C l diff t l

0 2

0.4

0.6 Colors: different values of -parameter

B/Bmagnetic field5 10 15 20

0.2

2slope ~

non universalB/Bwmagnetic field1

non-universal

universalcurrent inunits of

0.5 1.0 1.5 2.0 2.5 3.0

-1

universal regime

2 /L e K

units ofmagnetic field B/Bw

R lt f t d i llResults for vortex domain wall

E i tExperiments

To be continued!?

O tliOutline

Semi-classical approach to: Spin transfer Spin transfer Spin motive forces + topological Hall effect Anomalous Hall effect C ld t t i d Cold-atom systems: spin drag

T l i l H ll ff t (I)Nagaosa, Loss, Millis, ...1990s

Topological Hall effect (I)

Effective magnetic field, opposite for opposite spins:opposite spins:

fictitiousB x x x x

Skyrmion densitySkyrmion density,effective flux is

topological numberfictitious

topological number

Carsten Timm/Alan Stonebraker, Physics 2, 35 (2009)

T l i l H ll ff t (II)Topological Hall effect (II)Science (2009)

• A-phase of spiralA phase of spiral magnet MnSi• Inferred fromInferred fromneutron scattering• Electron transportElectron transportequivalent to electronsin fictitious magneticin fictitious magneticfield -> Hall effect

T l i l H ll ff t (III)Topological Hall effect (III)

PRL (2009)

fictitiousPR B 0xy PR B

S i t i M SiSpin torques in MnSi

S i 330 1648 (2010)Science 330, 1648 (2010)

O tliOutline

Semi-classical approach to: Spin transfer Spin transfer Spin motive forces + topological Hall effect Anomalous Hall effect C ld t t i d Cold-atom systems: spin drag

A l H ll ff t (I)Anomalous Hall effect (I)Conduction electron

( , , )x p s

x tp Time- and position-dependent , ,x tp

magnetization texture+strong spin orbit couplingAdiabatic eq For spin:Adiabatic eq. For spin:

, ,d x pt ts t x t

tp t t x t tp t

, , , ,s t x t p t t x t tdt

p t

A l H ll ff t (II)Anomalous Hall effect (II) Equation for electron (cf. Niu/MacDonald/Sinova)

dx t so

dx tp B

dt

Band-structure (intrinsic) contribution

to anomalous Hall

To do: spin transfer/motive forces with strong spin orbit coupling (Rashba fields )orbit coupling (Rashba fields, ...)

O tliOutline

Semi-classical approach to: Spin transfer Spin transfer Spin motive forces + topological Hall effect Anomalous Hall effect C ld t t i d Cold-atom systems: spin drag

S i dD’Amico/Vignale, expt: Weber et al. (2005)

Spin drag

j E E j EE“Spin drag”

s sj E E s sj EE

Problem: spin drag is small effect, hard to observe directly

E t ld tEnter cold atoms...

Tunable system: potential,b f i 2F 1number of spin states 2F+1,

external potentials, interactions, bosons, fermions...

But: no charge (real E-field does nothing) But: no charge (real E-field does nothing), and no disorder (no relaxation of center-of-

) l f t t h d!mass)– analogues of transport hard! Spin drag only relaxation effect!p g y

S i t t i ld

Berger (1983), Barnes/Maekawa (2007), Saslow (2007)

Spin transport in cold gases:

Charge and spin currents:“Spin drag”

s sj E E s sj EE

t d fi d

spin current:

, not defined

Spin drag relaxation time:

1

2s sj j j E p Spin drag relaxation time:

2 9

2 5

1~ (fermions "blocking" in 3D) 10 m

1 (b "l i " i i 1D) 10 ( h )

sd T

T

2 51~ ~ (bosons "lasing" in quasi-1D) 10 m (use charge= )sd T e

Experiments on cold fermionsExperiments on cold fermions

C l iConclusions

Interplay between current and magnetization dynamics: dissipationmagnetization dynamics: dissipation important

Semi-classical approach captures phenomenology; next step: strong spin-orbit

Cold atoms attractive for studying f d t lfundamentals

Utrecht Universityy

Postdoc & ph d. PositionsPostdoc & ph d. PositionsAvailable!

A diAppendix:

S i l i l h (A)Semi-classical approach (A)

,x t tdp s t

Newton’s equation for electron

2,

, , ,2

s tdt x

d x t tds ts t x t t s t x t t x t t

d d

precession of electron around magnetization

3

, , ,2

,,

2

dt dtx t

x t s t a x x tt

2t

Precession of magnetization around spin of electron

S i l i l h (B)Semi-classical approach (B)E ti f l t tiEquation for electron motion

in smoothly-varying magnetization texture

, , ,,

2 2x t t d x t t x t tdp s t x t t

dt x dt x

33 ,, d x t tx t ax t s t a x x t x x t

,2 2

x t s t a x x t x x tt dt

E ti f ti tiEquation for magnetization in presence of electrons

I l di i l ti (A)Including spin relaxation (A)

,x t tdp s t

Newton’s equation for electron

2s t

dt x

d d

precession of electron

,2 ss

ds t ds ts t x t t s t

dt dtd x t t d x t t

p faround magnetization

+ relaxation

, ,, , ss

d x t t d x t ts t x t t x t t

dt dt

3,,

2x t

x t s t a x x tt

Precession of magnetization around spin of electron

I l di i l ti (B)Including spin relaxation (B)E ti f l t tiEquation for electron motion

in smoothly-varying magnetization texture

, , , ,,

2 2 ss

d x t t x t t d x t t x t tdp x t tdt dt x dt x

3 3, ,, d x t t d x t tx t a a

Effective electric and magnetic field , ,,

,2 2 ss

x t a ax x t x x t x tt dt dt

E ti f ti tiEquation for magnetization in presence of electrons

S i t fSpin transferEquation for magnetization

in presence of electrons

3 3, ,,,

2 2 ss

d x t t d x t tx t a ax x t x x t x tt dt dt

3 3

3 3

2 2 ssa ax x t x x t

t t

3 3

2 2 ssa ax x t x t x x t x t

go from single to many electrons:go from single to many electrons: ( ) spin currentx x t x

( ) spin densityx x t