Anomalous Hall effects :  Merging The Semiclassical and Microscopic Theories

Karlsruhe, Germany, July 16th 2008

JAIRO SINOVA

Research fueled by:

NERC

OUTLINE

Anomalous Hall effect:

MπRBR sH 40

Simple electrical measurement Simple electrical measurement of magnetizationof magnetization

Spin-orbit coupling “force” deflects like-spinlike-spin particles

I

_ FSO

FSO

_ __

majority

minority

VInMnAs

controversial theoretically: three contributions to the AHE (intrinsic deflection, skew scattering, side jump scattering)

(thanks to P. Bruno– CESAM talk)

A history of controversy

Anomalous Hall effect: what is necessary to see the effects?

I

_ FSO

FSO

_ __

majority

minority

V

Necessary condition for AHE: TIME REVERSAL SYMMETRY MUST BE BROKEN

),(),( MBMB xyxy

Need a magnetic field and/or magnetic order

BUT IS IT SUFFICIENT? (P. Bruno– CESAM 2005)

Local time reversal symmetry being broken does not always mean AHE present

Staggered flux with zero average flux:

- -

-

Is xy zero or non-zero?

-

--

Translational invariant so xy =0

Similar argument follows for antiferromagnetic ordering

Does zero average flux necessary mean zero xy ?

--

- 3--

- 3No!! (Haldane, PRL 88)

(P. Bruno– CESAM 2005)

Is non-zero collinear magnetization sufficient?

(P. Bruno– CESAM 2005)

In the absence of spin-orbit coupling a spin rotation of restores TR symmetry and xy=0

If spin-orbit coupling is present there is no invariance under spin rotation and xy≠0

(P. Bruno– CESAM July 2005)

Collinear magnetization AND spin-orbit coupling → AHE

Does this mean that without spin-orbit coupling one cannot get AHE?

Even non-zero magnetization is not a necessary condition

No!! A non-trivial chiral magnetic structure WILL give AHE even without spin-orbit coupling

Mx=My=Mz=0 xy≠0

Bruno et al PRL 04

COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING vs. CHIRAL MAGNET STRUCTURES

AHE is present when SO coupling and/or non-trivial spatially varying magnetization (even if zero in average)

SO coupled chiral states: disorder and electric fields lead to AHE/SHE through both intrinsic and extrinsic contributions

Spatial dependent magnetization: also can lead to AHE. A local transformation to the magnetization direction leads to a non-abelian gauge field, i.e. effective SO coupling (chiral magnets), which mimics the collinear+SO effective Hamiltonian in the adiabatic approximation

So far one or the other have been considered but not both together, in the following we consider only collinear magnetization + SO coupling

Intrinsic deflection

Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling.

Movie created by Mario Borunda

Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure)

Skew scattering

Movie created by Mario Borunda

Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators.

Side-jump scattering

Movie created by Mario Borunda

Related to the intrinsic effect: analogy to refraction from an imbedded medium

Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step.

Intrinsic deflection

Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling.

Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure)

E

Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step.

Related to the intrinsic effect: analogy to refraction from an imbedded medium

Side jump scattering

Skew scattering

Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators.

Need to match the Kubo, Boltzmann, and Keldysh

Kubo: systematic formalism Boltzmann: easy physical interpretation of

different contributions (used to define them) Keldysh approach: also a systematic kinetic

equation approach (equivelnt to Kubo in the linear regime). In the quasiparticle limit it must yield Boltzmann eq.

Microscopic vs. Semiclassical

THE THREE CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACH

Skew scattering

Side-jump scattering

Intrinsic AHE: accelerating between scatterings

SkewσH

Skew (skew)-1 2~σ0 S where

S = Q(k,p)/Q(p,k) – 1~

V0 Im[<k|q><q|p><p|k>]

Vertex Corrections

σIntrinsic

Intrinsicσ0 /εF

n, q

n, q m, p

m, pn’, k

n, q

n’n, q

FOCUS ON INTRINSIC AHE (early 2000’s): semiclassical and Kubo

K. Ohgushi, et al PRB 62, R6065 (2000); T. Jungwirth et al PRL 88, 7208 (2002);T. Jungwirth et al. Appl. Phys. Lett. 83, 320 (2003); M. Onoda et al J. Phys. Soc. Jpn. 71, 19 (2002); Z. Fang, et al, Science 302, 92 (2003).

'

2

'

'

2

)(

'ˆˆ'Im]Re[

nnk knkn

yx

nkknxy EE

knvknknvknff

V

e

nk

nknxy kfV

e

)(]Re[ '

2

Semiclassical approach in the “clean limit”

Kubo:

n, q

n’n, q

STRATEGY: compute this contribution in strongly SO coupled ferromagnets and compare to experimental results, does it work?

Success of intrinsic AHE approach in comparing to experiment: phenomenological

“proof”• DMS systems (Jungwirth et al PRL 2002, APL 03)• Fe (Yao et al PRL 04)• layered 2D ferromagnets such as SrRuO3 and

pyrochlore ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19 (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003), Shindou and Nagaosa, Phys. Rev. Lett. 87, 116801 (2001)]

• colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999).

• CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004)

Berry’s phase based AHE effect is quantitative-successful in many instances BUT still not a theory that treats systematically intrinsic and

extrinsic contribution in an equal footing

Experiment AH 1000 (cm)-1

TheroyAH 750 (cm)-1

AHE in GaMnAs

AHE in Fe

INTRINSIC+EXTRINSIC: REACHING THE END OF A DECADES LONG

DEBATE AHE in Rashba systems with weak disorder:AHE in Rashba systems with weak disorder: Dugaev et al (PRB 05)Dugaev et al (PRB 05)

Sinitsyn et al (PRB 05, PRB 07)Sinitsyn et al (PRB 05, PRB 07) Inoue et al (PRL 06)Inoue et al (PRL 06)

Onoda et al (PRL 06, PRB 08)Onoda et al (PRL 06, PRB 08)Borunda et al (PRL 07), Nuner et al (PRB 07, PRL 08)Borunda et al (PRL 07), Nuner et al (PRB 07, PRL 08)

Kovalev et al (PRB 08)Kovalev et al (PRB 08)

All are done using same or equivalent linear response formulation–different or not obviously

equivalent answers!!!

The only way to create consensus is to show (IN DETAIL) agreement between ALL the different equivalent linear

response theories both in AHE and SHE

Semiclassical Boltzmann equation

' ''

( )l ll l l l

l

f feE f f

t k

2' ' '

' '' ''' '

' ''

2| | ( )

...

l l l l l l

l l l ll l l l

l l

T

V VT V

i

Golden rule:

' 'l l ll J. Smit (1956):Skew Scattering

In metallic regime:

Kubo-Streda formula summary

2 R+II Rxy x y-

R A AR A A

x y x y x y

e dGσ = dεf(ε)Tr[v G v -

4π dε

dG dG dG-v v G -v G v +v v G ]

dε dε dε

I IIxy xy xyσ =σ +σ

2 +I R A Axy x y-

R R Ax y

e df(ε)σ =- dε Tr[v (G -G )v G -

4π dε

-v G v (G -G )]

Calculation done easiest in normal spin basis

2' ' '

2| | ( )l l l l l lV

Golden Rule:

Coordinate shift:

0 0' ' ' '

' '

( ) ( )v ( )l l l

l l l l l l l l ll ll l

f f feE eE r f f

t

ModifiedBoltzmannEquation:

( , )l k

Iml l l l lz

y x x y

u u u uF

k k k k

Berry curvature:

' ' ' '',ˆ arg

'l l l l l l l lk k

r u i u u i u D Vk k

' ''

lll l l l l

l

v F eE rk

velocity: l l

l

J e f v

current:

' 'l l l lV T

Semiclassical approach II:

Sinitsyn et al PRB 06

Armchair edge

Zigzag edge

EF

“AHE” in graphene

2 R+II Rxy x y-

R A AR A A

x y x y x y

e dGσ = dεf(ε)Tr[v G v -

4π dε

dG dG dG-v v G -v G v +v v G ]

dε dε dε

I IIxy xy xyσ =σ +σ

2 +I R A Axy x y-

R R Ax y

e df(ε)σ =- dε Tr[v (G -G )v G -

4π dε

-v G v (G -G )]

In metallic regime:IIxyσ =0

Kubo-Streda formula:

2 32 42 4

I so so FF Fxy 2 2 22 22 2 22 2 2

F soF so F so F so

e V-e Δ (vk )4(vk ) 3(vk )σ = 1+ +

(vk ) +4Δ 2πn V4π (vk ) +Δ (vk ) +4Δ (vk ) +4Δ

Single K-band with spin up

x x y y so zKH =v(k σ +k σ )+Δ σ

Sinitsyn et al PRB 07

Comparing Botlzmann to Kubo in the chiral basis

AHE in Rashba 2D system

km

kkk

m

kH xyyxk

0

22

0

22

2)(

2

Inversion symmetry no R-SO

Broken inversion symmetry R-SO

Bychkov and Rashba (1984)

(differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator)

n, q

n’n, q

AHE in Rashba 2D systemKubo and semiclassical approach approach: (Nuner et al PRB08, Borunda et al PRL 07)

Only when ONE both sub-band there is a significant contribution

When both subbands are occupied there is additional vertex corrections that contribute

AHE in Rashba 2D system

When both subbands are occupied the skew scattering is only obtained at higher Born approximation order AND the extrinsic contribution is unique (a hybrid between skew and side-jump)

Kovalev et al PRB 08

Keldysh and Kubo match analytically in the metallic limit

Numerical Keldysh approach (Onoda et al PRL 07, PRB 08)

GR G0 G0RGR

G0 1 R GR 1

G0R 1

ˆ G ˆ G G0A 1

ˆ R ˆ G ˆ G ˆ R ˆ ˆ G A ˆ G R ˆ

ˆ ˆ R ˆ G ˆ A

Solved within the self consistent T-matrix approximation for the self-energy

AHE in Rashba 2D system: “dirty” metal limit?

Is it real? Is it justified? Is it “selective” data chosing?

Can the kinetic metal theory be justified when disorder is larger than any other scale?

OTHER ANOMALOUS HALL EFFECTS

Spin Hall effect

Take now a PARAMAGNET instead of a FERROMAGNET: Spin-orbit coupling “force” deflects like-spinlike-spin particles

I

_ FSO

FSO

_ __

V=0

non-magnetic

Spin-current generation in non-magnetic systems Spin-current generation in non-magnetic systems without applying external magnetic fieldswithout applying external magnetic fields

Spin accumulation without charge accumulationSpin accumulation without charge accumulationexcludes simple electrical detectionexcludes simple electrical detection

Carriers with same charge but opposite spin are deflected by the spin-orbit coupling to opposite sides.

Spin Hall Effect(Dyaknov and Perel)

InterbandCoherent Response

(EF) 0

Occupation # Response

`Skew Scattering‘(e2/h) kF (EF )1

X `Skewness’

[Hirsch, S.F. Zhang] Intrinsic

`Berry Phase’(e2/h) kF

[Murakami et al,

Sinova et al]

Influence of Disorder`Side Jump’’

[Inoue et al, Misckenko et al, Chalaev et al…] Paramagnets

INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/0308167)

Sinova et al PRL 2004 (cont-mat/0307663)

as there is an intrinsic AHE (e.g. Diluted magnetic semiconductors), there should be an intrinsic spin-Hall

effect!!!

km

kkk

m

kH xyyxk

0

22

0

22

2)(

2

Inversion symmetry no R-SO

Broken inversion symmetry R-SO

Bychkov and Rashba (1984)

(differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator)

n, q

n’n, q

‘Universal’ spin-Hall conductivity

*22*

2

2

4

22*22

sH

for8

for8

DDD

D

DD

xy

nnn

ne

mnn

e

Color plot of spin-Hall conductivity:yellow=e/8π and red=0

n, q

n’n, q

SHE conductivity: all contributions– Kubo formalism perturbation theory

Skewσ0 S

Vertex Corrections σIntrinsic

Intrinsicσ0 /εF

n, q

n’n, q

= j = -e v

= jz = {v,sz}

Disorder effects: beyond the finite lifetime approximation for Rashba 2DEG

Question: Are there any other major effects beyond the finite life time broadening? Does side jump

contribute significantly?

Ladder partial sum vertex correction:

Inoue et al PRB 04Dimitrova et al PRB 05Raimondi et al PRB 04Mishchenko et al PRL 04Loss et al, PRB 05

~

the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05)

n, q

n’n, q

+ +…=0

For the Rashba example the side jump contribution cancels the intrinsic contribution!!

First experimental observations at the end of 2004

Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/0410295PRL 05

Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system

Co-planar spin LED in GaAs 2D hole gas: ~1% polarization Co-planar spin LED in GaAs 2D hole gas: ~1% polarization

-1

0

1

CP

[%]

Light frequency (eV)1.505 1.52

Kato, Myars, Gossard, Awschalom, Science Nov 04

Observation of the spin Hall effect bulk in semiconductors

Local Kerr effect in n-type GaAs and InGaAs: Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization ~0.03% polarization (weaker SO-coupling, stronger disorder)(weaker SO-coupling, stronger disorder)

OTHER RECENT EXPERIMENTS

“demonstrate that the observed spin accumulation is due to a transverse bulk electron spin current”

Sih et al, Nature 05, PRL 05

Valenzuela and Tinkham cond-mat/0605423, Nature 06

Transport observation of the SHE by spin injection!!

Saitoh et al APL 06

QUANTUM SPIN HALL EFFECT

(Physics Today, Feb 2008)

Anomalous Hall effect in cold atoms

Unitary transformation

State vector in the pseudospin basis:

Diagonalization with local unitary transformation:

Pure gauge:

Interaction of atoms with laser fields

U(m) adiabatic gauge field

We can then introduce

the adiabatic conditionadiabatic condition and reach the dynamical evolution of degenerate ground subspace

Berry phase, Spin-orbit coupling, etc.

U(m) adiabatic U(m) adiabatic gauge field:gauge field:

Two dark-states consist of the degenerate subspace (pseudospin-1/2 states):

Spin-orbit coupling and AHE in atoms

xx

yy

zz

U(2) adiabatic gauge field:U(2) adiabatic gauge field:

Mixing angle:

(see also Staunesco, Galitskii, et al PRL 07)

Effective trap potentials:

Employ Gaussian laser beams and set the trap:

0M

E

The effective Hamiltonian ( ):

AHE

yy

-y-y

xx

Branislav NikolicU. of DelawareAllan MacDonald

U of Texas

Tomas JungwirthInst. of Phys. ASCR

U. of Nottingham

Joerg WunderlichCambridge-Hitachi

Laurens MolenkampWuerzburg

Kentaro NomuraU. Of Texas

Ewelina HankiewiczU. of MissouriTexas A&M U.

Mario BorundaTexas A&M U.

Nikolai SinitsynLANL

Other collaborators: Bernd Kästner, Satofumi Souma, Liviu Zarbo, Dimitri Culcer , Qian Niu, S-Q Shen, Brian

Gallagher, Tom Fox, Richard Campton, Winfried Teizer, Artem Abanov

Xin LiuTexas A&M U.

Alexey KovalevTexas A&M U.

EXTRAS

Spin-orbit coupling interaction

(one of the few echoes of relativistic physics in the solid

state)

Ingredients: -“Impurity” potential V(r)

- Motion of an electron

)(1

rVe

E

Producesan electric field

In the rest frame of an electronthe electric field generates and effective magnetic field

Ecm

kBeff

This gives an effective interaction with the electron’s magnetic moment

LSdr

rdV

err

mc

k

mc

SeBeffH SO

)(1

CONSEQUENCES•If part of the full Hamiltonian quantization axis of the spin now

depends on the momentum of the electron !! •If treated as scattering the electron gets scattered to the left or to

the right depending on its spin!!

The new challenge: understanding spin accumulation

Spin is not conserved; analogy with e-h system

Burkov et al. PRB 70 (2004)

Spin diffusion length

Quasi-equilibrium

Parallel conduction

Spin Accumulation – Weak SO

Spin Accumulation – Strong SO

Mean FreePath?

Spin Precession

Length

?

SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION

STUDIES

so>>ħ/

Width>>mean free path

Nomura, Wundrelich et al PRB 06

Key length: spin precession length!!Independent of !!

-1

0

1

Pol

ariz

atio

n in

%

1.505 1.510 1.515 1.520

-1

0

1

Energy in eV

Pol

ariz

atio

n in

%

1.5mchannel

n

n

py

xz

LED1

LED2

10m channel

SHE experiment in GaAs/AlGaAs 2DHG

- shows the basic SHE symmetries

- edge polarizations can be separated over large distances with no significant effect on the magnitude

- 1-2% polarization over detection length of ~100nm consistent with theory prediction (8% over 10nm accumulation length)

Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05

Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '05

Non-equilibrium Green’s function formalism (Keldysh-LB)

Advantages:•No worries about spin-current definition. Defined in leads where SO=0•Well established formalism valid in linear and nonlinear regime•Easy to see what is going on locally•Fermi surface transport

3. Charge based measurements of SHE

PRL 05

H-bar for detection of Spin-Hall-Effect

(electrical detection through inverse SHE)

E.M. Hankiewicz et al ., PRB 70, R241301 (2004)

New (smaller) sample

1 m

200 nm

sample layout

-2 -1 0 1 2 3 4 5

0

1000

2000

3000

4000

5000

6000

7000

0

1

2

3

4

5R

nonl

ocal /

VGate

/ V

Q2198HI (3,6)U (9,11)

I /

nA

SHE-Measurement

no signal in the n-conducting regime

strong increase of the signal in the p-conducting regime, with pronounced features

insulating

p-conducting n-conducting

Mesoscopic electron SHE

L

L/6L/2

calculated voltage signal for electrons (Hankiewicz and Sinova)

Mesoscopic hole SHE

L

calculated voltage signal (Hankiweicz, Sinova, & Molenkamp)

L

L/6

L/2