JAIRO SINOVA

Research fueled by:

NERC

ICNM 2007, Istanbul, Turkey July 25th 2007

Anomalous and Spin-Hall effects in mesoscopic systemsAnomalous and Spin-Hall effects in mesoscopic systems

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 SinitsynTexas A&M U.

U. of Texas

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

Alexey KovalevTexas A&M U.

OUTLINEOUTLINE The three spintronic Hall effectsThe three spintronic Hall effects Anomalous Hall effect and Spin Hall effect Anomalous Hall effect and Spin Hall effect

AHE phenomenology and its long historyAHE phenomenology and its long history Three contributions to the AHEThree contributions to the AHE Microscopic approach: focus on the intrinsic AHEMicroscopic approach: focus on the intrinsic AHE Application to the SHEApplication to the SHE SHE in Rashba systems: a lesson from the pastSHE in Rashba systems: a lesson from the past Recent experimental resultsRecent experimental results

Spin Hall spin accumulation: bulk and Spin Hall spin accumulation: bulk and mesoscopic regimemesoscopic regime

Mesoscopic spin Hall effect: non-equilibrium Mesoscopic spin Hall effect: non-equilibrium Green’s function formalism and recent Green’s function formalism and recent experimentsexperiments

SummarySummary

The spintronics Hall effectsThe spintronics Hall effects

AHE

SHEcharge current

gives spin current

polarized charge current gives charge-spin

current

SHE-1

spin current gives

charge current

Anomalous Hall Anomalous Hall transporttransport

Commonalities:

•Spin-orbit coupling is the key•Same basic (semiclassical) mechanisms

Differences:

•Charge-current (AHE) well define, spin current (SHE) is not•Exchange field present (AHE) vs. non-exchange field present (SHE-1)

Difficulties:

•Difficult to deal systematically with off-diagonal transport in multi-band system•Large SO coupling makes important length scales hard to pick•Conflicting results of supposedly equivalent theories•The Hall conductivities tend to be small

Spin-orbit coupling interactionSpin-orbit coupling interaction

(one of the few echoes of relativistic physics in the solid state)(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!!

Anomalous Hall effect: where things Anomalous Hall effect: where things started, the long debatestarted, the long debate

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: semiclassical theory identifies three contributions (intrinsic deflection, skew scattering, side jump scattering)

(thanks to P. Bruno–

CESAM talk)

A history of controversy

Intrinsic Intrinsic deflectiondeflection

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 Skew scatteringscattering

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 Side-jump scatteringscattering

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.

THE THREE CONTRIBUTIONS TO THE AHE: THE THREE CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACHMICROSCOPIC KUBO APPROACH

Skew scattering

Side-jump scattering

Intrinsic AHE

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

= -1 / 0

Averaging procedures:

= 0

FOCUS ON INTRINSIC AHE: semiclassical and KuboFOCUS ON INTRINSIC AHE: 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 approachSuccess of intrinsic AHE approach

• DMS systems (Jungwirth et al PRL 2002)• 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).

• Ferromagnetic Spinel CuCrSeBr: Wei-Lee et al, Science (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

Spin Hall effectSpin 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: 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

*22*

2

2

4

22*22

sH

for8

for8

DDD

D

DD

xy

nnn

ne

mnn

e

SHE conductivity: 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 Disorder effects: beyond the finite lifetime approximation for Rashba lifetime approximation for Rashba

2DEG2DEGQuestion: 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 04Raimondi 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); verified numerically by Normura

et al PRB 2006

n, q

n’n, q

+ +…=0

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

First experimentalFirst experimental observations at the end of 2004observations at the end of 2004

Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/0410295 PRL 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: Co-planar spin LED in GaAs 2D hole gas: ~1% polarization ~1% polarization

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

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

pyx

z

LED1

LED2

10m channel

SHE experiment in SHE experiment in GaAs/AlGaAs 2DHGGaAs/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 '06

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

SHE in the mesoscopic regime

Nonequilibrium Spin Hall Accumulation in Rashba 2DEG

Spin density (Landauer –Keldysh):

+eV/2 -eV/2eV=0

x

y

z

Y. K. Kato, R. C. Myers, A. C. Gossard, and D.D. Awschalom, Science 306, 1910 (2004).

PRL 95, 046601 (2005)

band structure

D.J. Chadi et al. PRB, 3058 (1972)

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

-1.0 -0.5 0.0 0.5 1.0k (0.01 )

-1500

-1000

-500

0

500

1000

E(m

eV) 8

6

7

fundamental energy gap

meV 30086 EE meV 30086 EE

semi-metal or semiconductor

HgTe

-1 .0 -0 .5 0 .0 0 .5 1 .0

k (0 .0 1 )

-1 5 0 0

-1 0 0 0

-5 0 0

0

5 0 0

1 0 0 0

E(m

eV)

H g Te

8

6

7

-1 .0 -0 .5 0 .0 0 .5 1 .0

k (0 .0 1 )

H g 0 .3 2 C d 0 .6 8Te

-1 5 0 0

-1 0 0 0

-5 0 0

0

5 0 0

1 0 0 0

E(m

eV)

6

8

7

V B O

-1 .0 -0 .5 0 .0 0 .5 1 .0

k (0 .0 1 )

-1 5 0 0

-1 0 0 0

-5 0 0

0

5 0 0

1 0 0 0

E(m

eV)

H g Te

8

6

7

-1 .0 -0 .5 0 .0 0 .5 1 .0

k (0 .0 1 )

H g 0 .3 2 C d 0 .6 8Te

-1 5 0 0

-1 0 0 0

-5 0 0

0

5 0 0

1 0 0 0

E(m

eV)

6

8

7

V B O

BarrierBarrierQWQW

VBO = 570 meV

HgTe-Quantum Well Structures

Typ-III QWTyp-III QW

VBO = 570 meV

HgCdTeHgCdTeHgTe

HgCdTe

HH1E1

QW < 55 Å

HgTe

inverterted normal

band structure

conduction band

valence band

HgTe-Quantum Well Structures

High Electron Mobility

H-bar for detection of Spin-Hall-H-bar for detection of Spin-Hall-EffectEffect

(electrical detection through inverse SHE)

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

Actual gated H-bar sample

HgTe-QW

R = 5-15 meV

5 m

ohmic Contacts

Gate-Contact

First Data

HgTe-QW

R = 5-15 meV

Signal due to depletion?

Results...

Symmetric HgTe-QW

R = 0-5 meV

Signal less than 10-4

-2 -1 0 1 2 3

-3.00E-007

-2.50E-007

-2.00E-007

-1.50E-007

-1.00E-007

-5.00E-008

U_

7-1

0 [V

]

-V_gate14 [V]

I: 1->4U:7-10

Sample is diffusive:

Vertex correction kills SHE (J. Inoue et al., Phys. Rev. B 70, 041303 (R) (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 signalin the n-conductingregime

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

insulating

p-conducting n-conducting

Mesoscopic electron SHEMesoscopic electron SHE

L

L/6L/2

calculated voltage signal for electrons (Hankiewicz and Sinova)

Mesoscopic hole SHEMesoscopic hole SHE

L

calculated voltage signal (Hankiweicz, Sinova, & Molenkamp)

L

L/6

L/2

Scaling of H-samples with the system size

L

L/6

0.000 0.004 0.008 0.0122.0

2.5

3.0

3.5

4.0

4.5

5.0

L=150nm

n=1*1011cm-2

Spin orbit coupling = 72meVnm

L=90nm

L=120nmL=200nm

L=240nm

chan

ge o

f vol

tage

[V

]

1/L [nm]

Oscillatory character of voltage difference with the system size.

Extrinsic (1971,1999) and Intrinsic (2003) SHE predicted and observed (2004): back to the beginning on a higher level

Extrinsic + intrinsic AHE in graphene:two approaches with the

same answer

SUMMARY

Optical detection of current-induced polarizationphotoluminescence (bulk and edge 2DHG)Kerr/Faraday rotation (3D bulk and edge, 2DEG)

Transport detection of the mesoscopic SHE in semiconducting systems: HgTe preliminary results agree with theoretical calculations

Experimental achievements

Experimental (and experiment modeling) challenges:

Photoluminescence cross sectionedge electric field vs. SHE induced spin accumulationfree vs. defect bound recombinationspin accumulation vs. repopulationangle-dependent luminescence (top vs. side emission)hot electron theory of extrinsic experiments

Optical detection of current-induced polarizationphotoluminescence (bulk and edge 2DHG)Kerr/Faraday rotation (3D bulk and edge, 2DEG)

Transport detection of the SHE

Generaledge electric field (Edelstein) vs. SHE induced spin accumulation

SHE detection at finite frequenciesdetection of the effect in the “clean” limit

WHERE WE ARE GOING (EXPERIMENTS)

INTRINSIC+EXTRINSIC: STILL CONTROVERSIAL!INTRINSIC+EXTRINSIC: STILL CONTROVERSIAL!

AHE in Rashba systems with disorder:AHE in Rashba systems with disorder: Dugaev et al PRB 05Dugaev et al PRB 05 Sinitsyn et al PRB 05Sinitsyn et al PRB 05 Inoue et al (PRL 06)Inoue et al (PRL 06) Onoda et al (PRL 06)Onoda et al (PRL 06)

Borunda et al (cond-mat 07)Borunda et al (cond-mat 07)

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 the different equivalent

linear response theories both in AHE and SHE

Need to match the Kubo to the Need to match the Kubo to the BoltzmannBoltzmann

Kubo: systematic formalismKubo: systematic formalism Botzmann: easy physical Botzmann: easy physical

interpretation of different interpretation of different contributionscontributions

Connecting Microscopic and Connecting Microscopic and Semiclassical approachSemiclassical approach

Sinitsyn et al PRL 06, PRB 06

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 )]

2' ' '

2| | ( )l l l l l lV

Golden Rule:

Coordinate shift:

0 0' ' ' '

' '

( ) ( )v ( )l l ll l l l l l l l l

l 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 kr u i u u i u D V

k k

' ''

lll l l l l

l

v F eE rk

velocity:

l ll

J e f v

current:

' 'l l l lV T

Semiclassical approach II: Sinitsyn et al PRB 06

Sinitsyn et al PRB 06

Armchair edge

Zigzag edge

EF

Some success 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 PRL 06, PRB 06 SAME RESULT OBTAINED USING BOLTMANN!!!

Comparing Boltzmann to Kubo in the chiral basis

0 0' ' ' '

' '

( ) ( )v ( )l l ll l l l l l l l l

l ll l

f f feE eE r f f

t

Intrinsic deflectionIntrinsic 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.