SpinSpin--Hall effect and anomalous transport in strongly Hall effect and anomalous transport in strongly spinspin--orbit coupled systemsorbit coupled systems

APCTP, August 8th 2005

Collaborators: Allan MacDonald, Kentaro Nomura, JoergWunderlich, Bernd Kästner, Tomas Jungwirth, Nikolai Sinitsyn, Laurens Molenkamp, Branislav

Nikolic, Satofumi Souma, Liviu Zarbo, Dimitri Culcer, Ewelina Hankeiwc, Qian Niu, S-Q Shen

JAIRO SINOVA

OUTLINEOUTLINEAnomalous Hall effect and Spin Hall effectAnomalous Hall effect and Spin Hall effect

Parsing the AHEParsing the AHEPerturbation theory treatmentPerturbation theory treatment

Parsing the SHE (example: 2D systems)Parsing the SHE (example: 2D systems)Perturbation theory treatmentPerturbation theory treatmentExact diagonalization calculationsExact diagonalization calculations

Spin Hall spin accumulationSpin Hall spin accumulationExact diagonalization studies (bulk)Exact diagonalization studies (bulk)Mesoscopic systemsMesoscopic systems

Conclusions and things for thoughtConclusions and things for thought

OUTLINEOUTLINESpinSpin--orbit coupling in semiconducting systemsorbit coupling in semiconducting systemsAnomalous Hall effect and Spin Hall effectAnomalous Hall effect and Spin Hall effect

Parsing the AHEParsing the AHEPerturbation theory treatmentPerturbation theory treatment

Parsing the SHE (example: 2D systems)Parsing the SHE (example: 2D systems)Perturbation theory treatmentPerturbation theory treatmentExact diagonalization calculationsExact diagonalization calculations

Spin Hall spin accumulationSpin Hall spin accumulationExact diagonalization studies (bulk)Exact diagonalization studies (bulk)Mesoscopic systemsMesoscopic systems

ConclusionsConclusions

Anomalous Hall effectAnomalous Hall effect

MπRBR sH 40 +=ρ

Simple electrical measurement Simple electrical measurement of magnetizationof magnetization

Spin-orbit coupling “force” deflects likelike--spinspin particles

I

_ FSO

FSO

_ __majority

minority

VInMnAs

Anomalous Hall EffectLinear Response

Theory

Order ofLimits

(thanks to P. Bruno– CESAM talk)

(thanks to P. Bruno– CESAM talk)

(thanks to P. Bruno– CESAM talk)

Perturbation Theory

= +Bloch Electron

Real Eigenstates

Averaging procedures:

= τ-1 / ν0 = ν0 / τ-1

Perturbation Theory: conductivity

n, qVertex Corrections

∼ 1-cos(θ)

n, qDrude Conductivity

σ = ne2τ/m*

= jν = -evν

AHE – Perturbation TheorySkew

σHSkew ∼ (τskew)-1 τ2~σ0 S

where S = Q(k,p)/Q(p,k) – 1~

Vν0 Im[<k|q><q|p><p|k>]

Vertex Corrections∼ σIntrinsic

Intrinsic∼σ0 /εFτ

n, q

n’≠n, q

n, q

n, q m, p

m, pn’, k

Anomalous Hall Effect

InterbandCoherent Response

∼ (EFτ) 0

Occupation # Response

`Skew Scattering‘∼(e2/h) kF (EF τ)1X `Skewness’

[Smidt] Intrinsic

`Berry Phase’∼(e2/h) kF

∼[Luttinger, Niu]

Influence of Disorder`Side Jump’’

[Berger]

Ferromagnets

INTRINSIC ANOMALOUS HALL EFFECTINTRINSIC ANOMALOUS HALL EFFECT

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).

In ferromagnetic systems where SO coupling is intrinsic and strong the Kubo formalism in the weak scattering limit and a semiclassical

treatment of Bloch wave-packet dynamics captures a nonzero anomalous Hall CONDUCTIVITY !! (The Berry’s phase in k-space approach)

( )[ ]

∑≠ −

−−='

2'

'

2

)(

'ˆˆ'Im]Re[

nnk knkn

yxnkknxy EE

knvknknvknff

Ve

r rv

rrrrhσ

∑ Ω−=nk

nknxy kfVe

r

r

h)(]Re[ '

2

σ

semiclassical

Kubo:

n, q

n’≠n, q

Success of intrinsic AHE approachSuccess of intrinsic AHE approach

Science 303, 1647 (2004)

• 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).

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.

ExperimentσAH ∼ 1000 (Ω cm)-1

TheroyσAH ∼ 750 (Ω cm)-1

OUTLINEOUTLINEMetal and semiconductor based spintronicsMetal and semiconductor based spintronicsSpinSpin--orbit coupling in semiconducting systemsorbit coupling in semiconducting systemsAnomalous Hall effect and Spin Hall effectAnomalous Hall effect and Spin Hall effect

Parsing the AHEParsing the AHEPerturbation theory treatmentPerturbation theory treatment

Parsing the SHE (example: 2D systems)Parsing the SHE (example: 2D systems)Perturbation theory treatmentPerturbation theory treatmentExact diagonalization calculationsExact diagonalization calculations

Spin Hall spin accumulationSpin Hall spin accumulationExact diagonalization studies (bulk)Exact diagonalization studies (bulk)Mesoscopic systemsMesoscopic systems

ConclusionsConclusions

Spin Hall effectSpin Hall effect

Spin-orbit coupling “force” deflects likelike--spinspin particles

I

_ FSO

FSO

_ __

V=0

non-magnetic

SpinSpin--current generation in noncurrent generation in non--magnetic systems 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

Take now a PARAMAGNET instead of a FERROMAGNET:

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

Refs: Dyakonov and Perel (71), J. E. Hirsch (99)

Spin Hall Effect(Dyaknov and Perel)

InterbandCoherent Response

∼ (EFτ) 0

Occupation # Response

`Skew Scattering‘∼(e2/h) kF (EF τ)1X `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

SHE conductivity– Perturbation Theory

Skew∼σ0 S

Vertex Corrections∼ σIntrinsic

Intrinsic∼σ0 /εFτ

n, q

n’≠n, q

= jν = -e vν

= jνz = vν,sz

INTRINSIC SPININTRINSIC SPIN--HALL EFFECT: 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!!!

kmkkk

mkH xyyxk

rrhh×+=−+= σλσσσλσ 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’ spinUniversal’ spin--Hall conductivityHall conductivity

<

=>=

*22*

2

2

4

22*22

sH

for8

for8

DDD

D

DD

xynn

nne

mnne

π

πλ

πσ h

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

mkkk

mkH xyyxk

rrhh×+=−+= σλσσσλσ 0

22

0

22

2)(

2

n, q

n’≠n, q

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

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

Inoue, Bauer, Molenkamp PRB 04

Ladder partial sum vertex correction:

Also:Mishchenko et al, PRL 04Raimondi et al, PRB 04, Dimitrova PRB05, Loss et al, PRB 05

λλ ~→

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

Generalized Rashba Models k1 Rashba: g=constant α = 1k3 Rashba: g=constant α = 3

Modified k1 Rashba: g∼k2 α = 1

ν

Nomura et al. cond-mat/0506189,

2DEG+Rahsba

2DHG+RahsbaSchliemann et al PRB 05

SHE conductivity 2D Rashba systems–Perturbation Theory

Skew∼σ0 S

Vertex Corrections∼ σIntrinsic

Intrinsic∼σ0 /εFτ

n, q

n’≠n, q

= jν = -e vν

= jνz = vν,sz

Kubo formula calculations

⟩=⟩+ nEnVH nR ||)(

ησ νµ

µν iEEnjnnjn

EEEfEf

Vi

nnnn

nn

nn +−

⟩⟩⟨⟨

−−

−= ∑''

'

',

||''||)()(

k^1 Rashba model

),(limlim0

ησ µνηL

L ∞→→

Nomura et al. cond-mat/0506189, supersedes PRB 71, 041304 (2005).

Spin-Hall Conductivities

4.6=∆τ

k^3 Rashba modelk^1 Rashba model

Numerical k1 Rashba Model

Data Collapse

Numerical k3 Rashba Model

Is SHE conductivity is zero for 2DEG-Rashba for any τ: reason linked to spin-dynamics for systems with linear SO coupling (Dimitrova 05, Loss et al 05) which demand a zero spin-current in the steady state.

σSHE=0 for 2DEG-Rashba σSHE≠0 for 2DHG-Rashba

OUTLINEOUTLINEMetal and semiconductor based spintronicsMetal and semiconductor based spintronicsSpinSpin--orbit coupling in semiconducting systemsorbit coupling in semiconducting systemsAnomalous Hall effect and Spin Hall effectAnomalous Hall effect and Spin Hall effect

Anomalous Hall effect and spin Hall effect (SHE)Anomalous Hall effect and spin Hall effect (SHE)Extrinsic and Intrinsic SHE Extrinsic and Intrinsic SHE

Systematic view of the AHE and SHESystematic view of the AHE and SHEParsing the AHEParsing the AHE

Perturbation theory treatmentPerturbation theory treatmentParsing the SHE (example: isotropic 2D models)Parsing the SHE (example: isotropic 2D models)

Perturbation theory treatmentPerturbation theory treatmentExact diagonalization calculationsExact diagonalization calculations

Spin Hall spin accumulationSpin Hall spin accumulationExact diagonalization studies (bulk)Exact diagonalization studies (bulk)Mesoscopic systemsMesoscopic systems

ConclusionsConclusions

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 PrecessionLength

?

SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION STUDIES

∆so>>ħ/τ

Width>>mean free path

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

SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION STUDIES

OUTLINEOUTLINEMetal and semiconductor based spintronicsMetal and semiconductor based spintronicsSpinSpin--orbit coupling in semiconducting systemsorbit coupling in semiconducting systemsAnomalous Hall effect and Spin Hall effectAnomalous Hall effect and Spin Hall effect

Anomalous Hall effect and spin Hall effect (SHE)Anomalous Hall effect and spin Hall effect (SHE)Extrinsic and Intrinsic SHE Extrinsic and Intrinsic SHE

Systematic view of the AHE and SHESystematic view of the AHE and SHEParsing the AHEParsing the AHE

Perturbation theory treatmentPerturbation theory treatmentParsing the SHE (example: isotropic 2D models)Parsing the SHE (example: isotropic 2D models)

Perturbation theory treatmentPerturbation theory treatmentExact diagonalization calculationsExact diagonalization calculations

Spin Hall spin accumulationSpin Hall spin accumulationExact diagonalization studies (bulk)Exact diagonalization studies (bulk)Mesoscopic systemsMesoscopic systems

Conclusions and things to think aboutConclusions and things to think about

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

Spin Hall effect in the mesoscopic regime, simplifying the debate

Hankiewicz, Molenkamp, Jungwirth, Sinova, PRB 70, 241301(R) (2004).

Also: Sheng et al, PRL 05Nikolic et al, PRB 05, PRL 05

PRL 05

ConclusionConclusion• SHE is now a reality.•Intrinsic SHE: things are clearing

•2DEG Rashba model SHE is zero in the bulk (no more controversy); but is finite in mesoscopic systems.•Intrinsic spin Hall spin accumulation in 2DHG system is consistent with experiments; 2DHG systems are promising.•SHE is complicated because it is small.

NEXTNEXTWhat about interactions (spin Coulomb drag)?What about interactions (spin Coulomb drag)?What is the most appropriate definition of spin What is the most appropriate definition of spin current (in what regime)?current (in what regime)?What is the relation between the mesoscopic What is the relation between the mesoscopic regime and bulk regime (similar to Stone et al?)regime and bulk regime (similar to Stone et al?)How can we measure SHE it in transport?How can we measure SHE it in transport?How dissipative can this transport really be?How dissipative can this transport really be?What is its relation (if any) to What is its relation (if any) to AharanovAharanov--Cashier Cashier effect or Aeffect or A--AnandanAnandan effect?effect?How does sideHow does side--jump affect SHE in other systems jump affect SHE in other systems besides systems with SO coupling linear in besides systems with SO coupling linear in momentum?momentum?……

WHAT IS YOUR FEELING ABOUT THE SHE? 1 Year ago

WHAT IS YOUR FEELING ABOUT THE SHE? today