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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.
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