Spin-Hall effect and anomalous transport in strongly
spin-orbit coupled systems
JAIRO SINOVA
Collaborators: Allan MacDonald, Kentaro Nomura, Joerg
Wunderlich, Bernd Kästner, Tomas Jungwirth,
Nikolai Sinitsyn, Laurens Molenkamp, Branislav
Nikolic, Satofumi Souma, Liviu Zarbo, Dimitri Culcer
, Ewelina Hankeiwc, Qian Niu, S-Q Shen
APCTP, August 8th 2005
OUTLINE
„
Anomalous Hall effect and Spin Hall effect
„ Parsing the AHE
„
„
Parsing the SHE (example: 2D systems)
„
„
„
Perturbation theory treatment
Exact diagonalization calculations
Spin Hall spin accumulation
„
„
„
Perturbation theory treatment
Exact diagonalization studies (bulk)
Mesoscopic systems
Conclusions and things for thought
OUTLINE
„
„
Spin-orbit coupling in semiconducting systems
Anomalous Hall effect and Spin Hall effect
„ Parsing the AHE
„
„
Parsing the SHE (example: 2D systems)
„
„
„
Perturbation theory treatment
Exact diagonalization calculations
Spin Hall spin accumulation
„
„
„
Perturbation theory treatment
Exact diagonalization studies (bulk)
Mesoscopic systems
Conclusions
Anomalous Hall effect
Spin-orbit coupling “force” deflects like-spin particles
majority
__ FSO
_
FSO
I
ρH = R0 B + 4πRs M
minority
V
InMnAs
Simple electrical measurement
of magnetization
Anomalous Hall Effect
Linear Response
Theory
Order of
Limits
(thanks to P. Bruno– CESAM talk)
(thanks to P. Bruno– CESAM talk)
(thanks to P. Bruno– CESAM talk)
Perturbation Theory
=
Real Eigenstates
+
Bloch Electron
Averaging procedures:
= τ-1 / ν0
= ν0 / τ-1
Perturbation Theory: conductivity
n, q
n, q
Drude Conductivity
σ = ne2τ/m*
Vertex Corrections
∼ 1-cos(θ)
= jν = -evν
AHE – Perturbation Theory
n, q n’, k m, p
m, p
n, q
n, q
σHSkew
Skew
∼ (τskew)-1 τ2~σ0 S
where
S = Q(k,p)/Q(p,k) – 1~
Vν0 Im[<k|q><q|p><p|k>]
Intrinsic
∼σ0 /εFτ
n’≠n, q
Vertex Corrections
∼ σIntrinsic
Anomalous Hall Effect
Interband
Coherent Response
Occupation #
Response
∼ (EFτ) 0
`Skew Scattering‘
∼(e2/h) kF (EF τ)1
X `Skewness’
Intrinsic
`Berry Phase’
∼(e2/h) kF
∼
[Luttinger, Niu]
[Smidt]
Influence of Disorder
`Side Jump’’
[Berger]
Ferromagnets
INTRINSIC ANOMALOUS HALL EFFECT
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)
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).
semiclassical
r
e2
f n 'k Ω n ( k )
Re[σ xy ] = −
∑
r
Vh k n
n, q
Kubo:
Im
e 2h
Re[σ xy ] = −
∑ ( f n 'k − f nk )
V kr n ≠ n '
[
r
r r
r
n' k vˆ x nk nk vˆ y n' k
( Enkv − En 'kr ) 2
]
n’≠n, q
Success of intrinsic AHE approach
• DMS systems (Jungwirth et al PRL 2002)
Experiment
• Fe (Yao et al PRL 04)
σAH ∼ 1000 (Ω cm)-1
• layered 2D ferromagnets such as SrRuO3 and pyrochlore
Theroy
ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19
σ ∼ 750 (Ω cm)-1
•
(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).
AH
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.
OUTLINE
„
„
„
Metal and semiconductor based spintronics
Spin-orbit coupling in semiconducting systems
Anomalous Hall effect and Spin Hall effect
„ Parsing the AHE
„
„
Parsing the SHE (example: 2D systems)
„
„
„
Perturbation theory treatment
Exact diagonalization calculations
Spin Hall spin accumulation
„
„
„
Perturbation theory treatment
Exact diagonalization studies (bulk)
Mesoscopic systems
Conclusions
Spin Hall effect
Take now a PARAMAGNET instead of a FERROMAGNET:
Spin-orbit coupling “force” deflects like-spin particles
_
FSO
__
FSO
non-magnetic
I
V=0
Carriers with same charge but opposite spin are deflected
by the spin-orbit coupling to opposite sides.
Spin-current generation in non-magnetic systems
without applying external magnetic fields
Spin accumulation without charge accumulation
excludes simple electrical detection
Refs: Dyakonov and Perel (71), J. E. Hirsch (99)
Spin Hall Effect
(Dyaknov and Perel)
Interband
Coherent Response
Occupation #
Response
∼ (EFτ) 0
`Skew Scattering‘
∼(e2/h) kF (EF τ)1
X `Skewness’
Intrinsic
`Berry Phase’
∼(e2/h) kF
∼
[Murakami et al,
Sinova et al]
[Hirsch, S.F. Zhang]
Influence of Disorder
`Side Jump’’
[Inoue et al, Misckenko et
al, Chalaev et al…]
Paramagnets
SHE conductivity– Perturbation Theory
Skew
∼σ0 S
n, q
n’≠n, q
Intrinsic
∼σ0 /εFτ
Vertex Corrections
∼ σIntrinsic
= jν = -e vν
= jνz = {vν,sz}
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!!!
n, q
(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’≠n, q
Inversion symmetry
⇒ no R-SO
Broken inversion symmetry
⇒ R-SO
h2k 2
h2k 2
r r
Hk =
σ 0 + λ (k xσy − k yσ x ) =
σ 0 + λσ × k
2m
2m
Bychkov and Rashba (1984)
‘Universal’ spin-Hall conductivity
n, q
n’≠n, q
σ xysH
Color plot of spin-Hall conductivity:
yellow=e/8π and red=0
 e
m 2 λ2
*
 for n2 D > n2 D =
4
8
π
π
h
=
e n2 D
*

for
n
n
<
2D
2D
 8π n2* D
h2k 2
h2k 2
r r
Hk =
σ 0 + λ (k xσy − k yσ x ) =
σ 0 + λσ × k
2m
2m
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?
Inoue, Bauer, Molenkamp PRB 04
Ladder partial sum vertex correction:
Also:
Mishchenko et al, PRL 04
Raimondi et al, PRB 04, Dimitrova
PRB05, Loss et al, PRB 05
~
λ →λ
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 = 1
k3 Rashba: g=constant α = 3
Modified k1 Rashba: g∼k2 α = 1
ν
Nomura et al. cond-mat/0506189,
2DEG+Rahsba
2DHG+Rahsba
Schliemann et al PRB 05
SHE conductivity 2D Rashba systems–
Perturbation Theory
Skew
∼σ0 S
n, q
n’≠n, q
Intrinsic
∼σ0 /εFτ
Vertex Corrections
∼ σIntrinsic
= jν = -e vν
= jνz = {vν,sz}
Kubo formula calculations
σ µν
i
=−
V
∑
n ,n '
f ( E n ) − f ( E n ' ) ⟨ n | j µ | n ' ⟩⟨ n ' | jν | n ⟩
E n − E n'
E n − E n ' + iη
( H R + V ) | n⟩ = E n | n⟩
lim lim σ
η → 0 L→ ∞
Nomura et al. cond-mat/0506189,
supersedes PRB 71, 041304 (2005).
µν
( L ,η )
k^1 Rashba model
Spin-Hall Conductivities
k^1 Rashba model
k^3 Rashba model
∆τ = 6.4
Numerical k1 Rashba Model
Numerical k3 Rashba Model
Data Collapse
σSHE=0 for 2DEG-Rashba
σSHE≠0 for 2DHG-Rashba
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.
OUTLINE
„
Metal and semiconductor based spintronics
Spin-orbit coupling in semiconducting systems
Anomalous Hall effect and Spin Hall effect
„
Anomalous Hall effect and spin Hall effect (SHE)
„ Extrinsic and Intrinsic SHE
Systematic view of the AHE and SHE
„
„
„
„
Parsing the AHE
„
„
Parsing the SHE (example: isotropic 2D models)
„
„
„
Perturbation theory treatment
Exact diagonalization calculations
Spin Hall spin accumulation
„
„
„
Perturbation theory treatment
Exact diagonalization studies (bulk)
Mesoscopic systems
Conclusions
Spin Accumulation
Spin is not conserved;
analogy with e-h system
Spin Accumulation – Weak SO
Quasi-equilibrium
Parallel conduction
Spin diffusion length
Burkov et al. PRB 70 (2004)
Spin Accumulation – Strong SO
?
Mean Free
Path?
Spin Precession
Length
SPIN ACCUMULATION IN 2DHG:
EXACT DIAGONALIZATION STUDIES
∆so>>ħ/τ
Width>>mean free path
SPIN ACCUMULATION IN 2DHG:
EXACT DIAGONALIZATION STUDIES
Key length: spin precession length!!
Independent of τ !!
OUTLINE
„
Metal and semiconductor based spintronics
Spin-orbit coupling in semiconducting systems
Anomalous Hall effect and Spin Hall effect
„
Anomalous Hall effect and spin Hall effect (SHE)
„ Extrinsic and Intrinsic SHE
Systematic view of the AHE and SHE
„
„
„
„
Parsing the AHE
„
„
Parsing the SHE (example: isotropic 2D models)
„
„
„
Perturbation theory treatment
Exact diagonalization calculations
Spin Hall spin accumulation
„
„
„
Perturbation theory treatment
Exact diagonalization studies (bulk)
Mesoscopic systems
Conclusions and things to think about
SHE in the mesoscopic regime
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
Spin Hall effect in the mesoscopic
regime, simplifying the debate
Hankiewicz, Molenkamp, Jungwirth, Sinova, PRB 70, 241301(R) (2004).
Also: Sheng et al, PRL 05
Nikolic et al, PRB 05, PRL 05
PRL 05
Conclusion
• 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.
NEXT
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„
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What about interactions (spin Coulomb drag)?
What is the most appropriate definition of spin
current (in what regime)?
What is the relation between the mesoscopic
regime and bulk regime (similar to Stone et al?)
How can we measure SHE it in transport?
How dissipative can this transport really be?
What is its relation (if any) to Aharanov-Cashier
effect or A-Anandan effect?
How does side-jump affect SHE in other systems
besides systems with SO coupling linear in
momentum?
…
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