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.