Transcript INTRINSIC SPIN

Anomalous and Spin-Hall effects in mesoscopic systems
JAIRO SINOVA
ICNM 2007, Istanbul, Turkey July 25th 2007
Research fueled by:
NERC
Mario Borunda
Texas A&M U.
Tomas Jungwirth
Inst. of Phys. ASCR
U. of Nottingham
Allan MacDonald
U of Texas
Alexey Kovalev
Texas A&M U.
Nikolai Sinitsyn
Texas A&M U.
U. of Texas
Joerg Wunderlich Laurens Molenkamp
Wuerzburg
Cambridge-Hitachi
Ewelina Hankiewicz
U. of Missouri
Texas A&M U.
Kentaro Nomura
U. Of Texas
Branislav Nikolic
U. of Delaware
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
OUTLINE


The three spintronic Hall effects
Anomalous Hall effect and Spin Hall effect









AHE phenomenology and its long history
Three contributions to the AHE
Microscopic approach: focus on the intrinsic AHE
Application to the SHE
SHE in Rashba systems: a lesson from the past
Recent experimental results
Spin Hall spin accumulation: bulk and mesoscopic
regime
Mesoscopic spin Hall effect: non-equilibrium Green’s
function formalism and recent experiments
Summary
The spintronics Hall effects
SHE
charge current
gives
spin current
AHE
polarized charge
current gives
charge-spin
current
SHE-1
spin current
gives
charge current
Anomalous Hall transport
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. nonexchange 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 interaction
(one of the few echoes of relativistic physics in the solid state)
Ingredients: -“Impurity” potential V(r)
- Motion of an electron
Produces
an electric field

1
E   V (r )
e
In the rest frame of an electron
the electric field generates and
effective magnetic field


 k  
 E
Beff  

 cm 
This gives an effective interaction with the electron’s magnetic moment
H SO


 
 eS   k  1 dV (r ) 
 r
    Beff  
   S  L

 mc   mc  er dr 
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
started, the long debate
Spin-orbit coupling “force” deflects like-spin particles
majority
__ FSO
_
FSO
I
 H  R0 B  4πRs M
minority
V
Simple electrical measurement
of magnetization
InMnAs
controversial theoretically: semiclassical theory identifies three
contributions (intrinsic deflection, skew scattering, side jump scattering)
A history of controversy
(thanks to P. Bruno–
CESAM talk)
Intrinsic deflection
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)
Movie created
by Mario
Borunda
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.
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.
Movie created by Mario Borunda
Side-jump scattering
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
Movie created by Mario Borunda
THE THREE CONTRIBUTIONS TO THE AHE:
MICROSCOPIC KUBO APPROACH
Skew scattering
n, q n’, k m, p
n, q
m, p
Skew
σHSkew  (skew)-1 2~σ0 S where
S = Q(k,p)/Q(p,k) – 1~
V0 Im[<k|q><q|p><p|k>]
Averaging procedures:
Side-jump scattering
Intrinsic AHE
Vertex Corrections
 σIntrinsic
n, q
n’n, q
Intrinsic
σ0 /εF
= -1 / 0
= 0 
FOCUS ON INTRINSIC AHE: semiclassical and Kubo
STRATEGY: compute this contribution in strongly SO coupled
ferromagnets and compare to experimental results, does it work?
n, q
Kubo:
Im
e
Re[ xy ]  
  f n'k  f nk 
V k n  n '
2


 

ˆ
ˆ
n' k vx nk nk v y n' k
( Enk  En 'k ) 2
n’n, q
Semiclassical approach in the “clean limit”

e2
Re[ xy ]   
f n 'k  n ( k )

V k n
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).

Success 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
Theroy
AH  750 ( cm)-1
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
Spin Hall Effect
(Dyaknov and Perel)
Interband
Coherent Response
Occupation #
Response
 (EF) 0
`Skew Scattering‘
(e2/h) kF (EF )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
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
n’n, q
Inversion symmetry
 no R-SO
(differences: spin is a
non-conserved quantity,
define spin current as the
gradient term of the
continuity equation. SpinHall conductivity: linear
response of this operator)
2 2
e
m

*
for
n

n

2D
2D

4
sH
8



 xy  
e n2 D
*

for
n

n
2D
2D
 8 n2* D
Broken inversion symmetry
 R-SO
2k 2
2k 2
 
Hk 
 0   (k xy  k y x ) 
 0    k
2m
2m
Bychkov and Rashba (1984)
SHE conductivity: Kubo formalism
perturbation theory
Skew
σ0 S
n, q
n’n, q
Intrinsic
σ0 /εF
Vertex Corrections
 σIntrinsic
= j = -e v
= jz = {v,sz}
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?
n, q
+…=0
+
n’n, q
For the Rashba example the side jump contribution cancels the intrinsic contribution!!
Inoue et al PRB 04
Raimondi et al PRB 04
Mishchenko et al PRL 04
Loss et al, PRB 05
Ladder partial sum vertex correction:
~
 
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
First experimental observations 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:
~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:
~0.03% polarization (weaker SO-coupling, stronger disorder)
OTHER RECENT EXPERIMENTS
Transport observation of the SHE by spin injection!!
Saitoh et al
APL 06
Sih et al, Nature 05, PRL 05
“demonstrate that the observed spin accumulation is due to a
transverse bulk electron spin current”
Valenzuela and
Tinkham condmat/0605423,
Nature 06
The new challenge: understanding 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/
Width>>mean free path
Nomura, Wundrelich
et al PRB 06
Key length: spin precession length!!
Independent of  !!
n
p
1.5m
channel
LED1
0
y
-1
z
n LED2
x
1
0
-1
1.505
1.510
1.515
1.520
Energy in eV
Wunderlich, Kaestner, Sinova,
Jungwirth, Phys. Rev. Lett. '05
10m channel
- 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)
Polarization in %
1
Nomura, Wunderlich, Sinova, Kaestner,
MacDonald, Jungwirth, Phys. Rev. B '06
Polarization in %
SHE experiment in
GaAs/AlGaAs 2DHG
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
Nonequilibrium Spin Hall Accumulation in Rashba 2DEG
+eV/2
eV=0
-eV/2
Y. K. Kato, R. C. Myers, A. C.
Gossard, and D.D. Awschalom,
Science 306, 1910 (2004).
y
z
x
PRL 95, 046601 (2005)
 Spin density (Landauer –Keldysh):
HgTe
band structure
semi-metal or semiconduct
1000
E (meV)
500
8
0
6
-500
-1000
7
-1500
-1.0
-0.5
0.0
k (0.01
0.5
1.0
)
fundamental energy gap
D.J. Chadi et al. PRB, 3058 (1972)
E  6  E 8  300 meV
HgTe-Quantum Well Structures
QW
Barrier
HgTe
Hg0.32Cd0.68Te
1000
1000
6
500
8
0
0
VBO
8
6
-500
-500
-1000
-1000
7
-1500
-1.0
-0.5
0.0
k (0.01 )
7
0.5
1.0-1.0
-0.5
0.0
0.5
1.0
-1500
k (0.01 )
VBO = 570 meV
E (meV)
E (meV)
500
HgTe-Quantum Well Structures
Typ-III QW
conduction band
6
6
HgTe
HgCdTe
HgCdTe
8
valence band
QW < 55 Å
HgTe
HgCdTe
E1
HH1
VBO = 570 meV
8
inverterted
normal
band structure
High Electron Mobility
15000
Q2134a_Gate
500
11
nHall=4.01*10 cm
6
-2
2
µ=1.06*10 cm (Vs)
400
10000
-1
300
0
200
-5000
100
-10000
0
-15000
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
B[T]
Graph1
3
4
5
6
7
8
Rxy[]
Rxx[]
5000
H-bar for detection of Spin-Hall-Effect
(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
GateContact
ohmic Contacts
First Data
HgTe-QW
R = 5-15 meV
Signal due to
depletion?
Results...
Symmetric HgTe-QW
R = 0-5 meV
I: 1->4
U:7-10
-1.00E-007
U_7-10 [V]
Signal less
than 10-4
-5.00E-008
-1.50E-007
-2.00E-007
-2.50E-007
-3.00E-007
-2
-1
0
1
2
3
-V_gate14 [V]
Sample is diffusive:
Vertex correction kills SHE (J. Inoue et al., Phys. Rev. B 70, 041303 (R) (2004)).
New (smaller) sample
sample layout
200 nm
1 m
SHE-Measurement
7000
5
insulating
n-conducting
p-conducting
Rnonlocal / 
5000
4
Q2198H
I (3,6)
U (9,11)
4000
3
3000
2
I / nA
6000
2000
1000
1
0
0
-2
-1
strong increase
of the signal in the
p-conducting regime,
with pronounced features
0
1
2
VGate / V
3
4
5
no signal
in the
n-conducting
regime
Mesoscopic electron SHE
calculated voltage signal for electrons
(Hankiewicz and Sinova)
L/2
L/6
L
Mesoscopic
hole
SHE
calculated voltage signal
(Hankiweicz, Sinova, & Molenkamp)
L
L/2
L/
6
L
Scaling of H-samples with the system size
L=90nm
change of voltage [V]
5.0
4.5
4.0
Spin orbit coupling
= 72meVnm
n=1*1011cm-2
L/6
3.5
L=150nm
3.0
L=240nm
L
2.5
2.0
0.000
L=120nm
L=200nm
0.004
0.008
0.012
1/L [nm]
Oscillatory character of voltage difference with the system size.
SUMMARY
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
Optical detection of current-induced polarization
photoluminescence (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
WHERE WE ARE GOING (EXPERIMENTS)
Experimental achievements
Optical detection of current-induced polarization
photoluminescence (bulk and edge 2DHG)
Kerr/Faraday rotation (3D bulk and edge, 2DEG)
Transport detection of the SHE
Experimental (and experiment modeling) challenges:
General
edge electric field (Edelstein) vs. SHE induced spin accumulation
Photoluminescence cross section
edge electric field vs. SHE induced spin accumulation
free vs. defect bound recombination
spin accumulation vs. repopulation
angle-dependent luminescence (top vs. side emission)
hot electron theory of extrinsic experiments
SHE detection at finite frequencies
detection of the effect in the “clean” limit
INTRINSIC+EXTRINSIC: STILL CONTROVERSIAL!
AHE in Rashba systems with disorder:
Dugaev et al PRB 05
Sinitsyn et al PRB 05
Inoue et al (PRL 06)
Onoda et al (PRL 06)
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
Connecting Microscopic and
Semiclassical approach
Sinitsyn et al PRL 06, PRB 06



Need to match the Kubo to the Boltzmann
Kubo: systematic formalism
Botzmann: easy physical interpretation of
different contributions
Kubo-Streda formula summary
σxy =σ +σ
I
xy
II
xy
Semiclassical Boltzmann equation
fl
fl
 eE
   l ' l ( f l  f l ' )
t
k
l'
e2 +  df(ε)
σ =-  dε
Tr[v x (G R -G A )v y G A 4π - 
dε
-v x G R v y (G R -G A )]
I
xy
Golden rule:
e2 + 
dG R
R
σ =
dεf(ε)Tr[v x G v y


4π
dε
dG R
dG A
dG A
R
A
-v x
v y G -v x G v y
+v x
v yG A ]
dε
dε
dε
II
xy
l 'l 
2
| Tl 'l |2  ( l '   l )
In metallic regime:
Vl 'l ''Vl ''l
Tl 'l  Vl 'l 
 ...
 l '   l ''  i
J. Smit (1956):   
l 'l
ll '
Skew Scattering
Semiclassical approach II: Sinitsyn et al PRB 06
Golden Rule:
l 'l 
2
| Vl 'l |2  ( l '   l )
l  ( , k )
Vl 'l  Tl 'l
Modified
Boltzmann fl  eEv f 0 ( l )   f 0 ( l ) eE r    ( f  f )


l
l 'l
l 'l
l 'l
l
l'
 l


Equation: t
l'
l'
l
 l
 F l  eE   l 'l rl 'l
velocity: vl 
k
l'
Sinitsyn et al PRB 06
 u u
ul ul
l
l
F  Im 

Berry curvature:
 k y k x
k x k y

l
z
Coordinate shift:

 current: J  e f l vl

l



 rl 'l  ul ' i
ul '  ul i
ul  Dˆ k ',k arg Vl 'l 
k '
k
Some success in graphene
EF
Armchair edge
Zigzag edge
Single K-band with spin up
H K  =v(k x σ x +k y σ y )+Δso σ z
Kubo-Streda
formula:
2
σxy =σ +σ
I
xy
II
xy
+
e
df(ε)
R
A
A
dε
Tr[v
(G
-G
)v
G
x
y
4π - 
dε
-v x G R v y (G R -G A )]
σ Ixy =-
In metallic regime:
-e 2 Δso
σ =
I
xy
4π
(vk F ) +Δ so
2
2
e2 + 
dG R
R
σ =
dεf(ε)Tr[v x G v y
4π -
dε
dG R
dG A
dG A
R
A
-v x
v y G -v x G v y
+v x
v yG A ]
dε
dε
dε
II
xy
σ IIxy =0

2
4
4(vk
)
3(vk
)
1+
F
F
+
2
 (vk ) 2 +4Δ
2
F
so
(vk F ) 2 +4Δ so



 e2 V3

2 
2
 2πn V

so (vk F ) 4
 (vk ) +4Δ 
2 2
2
F
so
Sinitsyn et al PRL 06, PRB 06 SAME RESULT OBTAINED USING BOLTMANN!!!
Comparing Boltzmann to Kubo in the chiral basis
fl
f 0 ( l )
f 0 ( l )
 eEvl
  l 'l
eE rl 'l   l 'l ( fl  fl ' )
t
 l
 l
l'
l'
Intrinsic deflection
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 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.
Side jump scattering
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.
Skew scattering
Asymmetric scattering due to the spinorbit coupling of the electron or the
impurity. This is also known as Mott
scattering used to polarize beams of
particles in accelerators.