Transcript Spin Hall Effect

Anomalous Hall effects :
Merging The Semiclassical and Microscopic Theories
JAIRO SINOVA
Karlsruhe, Germany, July 16th 2008
Research fueled by:
NERC
OUTLINE
Anomalous Hall effect:
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: three contributions to the AHE
(intrinsic deflection, skew scattering, side jump scattering)
A history of controversy
(thanks to P. Bruno–
CESAM talk)
Anomalous Hall effect: what is
necessary to see the effects?
majority
_
__
FSO
FSO
I
minority
V
Necessary condition for AHE: TIME REVERSAL SYMMETRY MUST BE BROKE
 
 
 xy ( B,M )   xy ( B, M )
Need a magnetic field and/or magnetic order
BUT IS IT SUFFICIENT?
(P. Bruno– CESAM 2005)
Local time reversal symmetry being broken
does not always mean AHE present
Staggered flux with zero average flux:
-  -
 - 
 - 
-  -
Is xy zero or non-zero?
Translational invariant so xy =0
Similar argument follows for antiferromagnetic ordering
Does zero average flux necessary mean zero xy ?
- 3 - 3
- - - -
No!!
(Haldane, PRL 88)
(P. Bruno– CESAM 2005)
Is non-zero collinear magnetization sufficient?
In the absence of spin-orbit coupling a spin
rotation of  restores TR symmetry and xy=0
If spin-orbit coupling is present there is no
invariance under spin rotation and xy≠0
(P. Bruno– CESAM 2005)
Collinear magnetization AND spin-orbit coupling → AHE
Does this mean that without spin-orbit coupling one cannot get AHE?
No!! A non-trivial chiral magnetic structure WILL give AHE
even without spin-orbit coupling
Mx=My=Mz=0
xy≠0
(P. Bruno– CESAM
July 2005)
Even non-zero magnetization is not a necessary condition
Bruno et al PRL 04
COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING
vs. CHIRAL MAGNET STRUCTURES
AHE is present when SO coupling and/or non-trivial spatially
varying magnetization (even if zero in average)
SO coupled chiral states:
disorder and electric fields lead to
AHE/SHE through both intrinsic
and extrinsic contributions
Spatial dependent magnetization: also can
lead to AHE. A local transformation to the
magnetization direction leads to a nonabelian gauge field, i.e. effective SO
coupling (chiral magnets), which mimics
the collinear+SO effective Hamiltonian in
the adiabatic approximation
So far one or the other have been considered but not both together, in
the following we consider only collinear magnetization + SO coupling
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
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.
Microscopic vs. Semiclassical
 Need to match the Kubo, Boltzmann, and Keldysh
 Kubo: systematic formalism
 Boltzmann: easy physical interpretation of
different contributions (used to define them)
 Keldysh approach: also a systematic kinetic
equation approach (equivelnt to Kubo in the linear
regime). In the quasiparticle limit it must yield
Boltzmann eq.
THE THREE CONTRIBUTIONS TO THE AHE:
MICROSCOPIC KUBO APPROACH
Skew scattering
n, q n’, k m, p
σHSkew
Skew
 (skew)-1 2~σ0 S
where
S = Q(k,p)/Q(p,k) – 1~
m, p
n, q
V0 Im[<k|q><q|p><p|k>]
Side-jump scattering
Vertex Corrections
 σIntrinsic
Intrinsic AHE: accelerating between scatterings
n, q
n’n, q
Intrinsic
σ0 /εF
FOCUS ON INTRINSIC AHE (early 2000’s): 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 in comparing to
experiment: phenomenological “proof”
• DMS systems (Jungwirth et al PRL 2002, APL 03)
• Fe (Yao et al PRL 04)
• layered 2D ferromagnets such as SrRuO3 and pyrochlore
ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19
•
AHE in GaMnAs
(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).
• CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004)
Berry’s phase based AHE effect is quantitativesuccessful in many instances BUT still not a
theory that treats systematically intrinsic and
extrinsic contribution in an equal footing
AHE in Fe
Experiment
AH  1000 ( cm)-1
Theroy
AH  750 ( cm)-1
INTRINSIC+EXTRINSIC: REACHING THE
END OF A DECADES LONG DEBATE
AHE in Rashba systems with weak disorder:
Dugaev et al (PRB 05)
Sinitsyn et al (PRB 05, PRB 07)
Inoue et al (PRL 06)
Onoda et al (PRL 06, PRB 08)
Borunda et al (PRL 07), Nuner et al (PRB 07, PRL 08)
Kovalev et al (PRB 08)
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 ALL the different equivalent linear
response theories both in AHE and SHE
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
Calculation done easiest in normal spin basis
Semiclassical approach II:
Golden Rule:
Coordinate shift:
l 'l 
2
| Vl 'l |2  ( l '   l )
l  ( , k )
Vl 'l  Tl 'l


 rl 'l  ul ' i
ul '  ul i
ul  Dˆ k ',k arg Vl 'l 
k '
k
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
Sinitsyn et al PRB 06
Berry curvature:
velocity: vl 
 u u
ul ul
l
l
F  Im 

 k y k x
k x k y

l
z
 l
 F l  eE   l 'l rl 'l
k
l'




current: J  e fl vl
l
“AHE” 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
Sinitsyn et al PRB 07
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
Comparing Botlzmann to Kubo in the chiral basis
AHE in Rashba 2D system
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
2k 2
2k 2
 
Hk 
 0   (k xy  k y x ) 
 0    k
2m
2m
Bychkov and Rashba (1984)
AHE in Rashba 2D system
Kubo and semiclassical approach approach: (Nuner et al PRB08, Borunda et al PRL 07)
Only when ONE both sub-band
there is a significant contribution
When both subbands are occupied there is additional vertex corrections that contribute
AHE in Rashba 2D system
Keldysh and Kubo match analytically in the metallic limit
When both subbands are occupied the skew scattering is only obtained at higher
Born approximation order AND the extrinsic contribution is unique (a hybrid between
skew and side-jump)
Kovalev et al PRB 08
Numerical Keldysh approach (Onoda et al PRL 07, PRB 08)
G R  G0  G0  R G R
1
R
G


G
 0 R 1
G 
R 1
0
ˆ  
ˆ R  Gˆ   
ˆA



A 1
ˆ
ˆ
 G  G  G0   ˆ R  Gˆ   Gˆ   ˆ R  ˆ   Gˆ A  Gˆ R  ˆ 
Solved within the self consistent T-matrix approximation for the self-energy
AHE in Rashba 2D system: “dirty” metal limit?
Is it real? Is it justified? Is it “selective” data chosing?
Can the kinetic metal theory be justified when disorder is larger than any other scale?
OTHER ANOMALOUS HALL EFFECTS
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
(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
2k 2
2k 2
 
Hk 
 0   (k xy  k y x ) 
 0    k
2m
2m
Bychkov and Rashba (1984)
‘Universal’ spin-Hall conductivity
n, q
n’n, q
 xysH
Color plot of spin-Hall conductivity:
yellow=e/8π and red=0
 e
m 2 2
*
 for n2 D  n2 D 
4
8




e n2 D
*

for
n

n
2D
2D
 8 n2* D
SHE conductivity: all contributions–
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
Dimitrova et al PRB 05
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)
First experimental observations at the end of 2004
Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/0410295
PRL 05
1
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)
CP [%]
0
-1
1.505
1.52
Light frequency (eV)
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
QUANTUM SPIN HALL EFFECT
(Physics Today, Feb 2008)
Anomalous Hall effect in cold atoms
Unitary transformation
State vector in the pseudospin basis:
Diagonalization with local unitary transformation:
Pure gauge:
Interaction of atoms
with laser fields
U(m) adiabatic gauge field
We can then introduce
the adiabatic condition and reach the dynamical evolution of degenerate ground subspace
U(m) adiabatic
gauge field:
Berry phase, Spin-orbit coupling, etc.
Spin-orbit coupling and AHE in atoms
y
z
x
Two dark-states consist of the degenerate subspace (pseudospin-1/2 states):
Mixing angle:
U(2) adiabatic gauge field:
(see also Staunesco, Galitskii, et al PRL 07)
Effective trap potentials:
y
Employ Gaussian laser beams and set the trap:
x
-y
The effective Hamiltonian (
):
M0
AHE
E
Mario Borunda
Texas A&M U.
Tomas Jungwirth
Inst. of Phys. ASCR
U. of Nottingham
Allan MacDonald
U of Texas
Xin Liu
Alexey Kovalev Nikolai Sinitsyn
Texas A&M U. Texas A&M U.
LANL
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
EXTRAS
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!!
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 '05
Polarization in %
SHE experiment in
GaAs/AlGaAs 2DHG
3. Charge based measurements of SHE
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
PRL 05
H-bar for detection of Spin-Hall-Effect
(electrical detection through inverse SHE)
E.M. Hankiewicz et al ., PRB 70, R241301 (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
0
1
2
3
4
5
VGate / V
strong increase of the signal in the
p-conducting regime, with
pronounced features
no signal in the nconducting 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