Transcript Spin-orbit coupling

Hall effects and weak localization in strong SO coupled systems :
merging Keldysh, Kubo, and Boltzmann approaches via the chiral basis.
JAIRO SINOVA
Texas A&M Univ. and Inst. Phys. ASCR
SPIE, San Diego, August 12th 2008
Other collaborators:
Bernd Kästner,
Satofumi Souma, Liviu
Zarbo, Dimitri Culcer ,
Qian Niu, S-Q
Shen,,Tom Fox,
Richard Campton,
Artem Abanov
Brian Gallagher
Research fueled by:
Mario Borunda
Texas A&M U.
Tomas Jungwirth
Inst. of Phys. ASCR
U. of Nottingham
SWAN-NRI
Allan MacDonald
U of Texas
Xin Liu
Alexey Kovalev Nikolai Sinitsyn
Texas A&M U. Texas A&M U.
LANL
Ewelina Hankiewicz
U. of Missouri
Texas A&M U.
Joerg WunderlichLaurens Molenkamp Kentaro Nomura Branislav Nikolic
Wuerzburg
U. Of Texas
U. of Delaware
Cambridge-Hitachi
I. Anomalous Hall Effect
1. History, semi-classical mechanism
2. Microscopic approach, IAHE
3. Merging the different linear theories
a. AHE in graphene
b. AHE in 2DEG+Rashba
II. Spin Hall Effect
1. Spin accumulation with strong SO
III. Weak Localization in GaMnAs
1. The experimental observations
2. Theory results
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)
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.
A history of controversy
(thanks to P. Bruno–
CESAM talk)
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
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.
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 50 YEAR OLD 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
and THEN test it experimentally
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
H K  =v(k x σ x +k y σ y )+Δso σ z
Kubo-Streda σ =σI +σII
xy
xy
xy
formula:
II
In metallic regime: σ xy =0
-e 2 Δso
σ =
I
xy
4π
(vk F ) +Δ so
2
2
Sinitsyn et al PRB 07

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
Sinitsyn et al
PRB 07
-e 2 Δso
σ =
I
xy
4π
(vk F ) 2 +Δ so
2

2
3(vk F ) 4
1+ 4(vk F )
+
 (vk ) 2 +4Δ 2
2
F
so
(vk F ) 2 +4Δ so



 e2 V3

2 
2
 2πn V


so (vk F ) 4
(vk F ) 2 +4Δ so

2 2
A more realistic test
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
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 Rapids 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
Testing the theory: in progress
AHE in Rashba 2D system: “dirty” metal limit?
Onoda et al 2008
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?
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
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”
Room temperature SHE in ZnSe ??? Stern et al 06
(signal same as GaAs but SO smaller????)
Valenzuela and
Tinkham condmat/0605423,
Nature 06
The challenge: understanding spin accumulation in
strongly spin-orbit coupled systems
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
Charge based measurements of ISHE
H-bar for detection of Spin-Hall-Effect
(electrical detection through inverse SHE)
(Numerical Keldysh calculation: no SO in leads)
E.M. Hankiewicz et al ., PRB 70, R241301 (2004)
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
New (smaller) sample
sample layout
200 nm
1 m
Experiments by Laruens Molenkamp group
SHE-Measurement
SUMMARY (AHE AND SHE)
•All linear theories treating disorder and non-trivial band structure have been
merged in agreement
•Clear identification of semi-clasical contributions from the microscopic theory
•Many strongly spin-orbit coupled systems are dominated by the intrinsic
contribution: old side-jump+intrinsic cancellations were an artifact of simple band
structure (e.g. constant Berry curvature)
•Intrinsic SHE can also be observed in strongly spin-orbit coupled system with
the induced spin-accumulation length scale in agreement with theory
•Charge based detection of intrinsic SHE seen in inverted semiconductor
systems
SWAN-NRI
Weak Localization in GaMnAs
Quantum driven localization of time reversed paths
interference. Each spin channel adds to the localization.
In the presence of spin-orbit coupling one decouples
channels in total angular momentum states. Singlet (zero
total spin) is the one not affected BUT contributes with a
negative sign to diffusion, i.e. Weak Antilocalization.
e.g. 2D
Matsukura et al Physica E 2004
Weak Localization at high
magnetic fields
Low field MR dominated by
complicated AMR effects
Kawabata A., Solid State Commun. 34 (1980) 432
•Focus is on low magnetic field region where AMR dominates
•Rely on subtracting e-e interaction contribution which they attribute to the 1-D
theory proportional to T-1/2 . However they ignore that e-e contribution depends
on the conductivity and strong AMR contributions will influence it.
•1-D dimensionality is not quite justified given the length scales at the
temperatures considered
•Lso seems too large to have real meaning. For a strongly spin-orbit coupled
system is should be lower.
•High field contribution ignored
B2?
Rokhinson et al observed a ~1%
negative MR at low temperature in a
GaMnAs film which saturates at
~20mT and “states” that it is isotropic
in field (ignoring the clear AMR in the
data).
? ? ? ? ? ?
The magnitude of the WL is stronger
than the largest expected from the
simples theory.
One expects saturation at very large
fields, not present in their experiment
But they still ascribe this feature to
weak localisation and furthermore
argue that the presence of weak
localisation is incompatible with the
Fermi level being in strongly spin
orbit coupled valence band ??!!
But is their main basis even right?
Success of metallic disorder valence band
theory seems unimportant
• Ferromagnetic transition temperatures 
 Magneto-crystalline anisotropy and coercively 
 Domain structure 
 Anisotropic magneto-resistance 
 Anomalous Hall effect 
 MO in the visible range 
 Non-Drude peak in longitudinal ac-conductivity 
• Ferromagnetic resonance 
• Domain wall resistance 
• TAMR 
Theory of WL in GaMnAs
Unlike the case of time-reversal symmetric systems there are no obvious
Invariant representation when the energy scales are similar (exchange field,
disorder, spin-orbit coupling, etc.)
Key result: for typical doping values and disorder WL is present!!!
The main point is b/c disorder affects most the inter-band correlations which in the
case of GaMnAs dominates the WAL contribution so the cross over from WAL to WL
occurs before Eso is of the order of exchange energy.
SUMMARY (Weak Localization in GaMnAs)
•Interpretation of low magnetic field MR effects do not support a clear signature of
WL (or WAL). Complicated AMR effects need to be taken into account more
carefully
•For moderate Mn doping GaMnAs should show WL due to the large disorder
scattering which limits the WAL corrections coming from interband correlations
•Interpretation of WL-> impurity band has no basis since the presence of SO
coupling in the model does not create a WAL regime for moderate Mn doping!!!
•Effects of e-e interactions at low fields should incorporate AMR effects to correctly
analyze the data
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!!
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
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
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)
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}
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 BROKEN
 
 
 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