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Anomalous Hall transport in metallic spin-orbit coupled
systems
November 11th 2008
JAIRO SINOVATexas A&M University
Institute of Physics ASCR
Research fueled by: 1
Hitachi CambridgeJorg Wunderlich, A.
Irvine, et al
Institute of Physics ASCRT. Jungwirth, K. Výborný, et al
University of Texas A. H. MaDonald, N. Sinitsyn
(LANL), et al.
University of NottinghamB. Gallagher, K. Edmonds, et
al.
Texas A&MA. Kovalev, M. Borunda, et
alWuerzburg Univ.
L. Molenkamp, E. Hankiweicz et al
2
Anomalous Hall transport: lots to think about
Taguchi et al Fang et al
Wunderlich et al
Kato et al
Valenzuela et al
SHE Inverse SHE
SHEIntrinsic AHE(magnetic monopoles?)
AHE
AHE in complex spin textures
The family of spintronics Hall effects
SHE-1
B=0spin current
gives charge current
Electrical detection
AHEB=0
polarized charge current
gives charge-spin
current
Electrical detection
SHEB=0
charge current gives
spin current
Optical detection
3
4
1) The basic phenomena of AHE2) When do we have AHE: necessary and sufficient conditions (P. Bruno)3) History of the anomalous Hall effect and major contributions4) Why is anomalous Hall transport difficult theoretically?5) Cartoon of mechanisms contributing to the AHE6) Semiclassical theory of AHE:
a) Boltzmann Eq. approach for diagonal transport: a warm-upb) Wave-packets of Bloch electrons: birth of Berry’s connectionc) Dynamics of Bloch electron wave-packets: birth of Berry’s curvatured) What does it all mean; does it make sense from Kubo formula?e) Boltmann Eq. for Bloch wave-packets: slowly does it.f) Back to the three main mechanisms: clarifying/correcting popular believes
7) Microscopic theory of AHE (Kubo approach)a) Kubo formula microscopic approach to transportb) Does it match the semiclassical approach?c) Other microscopic approaches
8) Spin-injection Hall effect: a new tool to explore spintronic Hall effects9) Spin Hall Effect
1) The basic idea2) Short history and developments3) Spin accumulation4) Inverse SHE in HgTe
OUTLINE
MRBR sH 40
Anomalous Hall effect
Simple electrical measurement
of magnetization
Spin dependent “force” deflects like-spin particles
I
_ FSO
FSO
_ __
majority
minority
V
InMnAs
sRR 0
5
y
x
xxxy
xyxx
y
x
E
E
j
j
xxxyxx
xxxx
122
22
222 xxxxxxxyxx
xy
xyxx
xyxy BA
xxxy AB
I
FSO
FSO
_
majority
minority
V
I
_ FSO
FSO
_ __
V=0non-magnetic
Ispin
FSO
FSO _
V
non-magnetic
IFSO
V
MzMz=0
Mz=0
non-magnetic
I=0
Mz=0
magnetic
optical detection
AHE
SHE SHE-1
SIHE
FSO
(a) (b)
(c) (d)
7
2xxxxxy BA xxxy AB
Anomalous Hall effect (scaling with ρ)
Dyck et al PRB 2005
Kotzler and Gil PRB 2005
Co films
Edmonds et al APL 2003
GaMnAs
Anomalous Hall effect: what is necessary to see the AHE?
Necessary condition for AHE: TIME REVERSAL SYMMETRY MUST BE BROKEN
Need a magnetic field and/or magnetic order
BUT IS IT SUFFICIENT?
(P. Bruno– CESAM 2005)
I
_ FSO
FSO
_ __
majority
minority
V
),(),( MBMB xyxy
Onsager relations (time reversal, general)
8
Local time reversal symmetry being broken does not always mean AHE present
Staggered flux with zero average flux in an electron gas:
- -
-
Is xy zero or non-zero?
Does zero average flux necessary mean zero xy ?
--
- 3--
- 3 B to –B does not leave the Hamiltonianinvariant so xy ≠0 (Haldane, PRL 88)
(P. Bruno– CESAM 2005)9
Similar argument follows for antiferromagnetic ordering
-
--
xy =0
B to –B plus a translation does NOT change the Hamiltonian so xy (B)= xy (-B) for this system BUT Onsager says xy (B)= -xy (-B)
Is non-zero collinear magnetization sufficient?
(P. Bruno– CESAM 2005)
In the absence of spin-orbit coupling a spin rotation of leaves the Hamiltonian the same so xy (B=0,M)= xy (B=0,-M) for this system and using Onsager relation one gets xy =0 (i.e. spins don’t know about left or right)
If spin-orbit coupling is present there is no invariance under spin rotation and xy≠0
10
(P. Bruno– CESAM July 2005)
Collinear magnetization AND spin-orbit coupling → AHE
Does this mean that without spin-orbit coupling one cannot get AHE?
Even non-zero magnetization is not a necessary condition
No!! A non-trivial chiral magnetic structure WILL give AHE even without spin-orbit coupling
Mx=My=Mz=0 xy≠0
Bruno et al PRL 04
11
COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING vs.
CHIRAL MAGNETIC STRUCTURES
AHE is present when SO coupling and/or non-trivial spatially varying magnetization (even if zero in
average)
SO coupled Bloch 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 an 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
12
(recent exception: J. Shibata and H. Kohno arXiv:0810.0610)
13
OUTLINE
•1880-81: Hall discovers the Hall and the anomalous Hall effect
The tumultuous history of AHE
•1970: Berger reintroduces (and renames) the side-jump: claims that it does not vanish and that it is the dominant contribution, ignores intrinsic contribution. (problem: his side-jump is gauge dependent)
Berger
14
Luttinger
•1954: Karplus and Luttinger attempt first microscopic theory: they develop (and later Kohn and Luttinger) a microscopic theory of linear response transport based on the equation of motion of the density matrix for non-interacting electrons, ; run into problems interpreting results since some terms are gauge dependent. Lack of easy physical connection.
rEeVHi
dt
ddis
0,ˆ
ˆ
Hall
•1970’s: Berger, Smit, and others argue about the existence of side-jump: the field is left in a confused state. Who is right? How can we tell? Three contributions to AHE are floating in the literature of the AHE: anomalous velocity (intrinsic), side-jump, and skew contributions.
•1955-58: Smit attempts to create a semi-classical theory using wave-packets formed from Bloch band states: identifies the skew scattering and notices a side-step of the wave-packet upon scattering and accelerating. .Speculates, wrongly, that the side-step cancels to zero.
knknkn
c uk
utk
Etkr
),(
The physical interpretation of the cancellation is based on a gauge dependent object!!
The tumultuous history of AHE: last three decades
15
•2004’s: Spin-Hall effect is revived by the proposal of intrinsic SHE (from two works working on intrinsic AHE): AHE comes to the masses, many debates are inherited in the discussions of SHE.
•1980’s: Ideas of geometric phases introduced by Berry; QHE discoveries
•2000’s: Materials with strong spin-orbit coupling show agreement with the anomalous velocity contribution: intrinsic contribution linked to Berry’s curvature of Bloch states. Ignores disorder contributions.
ckc
cnc Ee
k
kEr
)(1
•2004-8’s: Linear theories in simple models treating SO coupling and disorder finally merge: full semi-classical theory developed and microscopic approaches are in agreement among each other in simple models.
Intrinsic deflection
16
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.
~τ0 or independent of impurity density
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. They however come out in a different band so this gives rise to an anomalous velocity through scattering rates times side jump.
independent of impurity density
STRONG SPIN-ORBIT COUPLED REGIME (Δso>ħ/τ)
Side jump scattering
Vimp(r)
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.
~1/ni Vimp(r)
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
SO coupled quasiparticles
Why is AHE difficult theoretically
•AHE conductivity much smaller than σxx : many usual approximations fail
•Microscopic approaches: systematic but cumbersome; what do they mean; use non-gauge invariant quantities (final result gauge invariant)
•Multiband nature of band-structure (SO coupling) is VERY important; hard to see these effects in semi-classical description (where other bands are usually ignored).
•Simple semi-classical derivations give anomalous terms that are gauge dependent but are given physical meaning (dangerous and wrong)
•Usual “believes” on semi-classically defined terms do not match the full semi-classical theory (in agreement with microscopic theory)
•What happens near the scattering center does not stay near the scattering centers (not like Las Vegas)•T-matrix approximation (Kinetic energy conserved); no longer the case, adjustments have to be made to the collision integral term•Be VERY careful counting orders of contributions, easy mistakes can be made.
17
0)(
)(ˆ
ˆ ''
k
kHkv
k
Hv nn
nnk
18
What do we mean by gauge dependent?
Electrons in a solid (periodic potential) have a wave-function of the form
)(),(,
)(
rueetrnk
tkE
irki
k
n
Gauge dependent car
)(),(~
,
)()( rueeetr
nk
tkE
irkikia
k
n
BUT
is also a solution for any a(k)
Any physical object/observable must be independent of any a(k) we choose to put
Gauge wand (puts an exp(ia(k)) on the Bloch electrons)
Gauge invariant car
19
OUTLINE
Semi-classical theory of transport for free electrons: warm up I
Idea I: to interpret things in classical particles transform description from plane waves to wave-packets (which is a good basis too)
k
tm
kirrik
krkeekw
Vtr c
ccc
2)(
2
)(1
),(
~100 λF≈100 ÅValid for fields and disorder that vary in scales larger than the wavepackets
ckk
kEcc rv
)(
Eekc
Eqs of motion
0ck
)(kE
a
a
~1% of BZ
20
'
)('',
kkkkk
kkk ffk
fk
t
f
dt
df
Idea II: the distribution of the wave-packets obey Boltzmann classical dynamics. Connection to quantum mechanics is through the treatment of the collisions.
Warning: for non-interacting electrons there is NO (1-f)f terms
)(||'
2
',2
', kkkkkkEEV
(Born approximation usual: KE conserved;ignores what happens at the scattering center)
'
)('',
kkkkk
kkff
k
fEe
t
f
kkeqkgEff )(Let and look for a solution of the form kk
vEgg 0
'
)()(
'',0k
kkkkk
keq
kvEvEg
E
EfvEe
'
')cos1(
)(',0
kvvkkk
k
keq
k kkvEg
E
EfvEe
'
)()(
'',k
kkkkk
keq
kff
E
EfvEe
tr/1k
keqtr E
Efeg
)(
0 k
k
keqtrk
vEE
Efeg
)(
Step 2: Once we have the non-equilibrium distribution function we can calculate the current and from that deduce the conductivity
xk
kFxktr
kxkkx EEEv
V
evf
V
eJ )(2
,
2
m
ne trxx
2
Same results as in the microscopic calculations
2
0
1
Vnitr
Semi-classical theory of transport for free electrons: warm up continued
21
Step one: find the non-equilibrium part of the distribution function
Semi-classical theory of transport for Bloch electrons: the dynamics of wave-packets KNOW about the band structure
Idea I: to interpret things in classical particles transform description from plane waves to wave-packets (which is a good basis too)
knk
tkE
irrik
krkrueekw
Vtr
n
c
ccc
)()(
1),(
,
)()(
~100 λF≈100 ÅValid for fields and disorder that vary in scales larger than the wavepackets
0 ck
)(kE
a
a
~1% of BZ
Once the information of the periodic potential is imbedded into the wave-packet the choice of the phase factors are important to have it center at rc
22
ckk
kEcr
)(
Eekc
Are the Eqs of motion
Yes
No?
?
Building a wave-packet from Bloch electrons: the birth of the Berry’s connection
k
nk
rrik
krkruekw
Ntr c
ccc
)()(
1),(
,
)(
0cccc rkcrk
rr
We want to have wkc(k) such that
nckc
nckc
c
uk
iukki
ckekkwkw
,,)(
)()(
23
Berry’s phase connectionnk
ck cnckc
uk
iuA,,
Influences dynamics ofwave-packets
(velocities: 2nd step of semiclassical approach)
Influences scattering ofwave-packets
(collision integral term of Boltzmann eqn. in several parts)
Dynamics of wave-packets of Bloch electrons: the birth of the Berry’s curvature and the anomalous
velocityTask: build a Lagrangian from the new dynamic variables rc and kc
)()()(ˆ0 ccnkcccrkrk
reVkEAkrkreVHt
iLccccc
24
ckc
c
cnc
c
kk
kEr
Eek
)(1nk
cck cnckc
uk
iuk ,,
Applying Lagrange’s equations on the above Lagrangian:
Berry’s connection (gauge dependent)
k
nk
tkE
irrkiAkki
crkrueeekkw
Vtr
n
cckc
cc
)()(1
),(,
)()()(
nkc
k cnckcu
kiuA
,,
where
motion perpendicular to E, Hall type motion!
already LINEAR E The whole Fermi sea participateson this current contribution !!
k
yzkkk
cx EEfV
evf
V
eJ
c
)(0
2int
Berry’s curvature (gauge invariant)
“Anomalous velocity”
Does the intrinsic contribution to the current make sense microscopically?
From microscopic linear response to semiclassical linear response
25
''
2
''
2
'2
'
''
'
2
'2
'
'
2
'2
'
'
2
''Im2
''
)(
)('')(Im
)(
''Im
)(
'ˆˆ'Im]Re[
nk yxkn
yxnnknkkn
nnk knkn
knknyx
knkn
nkkn
nnk knkn
yx
nkkn
nnk knkn
yx
nkknxy
knk
knk
fV
ekn
kknknkn
kiff
V
e
EE
EEknk
knknknk
EE
ffV
e
EE
knkH
knknkH
kn
ffV
e
EE
knvknknvknff
V
e
k
yzkkx EEfV
eJ
c
)(0
2int
Intrinsic contribution matches Kubo linear response theory (clean limit) !!!
Kubo formula (1st order perturbation theory)
Dynamics of wave-packets of Bloch electrons: How do the Berry’s curvature dynamics affect scattering?
26
Tried to interpret it physically BUT it is gauge dependent (i.e. only gauge invariant quantities have measurable physical meaning) !
Early theories (Berge,Smit) noticed that Bloch electron wave-packets seem to experience a side-step upon scattering: (a dangerous way of doing dynamics)
knk
iknkd
dE
kwk
ikwkddt
d
k
Ekwkdekw
kiekwkd
dt
d
ekwk
ieekwkdkdV
rd
dt
d
eek
ieekwkwkdkdV
rd
dt
d
ereeekwkwkdkdV
rd
dt
dr
dt
dr
kn
ktiEtiE
tiEtiErkki
tiErkitiErki
tiErkitiErki
krkrc
kk
kk
kk
kk
cccc
)()()()()(
)()'('
)()'('
)()'('
*2//*
//)'(*
//'*
//'*
'
'
'
''''', ' knknknknnknku
kiuu
kiur
Dynamics of wave-packets of Bloch electrons: How do the Berry’s curvature dynamics affect scattering?
27
Tried to interpret it physically BUT it is gauge dependent (i.e. only gauge invariant quantities have measurable physical meaning) !
Early theories (Berge,Smit) noticed that Bloch electron wave-packets seem to experience a side-step upon scattering:
''''', ' knknknknnknku
kiuu
kiur
The gauge invariant expressions can be derived using the gauge invariant Lagrangian dynamics shown earlier (Sinitsyn et al 2006) l=(n,k)
'''', arg''
llllllll uukk
uk
iuuk
iur
Side jump scattering
How does side-jump affect transport?
28
'''', arg''
llllllll uukk
uk
iuuk
iur
Side jump scattering
The side-jump comes into play through an additional current and influencing the Boltzmann equation and through it the non-equilibrium distribution function
VERY STRANGE THING: for spin-independent scatterers side-jump is independent of scatterers!!
1st-It creates a side-jump current: ','
', lll
lljs
l rv
2nd-An extra term has to be added to the collision term of the Boltzmann eq. to account because upon elastic scattering some kinetic energy is transferred to potential energy.
'
)( '',l
llll ffI
)(|| '2
',2
', llllT
ll EET
full ωll’ does not assume KE conserved,T-matrix approximation of ωll’ (ωT
ll’) does.
'' llll rEeEE
''
0'', )
)((
lll
l
lll
Tll rEe
E
EfffI
29
Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current
As before we do this in two steps: first calculate steady state non-equilibrium distribution function and then use it to compute the current.
''
0'',
00 )
)((
)(
lll
l
lll
Tll
l
ll
l rEeE
Efff
E
EfvEe
t
f
Set to 0 for steady state solution
k
Ev l
l
0Only the normal velocity term, since we are looking for linear in E equation
)4(',
)3(',
)3(',
)2(','
2',
2', )(|| a
lls
lla
llllllllT
ll EET
order of the disorder potential strength and symmetric and anti-symmetric components
)(|| '2
',2)2(
', llllll EEV
)()(Im)2( '''''
','''','',2)3(
', lll
lldislllllla
ll EEEEVVV
1st Born approximation
2nd Born approximation (usual skew scattering contribution)
adisl
al
al
slleqk
ggggEff 43)(
To solve this equation we write the non-equilibrium component in various components that correspond to solving parts of the equation the corresponding order of disorder
30
Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current
'
'0
'',0
0 ))(
()(
lll
l
lll
Tll
l
ll rEe
E
Efff
E
EfvEe
adisl
al
al
slleqk
ggggEff 43)(
''
)2(',
00 )(
)(
l
sl
slll
l
ll gg
E
EfvEe
'
')3(
','
3'
3)2(', )()(0
l
sl
sl
all
l
al
alll gggg
'
')4(
','
4'
4)2(', )()(0
l
sl
sl
all
l
al
alll gggg
''
0)2(', )
)((0
'l
lll
ladisadisll rEe
E
Efgg
ll
~V0 1 isl ng
~V 13 ia
l ng
~V2 04i
al ng
~V2 0i
adisl ng
0,0
2int ~)( i
lzllxy nEf
V
e
2nd step: (after solving them) we put them into the equation for the current and identify from there the different contributions to the AHE using the full expression for the velocity
'
',',
1
llllll
ll r
Ee
k
Ev
00 ~ i
llx
y
adisladis
xy nvE
g
V
e
i
llx
y
alsk
xy nvE
g
V
e 10
31 ~ 0
0
42 ~ i
llx
y
alsk
xy nvE
g
V
e
0
'',', ~ i
l lllll
y
sljs
xy nrE
g
V
e
Intrinsic deflection
31
Popular believe: ~τ1 or ~1/ni WRONG
E
~ni0 or independent of impurity density
0,0
2int ~)( i
lzllxy nEf
V
e
i
llx
y
alsk
xy nvE
g
V
e 10
31 ~
00
42 ~ i
llx
y
alsk
xy nvE
g
V
e
Skew scattering (2 contributions)
term missed by many people using semiclassical approach
Side jump scattering (2 contributions)
Popular believe: ~ni0 or independent of impurity density
0
'',', ~ i
l lllll
y
sljs
xy nrE
g
V
e
00 ~ i
llx
y
adisladis
xy nvE
g
V
e
Origin is on its effect on the distribution function
32
OUTLINE
• Boltzmann semiclassical approach: easy physical interpretation of different contributions (used to define them) but very easy to miss terms and make mistakes. MUST BE CONFIRMED MICROSCOPICALLY! How one understands but not necessarily computes the effect.
• Kubo approach: systematic formalism but not very transparent.
• Keldysh approach: also a systematic kinetic equation approach (equivalent to Kubo in the linear regime). In the quasi-particle limit it must yield Boltzmann semiclassical treatment.
Microscopic vs. Semiclassical
33
Kubo microscopic approach to transport: diagrammatic perturbation
theory
Averaging procedures: = 1/ 0 = 0
= +
Bloch ElectronReal Eigenstates
l
jkFA
ikFR
ij vEGvEGV
e ˆ)(ˆˆ)(ˆ~
2
Tr
Need to perform disorder average (effects of scattering)
iVHEEG
disF
FR
ˆˆ1
)(ˆ0
n, q
Drude Conductivity
σ = ne2 /m*~1/ni
Vertex Corrections 1-cos(θ)
Perturbation Theory: conductivity
n, q
34
35
intrinsic AHE approach in comparing to experiment: phenomenological “proof”
Berry’s phase based AHE effect is reasonably successful in many instances BUT still not a
theory that treats systematically intrinsic and ext rinsic contribution in an equal footing
n, q
n’n, q• 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 et al (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003)
• colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999).
• CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004)
Experiment AH 1000 (cm)-
1
TheroyAH 750 (cm)-1
AHE in Fe
AHE in GaMnAs
36
“Skew scattering”
“Side-jump scattering”
Intrinsic AHE: accelerating between scatterings
n, q
n, q m, p
m, pn’, k
n, q
n’n, q
Early identifications of the contributions
Vertex Corrections
σIntrinsic ~ 0 or n0i
Intrinsic
σ0 /εF~ 0 or n0i
Kubo microscopic approach to AHE
n, q
n, q m, p
m, pn’, k
matrix in band index
m’, k’
Armchair edge
Zigzag edge
EF
“AHE” in graphene: linking microscopic and semiclassical theories
37
x x y y so zKH =v(k σ +k σ )+Δ σ
Single K-band with spin up
x x y y so zKH =v(k σ +k σ )+Δ σ
In metallic regime: IIxyσ =0
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
Sinitsyn et al PRB 0738
Kubo-Streda calculation of AHE in graphene Don’t be afraid of the equations,
formalism can be tedious but is systematic (slowly but steady does it)
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 )]
Kubo-Streda formula:A. Crépieux and P. Bruno (2001)
Comparing Boltzmann to Kubo (chiral basis)
39
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
intxy
jsxy
adisxy
1skxy
2skxy
Kubo identifies, without a lot of effort, the order in ni of the diagrams BUT not so much their physical interpretation according to semiclassical theory
Sinitsyn et al 2007
40
Next simplest example: AHE in Rashba 2D system
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)
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
41
AHE in Rashba 2D system
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
Keldysh and Kubo match analytically in the metallic limit
Numerical Keldysh approach (Onoda et al PRL 07, PRB 08)
GR G0 G0RGR
G0 1 R GR 1
G0R 1
ˆ G ˆ G G0A 1
ˆ R ˆ G ˆ G ˆ R ˆ ˆ G A ˆ G R ˆ
ˆ ˆ R ˆ G ˆ A
42
43
Recent progress: full understanding of simple models in each approach
Semi-classical approach:Gauge invariant formulation; shown to match microscopic approach in 2DEG+Rashba,
GrapheneSinitsyn et al PRB 05, PRL 06, PRB 07 Borunda et al PRL 07, Nunner et al PRB 08Sinitsyn JP:C-M 08
Kubo microscopic approach:
Results in agreement with semiclassical calculations 2DEG+Rashba, Graphene
Sinitsyn et al PRL 06, PRB 07, Nunner PRB 08, Inoue PRL 06, Dugaev PRB 05
NEGF/Keldysh microscopic approach:
Numerical/analytical results in agreement in the metallic regime with
semiclassical calculations 2DEG+Rashba, Graphene
Kovalev et al PRB 08, Onoda PRL 06, PRB 08
AHE in Rashba 2D system: “dirty” metal limit?
Is it real? Is it justified? what does it mean in the regions where sing of σAHE changes?
Can the kinetic metal theory be justified when disorder is larger than any other scale?
44
No localization physics included (but σ is below the minimum conductivity limit !)
Onoda et al 2007,2008
45
AHE in Rashba 2D system: “dirty” metal limit?
AHE in Rashba 2D system: “dirty” metal limit?
46
47
CONCLUSIONS (AHE)
•When creating a semiclassical theory be aware of the Gague Wand
•Usual identifications of contributions to AHE from semiclassical to microscopic theories need to be done carefully (σxx dependence not the one usually thought off)
•Now that the metallic theory is clear (and nothing missing), time to study more complex materials in detail (microscopic nano-engineering of AHE)
•Can we test the simple models?
•Strongly disorder limit??
48
OUTLINE
Spin-injection Hall effectSpin-injection Hall effect: A new member of the spintronic Hall
family
JAIRO SINOVATexas A&M University
Institute of Physics ASCR
Research fueled by: 49
Hitachi CambridgeJorg Wunderlich, A.
Irvine, et al
Institute of Physics ASCRTomas Jungwirth,, Vít Novák, et
al
Texas A&MA. Kovalev, M. Borunda,
et al
50
Towards a spin-based non-magnetic FET device:Towards a spin-based non-magnetic FET device: can we electrically measure the spin-polarization?
ISHE is now routinely used to detect other effects related to the generated spin-currents (Sitho et al Nature 2008)
Can we achieve direct spin polarization detection through an electrical measurement in an all paramagnetic semiconductor system?
Long standing paradigm: Datta-Das FET
Unfortunately it has not worked: •no reliable detection of spin-polarization in a diagonal transport configuration •No long spin-coherence in a Rashba SO coupled system
Alternative:Alternative: utilize technology developed to detect SHE in 2DHG and measure polarization via Hall
probes
J. Wunderlich, B. Kaestner, J. Sinova andT. Jungwirth, Phys. Rev. Lett. 94 047204 (2005)
Spin-Hall Effect
51
B. Kaestner, et al, JPL 02; B. Kaestner, et al Microelec. J. 03; Xiulai Xu, et al APL 04, Wunderlich et al PRL 05
Proposed experiment/device: Coplanar photocell in reverse bias with Hall probes along the 2DEG channelBorunda, Wunderlich, Jungwirth, Sinova et al PRL 07
Device schematic - materialmaterial
52
i pn
2DHG
-
2DHGi p
n
53
Device schematic - trenchtrench
i
p
n2DHG
2DEG
54
Device schematic – n-etchn-etch
Vd
VH
2DHG
2DEG
Vs
55
Device schematic – Hall Hall measurementmeasurement
2DHG
2DEG
e
h
ee
ee
e
hh
h
h h
Vs
Vd
VH
56
Device schematic – SIHE SIHE measurementmeasurement
0 30 60 90 120 150
-50
-40
-30
-20
-10
0
10
20
30
40
50
0
2
4
6
8
10
12
14
16
18
20
22
24R
H [
]
tm [s]
RL [k
]
0 30 60 90 120 150
-50
-40
-30
-20
-10
0
10
20
30
40
50
0
2
4
6
8
10
12
14
16
18
20
22
24R
H [
]
tm [s]
RL [k
]
0 30 60 90 120 150
-50
-40
-30
-20
-10
0
10
20
30
40
50
0
2
4
6
8
10
12
14
16
18
20
22
24R
H [
]
tm [s]
RL [k
]
0 30 60 90 120 150
-50
-40
-30
-20
-10
0
10
20
30
40
50
0
2
4
6
8
10
12
14
16
18
20
22
24R
H [
]
tm [s]
RL [k
]
0 30 60 90 120 1500
2
4
6
8
10
12
14
16
18
20
22
24
tm [s]
RL [k
]5m
Reverse- or zero-biased: Photovoltaic Photovoltaic CellCell
trans. signaltrans. signal
Red-shift of confined 2D hole free electron trans.due to built in field and reverse biaslight excitation with = 850nm
(well below bulk band-gap energy)
σσooσσ++σσ-- σσoo
VL
0.95
1.00
1.05
0.95
1.00
1.05
0 30 60 90 120 150
0.95
1.00
1.05
tm [s]
P/Pav.
I/Iav.
V/Vav.
Vav. = 9.4mV
Iav. = 525nA
(a)
(b)
(c)
57
-1/2
-1/2 +1/2
+1/2 +3/2-3/2
bulk
Transitions allowed for ħω>EgTransitions allowed for ħω<Eg
-1/2
-1/2 +1/2
+1/2+3/2-3/2
Band bending: stark effect
Transitions allowed for ħω<Eg
5m
-4 -2 0 2 4-100
-50
0
50
100
tm [s]
RH [
]
n2
+ -
Spin injection Hall effect: Spin injection Hall effect: experimental observation
-4 -2 0 2 4-100
-50
0
50
100
tm [s]
RH [
]
n1 (4)
n2
-4 -2 0 2 4-100
-50
0
50
100
tm [s]
RH [
]
n1 (4)
n2
n3 (4)
Local Hall voltage changes sign and magnitude along the stripe58
Spin injection Hall effect Anomalous Hall effect
-1.0 -0.5 0.0 0.5 1.0-2
-1
0
1
2
H [
10-3 ]
( ) / (
)
n1
-1.0 -0.5 0.0 0.5 1.0
-10
-5
0
5
10
H [
10-3 ]
( ) / (
)
n2
-1.0 -0.5 0.0 0.5 1.0
-0.5
0.0
0.5
H [
10-3 ]
( ) / (
)
p
-1.0 -0.5 0.0 0.5 1.0
-0.5
0.0
0.5
H [
10-3 ]
( ) / (
)
p
59
and high temperature operation
Zero bias-
-6 -3 0 3 6
-5
0
5
tm [s]
H [
10-3
]
n1 (10)
n3 (50)
n2 VB = 0V
T = 4K
+-
-6 -3 0 3 6
-1
0
1
tm [s]
H [
10-3
]
n1 (2)
n3
n2 (2)
T = 230K
VB = -10V
A
+-
Persistent Spin injection Hall effectPersistent Spin injection Hall effect
60
THEORY CONSIDERATIONSTHEORY CONSIDERATIONSSpin transport in a 2DEG with Rashba+Dresselhaus
SO
))(V(2 dis
*22
rkkkkkm
kH yyxxyxxy
2DEG
61
, AeV 0
02.0
AeV 03.001.00
)AeV/ (for0
03.001.0 ZE
For our 2DEG system:
067.0 emm
The 2DEG is well described by the effective Hamiltonian:
Hence
GaAs, for A 2o
3.5)(
11
3 22
2*
sogg EE
P GaAs, for AeV with 30
102 BkB z zE*
62
• spin along the [110] direction is conserved
• long lived precessing spin wave for spin perpendicular to [110]
What is special about ?
))((2
22
yxxy kkm
kH
2DEG ))(V( dis
* rk
Ignoring the term
for now
k k Q
• The nesting property of the Fermi surface:
2
4
m
Q
The long lived spin-excitation: “spin-helix”
0, , zQ Qk k Q k Q k k k k kk k k
S c c S c c S c c c c
0 0, 2 , ,z zQ Q Q QS S S S S S
ReD , 0k Q k k Q k
H c c k Q k c c
An exact SU(2) symmetry
Only Sz, zero wavevector U(1) symmetry previously known:
J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003).
K. C. Hall et. al., Appl. Phys. Lett 83, 2937 (2003).
• Finite wave-vector spin components
• Shifting property essential
63
• Spin configurations do not depend on the particle initial momenta.
• For the same x+ distance traveled, the spin precesses by exactly the same angle.
• After a length xP=h/4mα all the spins return exactly to the original configuration.
Physical Picture: Persistent Spin Helix
Thanks to SC Zhang, Stanford University
64
65
Persistent state spin helix verified by pump-probe experiments
Similar wafer parameters to ours
The Spin-Charge Drift-Diffusion Transport Equations
For arbitrary α,β spin-charge transport equation is obtained for diffusive regime
For propagation on [1-10], the equations decouple in two blocks. Focus on the one coupling Sx+ and Sz:
For Dresselhauss = 0, the equations reduce to Burkov, Nunez and MacDonald, PRB 70, 155308 (2004);
Mishchenko, Shytov, Halperin, PRL 93, 226602 (2004)
STTSCSC SDS
STSCnB SDS
STSCnB SDS
SBSBn Dn
zxxxxzzt
xzxxxxt
xzxxxxt
xxxxt
)( 21222
2212
1122
212
66
k
mTkB F
F 2
22
2/1222
2/1 )(2
,)()(2
DTCvD F 2/12
2/12 4,2/ and
STTSC SDS
STSC SDS
zxxzzt
xzxxxt
)( 2122
222
2~~
4~~~
arctan,)~~~
(||,)exp(|| 21
22
41
22
21
21414
22
22
1LL
LLLLLLqiqq
]exp[)( ]011[0/]011[/ xqSxS xzxz Steady state solution for the spin-polarization
component if propagating along the [1-10] orientation
22/1 ||2
~ mL
67
Steady state spin transport in diffusive regime
Spatial variation scale consistent with the one observed in SIHE
68
2xxxxxy BA xxxy AB
Anomalous Hall effect (scaling with ρ)
Dyck et al PRB 2005
Kotzler and Gil PRB 2005
Co films
Edmonds et al APL 2003
GaMnAs Strong SO coupled regime
Weak SO coupled regime
69
AHE contribution
zzi
H pxpnn
ex 3
]011[*
]011[ 101.1)(2)(
Type (i) contribution much smaller in the weak SO coupled regime where the SO-coupled bands are not resolved, dominant contribution from type (ii)
Crepieux et al PRB 01Nozier et al J. Phys. 79
Two types of contributions: i)S.O. from band structure interacting with the field (external and internal)ii)Bloch electrons interacting with S.O. part of the disorder
))(V(2 dis
*22
rkkkkkm
kH yyxxyxxy
2DEG
)(2
02
*2
nnnVe
xy
skew)(
2 *2
nne
xy
jump-side
4103.5 jump-sideH
Lower bound estimate of skew scatt. contribution
Spin injection Hall effect: Theoretical consideration
Local spin polarization calculation of the Hall signal Weak SO coupling regime extrinsic skew-scattering term is dominant
)(2)( ]011[*
]011[ xpnn
ex z
iH
70
Lower bound estimate
The family of spintronics Hall effects
SHE-1
B=0spin current
gives charge current
Electrical detection
AHEB=0
polarized charge current
gives charge-spin
current
Electrical detection
SHEB=0
charge current gives
spin currentOptical
detection
71
SIHEB=0
Optical injected polarized
current gives charge current
Electrical detection
72
SIHE: a new tool to explore spintronics
•nondestructive electric probing tool of spin propagation without magnetic elements
•all electrical spin-polarimeter in the optical range
•Gating (tunes α/β ratio) allows for FET type devices (high T operation)•New tool to explore the AHE in the strong SO coupled regime
73
CONCLUSIONS (SIHE)
Spin-injection Hall effect observed in a conventional
2DEG
- nondestructive electrical probing tool of spin propagation
- indication of precession of spin-polarization
- observations in qualitative agreement with theoretical
expectations
- optical spin-injection in a reverse biased coplanar pn-
junction: large and persistent Hall signal (applications
!!!)
74
OUTLINE
Spin Hall effectTake 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 without applying external magnetic
fields. Spin accumulation without charge accumulation excludes simple electrical
detection
Carriers with same charge but opposite spin are deflected due to SO to opposite
sides.
75
Spin Hall Effect(Dyaknov and Perel)
InterbandCoherent Response
(EF) 0
Occupation # Response`Skew Scattering‘
[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…]
INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/0308167)
Sinova et al PRL 2004 (cond-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
76
‘Universal’ spin-Hall conductivity
*22*
2
2
4
22*22
sH
for8
for8
DDD
D
DD
xy
nnn
ne
mnn
e
Color plot of spin-Hall conductivity:yellow=e/8π and red=0
n, q
n’n, q
77
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?
the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05)
Ladder partial sum vertex correction:
Inoue et al PRB 04Dimitrova et al PRB 05Raimondi et al PRB 04Mishchenko et al PRL 04Loss et al, PRB 05
~
n, q
n’n, q
+ +…=0
For the Rashba example the side jump contribution cancels the intrinsic contribution!!
78
First experimental observations at the end of 2004C
P [%
]
Light frequency (eV)1.505 1.52
Kato, et al Science Nov 04
Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder)
79
Wunderlich et al PRL Jan 05
SHE in two dimensional hole gas
Co-planar spin LED in GaAs 2D hole gas: ~1% polarization
Other experiments followed
Also reports of room temperature SHE
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
80
81
OUTLINE
Trying to understand 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
82
Spin Accumulation – Strong SO
Mean FreePath?
Spin Precession
Length
?
83
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 !!
84
-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
%
10m channel
SHE experiment in GaAs/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 '05
85
First experimental observations at the end of 2004C
P [%
]
Light frequency (eV)1.505 1.52
Kato, et al Science Nov 04
Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder)
86
Wunderlich et al PRL Jan 05
SHE in two dimensional hole gas
Co-planar spin LED in GaAs 2D hole gas: ~1% polarization
Other experiments followed
Also reports of room temperature SHE
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
87
-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
%
10m channel
SHE experiment in GaAs/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 '05
88
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
Charge based measurements of SHE: SHE-1
Non-equilibrium Green’s function formalism (Keldysh-LB)
89
90
H-bar structures for detection of Spin-Hall-Effect
(electrical detection through inverse SHE)
E.M. Hankiewicz et al ., PRB 70, R241301 (2004)
91
sample layout
Molenkamp et al (unpublished)
insu
lati
ng
p-c
onduct
ing
n-conducting
Strong SHE-1 in HgTe
theory
92
References for AHE (focus on theory; not complete, chosen for pedagogical purposes)
•Early theories: R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954): on-line;J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955): on-line; J. M. Luttinger, Phys. Rev. 112, 739 (1958): on-line•J Smit, Physica 21, 877 (1955): on-line, Physica 24, 39 (1958): on-line; L. Berger, Phys. Rev. B 2, 4559 (1970): on-line ; P. Leroux-Hugon and A. Ghazali, J. Phys. C 5, 1072 (1972): on-line•AHE in conduction electrons: P. Nozieres, C. Lewiner, J. Phys. France 34, 901 (1973): on-line•Dirac+Pauli approach: A. Crépieux and P. Bruno, Phys. Rev. B 64, 014416 (2001): on-line•“intrinsic” or Berry’s phase AHE: Y. Taguchi, et al Science 291, 5513 (2001): on-line; Jinwu Ye, et al Phys. Rev. Lett. 83, 3737 (1999): on-line;T. Jungwirth, et al PRL. 88, 207208 (2002): on-line, ibid, Appl. Phys. Lett. 83, 320 (2003): on-line•M(r) induced AHW: P. Bruno, al Phys. Rev. Lett. 93, 096806 (2004): on-line•AHE Fermi liquid properties: F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004): on-line•Kubo+Boltzmann: N.A. Sinitsyn et al, Phys. Rev. Lett. 97, 106804 (2006): on-line, Phys. Rev. B 72, 045346 (2005): on-line•Wave-packet dynamics: Ganesh Sundaram and Qian Niu, Phys. Rev. B 59, 14915 (1999): on-line; M. P. Marder, "Condensed Matter Physics", Wiley, New York, (2000)•AHE in 2DEG+Rashba:V. K. Dugaev, et al Phys. Rev. B 71, 224423 (2005): on-line; Jun-ichiro InouePhys. Rev. Lett. 97, 046604 (2006): on-line; N.A. Sinitsyn, Phys. Rev. B 75, 045315 (2007): on-line; Shigeki Onoda, et al , Phys. Rev. Lett. 97, 126602 (2006): on-line, Phys. Rev. B 77, 165103 (2008): on-line; N.A. Sinitsyn, et al Phys. Rev. B 75, 045315 (2007): on-line; Mario Borunda et al, Phys. Rev. Lett. 99, 066604 (2007): on-line; Tamara S. et al Phys. Rev. B 76, 235312 (2007): on-line; A. Kovalev, et al Phys. Rev. B 78, 041305 (2008): on-line•Reviews: L. Chien and C.R. Westgate, "The Hall Effect and Its Applications", Plenum, New York (1980); Jairo Sinova, et al Int. J. Mod. Phys. B 18, 1083 (2004): on-line; N. A. Sinitsyn, J. Phys.: Condens. Matter 20, 023201 (2008): on-line
DETAILED DERIVATIONS
93
The following slides include detailed derivations that we skipped in the previous slides where the results were stated. Some of these contain important tricks worth learning that appear continuously throughout the literature on anomalous transport.
Building a wave-packet from Bloch electrons: the birth of the Berry’s connection
k
nk
rrik
krkruekw
Ntr c
ccc
)()(
1),(
,
)(
0cccc rkcrk
rr
We want to have wkc(k) such that
nckc
nckc
c
c
ccnck
cnkcnkc
cnkc
c
c
nkc
c
c
nkc
c
cnkc
c
cnk
c
ccccc
uk
iukki
ckkk
knkc
kk
knkk
k
nkkk
k
nkkkk
rrkki
kkk
nkkkk
rrkki
k
rrkki
kknkkk
kknk
rrki
kcrrki
krkcrk
ekkwkwkwki
uk
iu
kwki
rukwrdruki
rukwrd
rukwki
rukwrd
rukwki
erukwrd
rukwki
erukwrdN
eki
rukwrukwrdN
ruekwrrruekwrdN
rr
,,
,
,,
,
',
',
',
',
)(
,
22
,
*2
,
**
,',
)()'(**
',
,',
)()'(**
)()'(
',,
**
',,
)(*)('*
)()()(ln
)(ln)()()()()(
)()()()(
)()()()'(
)()()()'(1
)()()()'(1
)()())(()'(1
)(
Berry’s phase connection
94
Integration by parts
Using periodicity and sum of
',
)'(1kk
i
Rkki ieN
Integration over a unit cell
Detailed derivation