New developments in the AHE: New developments in the AHE: phenomenological regime, unified linear theories, and a new member of the spintronic Hall family Low Dimensional Systems Workshop KITP, Santa Barbara, CA May 14 th , 2009 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research fueled by: Hitachi Cambridge Jorg Wunderlich , A. Irvine, et al Institute of Physics ASCR Tomas Jungwirth , Vít Novák, et al Texas A&M L. Zarbo Stanford University Shoucheng Zhang , Rundong Li, Jin Wang
Transcript
New developments in the AHE: New developments in the AHE: phenomenological regime, unified linear theories, and a new member of the
spintronic Hall family
Low Dimensional Systems Workshop
KITP, Santa Barbara, CAMay 14th , 2009
JAIRO SINOVATexas A&M University
Institute of Physics ASCR
Research fueled by:
Hitachi CambridgeJorg Wunderlich, A. Irvine, et
al
Institute of Physics ASCRTomas Jungwirth, Vít Novák, et
al
Texas A&M L. Zarbo
Stanford UniversityShoucheng Zhang, Rundong Li, Jin Wang
2
Anomalous Hall transport: lots to think about
Wunderlich et al
SHE
Kato et al
Fang et al
Intrinsic AHE(magnetic monopoles?)
AHE
Taguchi et al
AHE in complex spin textures
Valenzuela et al
Inverse SHE
Brune et al
OUTLINE• Introduction• SIHE experiment
– Making the device– Basic observation– Analogy to AHE– Photovoltaic and high T operation– The effective Hamiltonian– Spin-charge Dyanmcis
• AHE in spin injection Hall effect: – AHE basics– Strong and weak spin-orbit couple contributions of AHE– SIHE observations– AHE in SIHE
• Spin-charge dynamics of SIHE with magnetic field: – Static magnetic field steady state– Time varying injection
• AHE general prospective– Phenomenological regimes– New challenges
The family of spintronic Hall effectsThe family of spintronic Hall effects
4
AHEB=0
polarized charge current
gives charge-spin
current
Electrical detection
jqs––– – –– – –– – –
+ + + + + + + + + +AHE
Ferromagnetic(polarized charge current)
SHEB=0
charge current gives
spin current
Optical detection
jq
SHE
nonmagnetic(unpolarizedcharge current)
SHE-1
B=0spin current
gives charge current
Electrical detection
js–––––––––––
+ + + + + + + + + +iSHE
5
Towards a spin-based non-magnetic FET device:Towards a spin-based non-magnetic FET device: can we electrically measure the spin-polarization?
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
Spin-detection in semiconductors
Ohno et al. Nature’99, others
Crooker et al. JAP’07, others Magneto-optical imaging
non-destructive
lacks nano-scale resolution and only an optical lab tool MR Ferromagnet
electrical
destructive and requires semiconductor/magnet hybrid design & B-field to orient the FM
spin-LED
all-semiconductor
destructive and requires further conversion of emitted light to electrical signal
non-destructive
electrical
100-10nm resolution with current lithography
in situ directly along the SmC channel (all-SmC requiring no magnetic elements in the structure or B-field)
Wunderlich et al. arXives:0811.3486
Spin-injection Hall effect
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
8
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
i pn
2DHG
Device schematic - materialmaterial
-
2DHGi p
n
Device schematic - trenchtrench
i
p
n2DHG
2DEG
Device schematic – n-etchn-etch
Vd
VH
2DHG
2DEG
Vs
12
Device schematic – Hall Hall measurementmeasurement
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
Spin Currents 2009
28
WEAK SPIN-ORBIT COUPLED REGIME (Δso<ħ/τ)
Side jump scattering from SO disorder
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 λ*Vimp(r)
Skew scattering from SO disorder
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
λ*Vimp(r)
The terms/contributions dominant in the strong SO couple regime are strongly reduced (quasiparticles not well defined due to strong disorder broadening). Other terms, originating from the interaction of the quasiparticles with the SO-coupled part of the disorder potential dominate.
Better understood than the strongly SO couple regime
29
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
30
Lower bound estimate
32
Drift-Diffusion eqs. with magnetic field perpendicular to 110 and time varying spin-injection
Spin Currents 2009
σ+(t)
B
Similar to steady state B=0 case, solve above equations with appropriate boundary conditions: resonant behavior around ωL and small shift of oscillation period
Jing Wang, Rundong Li, SC Zhang, et al
Semiclassical Monte Carlo of SIHE
Numerical solution of Boltzmann equation
Spin-independent scattering:
Spin-dependent scattering:
•phonons,•remote impurities,•interface roughness, etc.
•side-jump, skew scattering.
AHE
•Realistic system sizes (m).•Less computationally intensive than other methods (e.g. NEGF).
Spin Currents 2009
Single Particle Monte Carlo
Spin Currents 2009
Spin-Dependent Semiclassical Monte CarloTemperature effects, disorder, nonlinear effects, transient regimes.Transparent inclusion of relevant microscopic mechanisms affecting spin transport (impurities, phonons, AHE contributions, etc.).Less computationally intensive than other methods(NEGF).Realistic size devices.
Effects of B field: current set-up
Spin Currents 2009
In-Plane magnetic fieldOut-of plane magnetic field
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
37
SIHEB=0
Optical injected polarized
current gives charge current
Electrical detection
38
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
39
AHE in the strong SO regime
40
•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
41
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
42
•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.
Why is AHE difficult theoretically in the strong SO couple regime?
•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.
43
0)(
)(ˆ
ˆ ''
k
kHkv
k
Hv nn
nnk
44
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
• 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. SemiclassicalAHE in the strongly SO couple regime
45
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
46
47
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
48
“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
49
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, JS, et al PRB 0750
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)
51
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
52
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
Comparing Boltzmann to Kubo (chiral basis)
53
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
Intrinsic deflection
54
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
55
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
– – – – – – – – – – – + + + + + + + + + +
jqsnonmagneticSpin-polarizer
current injected optically
Spin injection Hall effect (SIHE)Spin injection Hall effect (SIHE)
Up to now no 2DEG+R ferromagnetis: SIHE offers this possibility
56
Phenomenological regimes of AHE
Spin Currents 2009
Review of AHE (to appear in RMP 09), Nagaosa, Sinova, Onoda, MacDonald, Ong
1. A high conductivity regime for σxx>106 (cm)-1 in which AHE is skew dominated2. A good metal regime for σxx ~104-106 (cm) -1 in which σxy
AH~ const3. A bad metal/hopping regime for σxx<104 (cm) -1 for which σxy
AH~ σxyα with α>1
Skew dominated regime
Scattering independent regime
Spin Currents 2009
'
'
2int ''Im
2]Re[
nk yxknxy kn
kkn
kf
V
e
Q: is the scattering independent regime dominated by the intrinsic AHE?
intrinsic AHE approach in comparing to experiment: phenomenological “proof”
Berry’s phase based AHE effect is reasonably successful in many instances
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
Spin Currents 2009
Spin Currents 2009
Hopping conduction regime: terra incognita
•Approximate scaling seen as a function of T•No theory of approximate scaling
Spin Currents 2009
Nagaosa et al RMP 09
Tentative phase diagram of AHE
Spin Currents 2009
AHE Review, RMP 09, Nagaosa, Sinova, Onoda, MacDonald, Ong
EXTRA SLIDES
63
A2070Kelvin Nanotechnology, University of Glasgow
thickness composition doping function
5nm GaAs p=1E19 (Be) cap
2ML GaAs un
50nm AlxGa1-xAs, x=0.5 p=8E18 (Be)
3nm AlxGa1-xAs, x=0.3 un
90nm GaAs un channel
5nm AlxGa1-xAs, x=0.3 un spacer
2ML GaAs un
n=5E12 delta (Si) delta-doping
2ML GaAs un
300nm AlxGa1-xAs, x=0.3 un
50 period (9ML GaAs: 9ML AlGaAs, x=0.3) superlattice
1000nm GaAs un
GaAs SI substrate
64
65
810 820 830 840 850 8600
25
50
0
10
20
R
H []
[nm]
P
L [
103 c
ount
s ]
66
How does side-jump affect transport?
67
'''', 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
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)
