New developments in the AHE: New developments in the AHE: phenomenological regime, unified linear theories, and a new member of the
spintronic Hall family
Spin Currents 2009 Stanford Sierra Conference
Center at Fallen Leaf Lake, South Lake Tahoe
April 19th , 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
OUTLINE• Introduction• 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
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
Spin Currents 2009
The family of spintronic Hall effects
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
Spin Currents 2009
MRBR sH 40
Anomalous Hall effect basics and history
Simple electrical measurement of out of plane magnetization
Spin dependent “force” deflects like-spin particles
I
_ FSO
FSO
_ __
majority
minority
V
InMnAs
sRR 0
y
x
xxxy
xyxx
y
x
E
E
j
j
xxxyxx
xxxx
122
22222 xxxxxxxyxx
xy
xyxx
xyxy BA
xxxy AB Spin Currents 2009
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
Spin Currents 2009
•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
7
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
8
•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.
• 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.
AHE: Microscopic vs. Semiclassical
9
Intrinsic deflection
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
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
Spin Currents 2009
Consistent theory of anomalous transport in spin-orbit coupled systems
Mario F. Borunda, et al, "Absence of skew scattering in two-dimensional systems: Testing the origins of the anomalous Hall Effect", PRL 07
T. S. Nunner, et al“Anomalous Hall effect in a two-dimensional electron gas”, Phys. Rev. B 76, 235312 (2007)
A. A. Kovalev, et al “Hybrid skew scattering regime of the anomalous Hall effect in Rashba systems”, Phys. Rev. B Rapids 78, 041305(R) (2008).
Boltzmann approachKeldysh approach
Kubo approach
Spin injection Hall device is the perfect testing ground for these theory predictions
5m
Spin injection Hall effect: : experimental observation
-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 stripeSpin Currents 2009
-1.0 -0.5 0.0 0.5 1.0
-10
-5
0
5
10
H [
10-3 ]
( ) / (
)
n2
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()(V2 dis
*dis
22
rkrkkkkm
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 Currents 2009
GaAs, for A 2o
3.5* zE* , AeV 0
02.0 AeV 0
03.001.0
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
Lower bound estimate
Spin Currents 2009
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
SIHEB=0
Optical injected polarized
current gives charge current
Electrical detection
Spin Currents 2009
OUTLINE• Introduction• AHE in spin injection Hall effect:
– SIHE observations– AHE basics– Strong and weak spin-orbit couple contributions of AHE– AHE in SIHE
• Spin-charge dynamics of SIHE with magnetic field: – Static magnetic field steady state– Time varying injection
• AHE prospectives– Phenomenological regimes– New challenges
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
Steady state spin transport in diffusive regime
Spatial variation scale consistent with the one observed in SIHE
Spin Currents 2009
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
OUTLINE• Introduction• AHE in spin injection Hall effect:
– SIHE observations– AHE basics– Strong and weak spin-orbit couple contributions of AHE– 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
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
Spin Currents 2009 Sanford Sierra Conference Center at Fallen Leaf Lake, South Lake
TahoeApril 19th , 2009
Research fueled by:
Hitachi CambridgeJorg Wunderlich, A.
Irvine, et alInstitute of Physics ASCR
Tomas Jungwirth,, Vít Novák, et al
Texas A&M J. Sinova. L. ZarboStanford UniversityShoucheng Zhang, Rundong Li, Jin Wang
CONCLUSION
•Predicted resonant behavior for time varying polarization injection with perpendicular magnetic field•New tool to explore the AHE in the strong SO coupled regime
•AHE in SIHE signals in agreement with theory expectations
•Three rough AHE regimes appear when reviewing a large range of materials •AHE linear response regimes in metallic regime are now well understood in simple models
Spin Currents 2009
32
33
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
The Spin-Charge Drift-Diffusion Transport EquationsFor 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:
STTSCSC SDS
STSCnB SDS
STSCnB SDS
SBSBn Dn
zxxxxzzt
xzxxxxt
xzxxxxt
xxxxt
)( 21222
2212
1122
212
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
Spin Currents 2009
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.
35
0)(
)(ˆ
ˆ ''
k
kHkv
k
Hv nn
nnk