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