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