JAIRO SINOVA

Research fueled by:

New Horizons in Condensed Matter PhysicsAspen Center for Physics February 4th 2008

Theory challenges of semiconducting spintronics: spin-Hall effect and spin-dependent transport in

spin-orbit coupled systems

NERC

The challenges ahead in semiconductor spintronics

• Spin/charge transport in multi-band systems with inter-band coherence (Berry’s phase dependent transport): • SHE and AHE in strongly SO coupled systems, etc.• What are the relevant length scales: spin-current connection to spin-

accumulation.• QSHE: transport in Z2 systems.• Technological issues: how dissipative is it?

• Interplay between quasiparticle and collective degrees of freedom in a multiband system:• Carrier mediated ferromagnetism: diluted magnetic semiconductors• Magnetization dynamics: obtaining phenomenological LLG coefficients through

microscopic calculations

Anomalous Hall effect: where things started, the long debate

MπRBR sH 40 Simple electrical measurement of magnetization

Spin-orbit coupling “force” deflects like-spin particles

I

_ FSO

FSO

_ __majority

minority

V

controversial theoretically: semiclassical theory identifies three contributions (intrinsic deflection, skew scattering, side jump scattering)

Spin Hall effect

I

_ FSO

FSO

_ __

V=0

non-magneticSpin-current generation in non-magnetic systems

without applying external magnetic fieldsSpin accumulation without charge accumulation

excludes simple electrical detection

Carriers with opposite spin are deflected by the SOC to opposite sides.

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.

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

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.

Related to the intrinsic effect: analogy to refraction from an imbedded medium

Side jump scattering

Skew scatteringAsymmetric 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.

Spin Hall Effect(Dyaknov and Perel)

InterbandCoherent Response

(EF) 0Occupation #

Response

`Skew Scattering‘ (e2/h) kF (EF )1

X `Skewness’

[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.]

Paramagnets Quantum Spin Hall Effect

(Kane et al and Zhang et al)

Future challenges in anomalous transport theory

1. Reaching agreement between different approaches (mostly AHE)

2. Connect spin current to spin accumulation for strongly SO system

3. Connect SHE to the inverse SHE in strongly SO coupled regime (charge based measurements of SHE)

4. Understanding weak localization corrections in SO coupled systems for Hall transport

5. Systematic treatment of microscopic calculations of AHE in strongly SO coupled ferromagnet (e.g. DMS) with complex band structure

6. Dissipation: answer the questions if a spin based device can really beat the kBTln2 limit of dissipation

Need to match Kubo to Boltzmann to Keldysh

Kubo: systematic formalism Boltzmann: easy physical interpretation of

different contributions Keldysh: microscopic version of Boltzmann

+ more

1. Intrinsic + Extrinsic:Connecting Microscopic and Semiclassical

approachesSinitsyn et al PRL 06, PRB 07

AHE in Rashba systems with disorder: Dugaev et al PRB 05 Sinitsyn et al PRB 05 Inoue et al (PRL 06) Onoda et al (PRL 06)

Borunda et al (PRL 07)

All are done using same or equivalent linear response formulation–different or not obviously equivalent answers!!!

The new challenge: understanding 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

2. From spin current to spin accumulation

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 !!

-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

%

1.5mmchannel

n

n

pyx

z

LED1

LED2

10mm 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

Non-equilibrium Green’s function formalism (Keldysh-LB)

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

3. Charge based measurements of SHE

PRL 05

H-bar for detection of Spin-Hall-Effect

(electrical detection through inverse SHE)

E.M. Hankiewicz et al ., PRB 70, R241301 (2004)

New (smaller) sample

1 mm

200 nm

sample layout

-2 -1 0 1 2 3 4 5

0

1000

2000

3000

4000

5000

6000

7000

0

1

2

3

4

5R

nonl

ocal /

VGate

/ V

Q2198HI (3,6)U (9,11)

I / n

A

SHE-Measurement

no signalin the n-conductingregime

strong increase of the signal in thep-conducting regime,with pronounced features

insulatingp-conducting n-conducting

Mesoscopic electron SHE

L

L/6L/2

calculated voltage signal for electrons (Hankiewicz and Sinova)

Mesoscopic hole SHE

L

calculated voltage signal (Hankiweicz, Sinova, & Molenkamp)

L

L/6

L/2

Theoretical achievements:

Theoretical challenges:GUT the bulk (beyond simple graphene)

intrinsic + extrinsic SHE+AHE+AMR

Obtain the same results for different equivalent approaches (Keldysh and Kubo must agree)

Othersmaterials and defectscoupling with the latticeeffects of interactions (spin Coulomb drag)spin accumulation -> SHE conductivity

Intrinsic SHEback to the beginning on a higher level

2003 2006Extrinsic SHE

approx microscopic modelingExtrinsic + intrinsic AHE in graphene:

two approaches with the same answer

WHERE WE ARE GOING (THEORY)

EXTRAS

Experimental achievements

Experimental (and experiment modeling) challenges:

Photoluminescence cross sectionedge electric field vs. SHE induced spin accumulationfree vs. defect bound recombinationspin accumulation vs. repopulationangle-dependent luminescence (top vs. side emission)hot electron theory of extrinsic experiments

Optical detection of current-induced polarizationphotoluminescence (bulk and edge 2DHG)Kerr/Faraday rotation (3D bulk and edge, 2DEG)

Transport detection of the SHE

Generaledge electric field (Edelstein) vs. SHE induced spin accumulation

SHE detection at finite frequenciesdetection of the effect in the “clean” limit

WHERE WE ARE GOING (EXPERIMENTS)

Scaling of H-samples with the system size

L

L/6

0.000 0.004 0.008 0.0122.0

2.5

3.0

3.5

4.0

4.5

5.0

L=150nm

n=1*1011cm-2

Spin orbit coupling = 72meVnm

L=90nm

L=120nmL=200nm

L=240nm

chan

ge o

f vol

tage

[mV

]

1/L [nm]

Oscillatory character of voltage difference with the system size.

Aharonov-Casher effect: corollary of Aharonov-Bohm effect

with electric fields instead

Control of conductance through a novel Berry’s phase effect induced by gate voltages instead of magnetic fields

• M. Koenig, et al, "Direct observation of the Aharonov-Casher phase", Phys. Rev. Lett. 96, 076804 (2006).

• Alexey A. Kovalev, et al "Aharonov-Casher effect in a two dimensional hole ring with spin-orbit interaction", pre-print: cond-mat/0701534, submitted to Phys. Rev. B

HgTe Ring-Structures

Three phase factors:

Aharonov-Bohm

Berry

Aharonov-Casher

effexttottotext

tot

BBBBBb

Bs

;,,for 1

toparallelanti and parallel ,and

THE THREE CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACH

Skew scattering

Side-jump scattering

Intrinsic AHE

SkewσH

Skew (skew)-1 2~σ0 S where S = Q(k,p)/Q(p,k) – 1~

V0 Im[<k|q><q|p><p|k>]

Vertex Corrections σIntrinsic

Intrinsicσ0 /εF

n, q

n, q m, p

m, pn’, k

n, q

n’n, q

= -1 / 0

Averaging procedures:

= 0

Success of intrinsic AHE approach in strongly SO

coupled systems• DMS systems (Jungwirth et al PRL 2002)• Fe (Yao et al PRL 04)• Layered 2D ferromagnets such as SrRuO3 and

pyrochlore ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19 (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003), Shindou and Nagaosa, Phys. Rev. Lett. 87, 116801 (2001)]

• Colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999).

• Ferromagnetic Spinel CuCrSeBr: Wei-Lee et al, Science (2004)

Berry’s phase based AHE effect is quantitative-

successful in many instances BUT still not a theory that

treats systematically intrinsic and extrinsic

contribution in an equal footing.

Experiment sAH 1000 ( cm)-1

TheroysAH 750 ( cm)-1

First experimental observations at the end of 2004

Wunderlich, Kästner, Sinova, Jungwirth, PRL 05

Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system

-1

0

1

CP [%

]

Light frequency (eV)1.505

1.52

Kato, Myars, Gossard, Awschalom, Science Nov 04

Observation of the spin Hall effect bulk in semiconductors

Local Kerr effect in n-type GaAs and InGaAs: (weaker SO-coupling, stronger disorder)

OTHER RECENT EXPERIMENTS

“demonstrate that the observed spin accumulation is due to a transverse bulk electron spin current”

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

SHE at room temperature in HgTe systems Stern et al PRL 06 !!!

Semiclassical Boltzmann equation

' ''

( )l ll l l l

l

f feE f ft k

2' ' '

' '' ''' '

' ''

2 | | ( )

...

l l l l l l

l l l ll l l l

l l

T

V VT Vi

Golden rule:

' 'l l ll J. Smit (1956):Skew Scattering

In metallic regime:

Kubo-Streda formula summary

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 )]

2' ' '

2 | | ( )l l l l l lV

Golden Rule:

Coordinate shift:

0 0' ' ' '

' '

( ) ( )v ( )l l ll l l l l l l l l

l ll l

f f feE eE r f ft

ModifiedBoltzmannEquation:

( , )l km

Iml l l l lz

y x x y

u u u uFk k k k

Berry curvature:

' ' ' '',ˆ arg

'l l l l l l l lk kr u i u u i u D Vk k

' ''

lll l l l l

l

v F eE rk

velocity:

l ll

J e f v

current:

' 'l l l lV T

Semiclassical approach II

Sinitsyn et al PRL 06, PRB 06

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 )]

In metallic regime: IIxyσ =0

Kubo-Streda formula:

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

Single K-band with spin up

x x y y so zKH =v(k σ +k σ )+Δ σ

Sinitsyn et al PRL 06, PRB 06 SAME RESULT OBTAINED USING BOLTMANN!!!

For single occupied linear Rashba band; zero for both occupied !!

Armchair edge

Zigzag edge

EF

Success in graphene

Comparing Boltzmann to Kubo in the chiral basis

0 0' ' ' '

' '

( ) ( )v ( )l l ll l l l l l l l l

l ll l

f f feE eE r f ft

The spintronics Hall effects: multi-band transport with inter-band

coherence

AHE

SHEcharge current

gives spin current

polarized charge current gives charge-spin

current

SHE-1

spin current gives

charge current

Anomalous Hall transport

Commonalities:

• Spin-orbit coupling is the key• Same basic (semiclassical)

mechanisms

Differences:

• Charge-current (AHE) well define, spin current (SHE) is not

• Exchange field present (AHE) vs. non-exchange field present (SHE-1)

Difficulties:

• Difficult to deal systematically with off-diagonal transport in multi-band system

• Large SO coupling makes important length scales hard to pick• Farraginous results of supposedly equivalent theories• The Hall conductivities tend to be small

Actual gated H-bar sample

HgTe-QW

R = 5-15 meV

5 mm

ohmic Contacts

Gate-Contact

First Data

HgTe-QW

R = 5-15 meV

Signal due to depletion?

Results...

Symmetric HgTe-QW

R = 0-5 meV

Signal less than 10-4

-2 -1 0 1 2 3

-3.00E-007

-2.50E-007

-2.00E-007

-1.50E-007

-1.00E-007

-5.00E-008

U_7

-10

[V]

-V_gate14 [V]

I: 1->4U:7-10

Sample is diffusive:Vertex correction kills SHE (J. Inoue et al., Phys. Rev. B 70, 041303 (R) (2004)).