The spin Hall effectThe quantum AHE and the SHEThe persistent spin helix

Shou-cheng Zhang, Stanford University

Les Houches, June 2006

Credits

Collaborators:

• Andrei Bernevig (Stanford)• Taylor Hughes (Stanford)• Shuichi Murakami (Tokyo)• Naoto Nagaosa (Tokyo)• Xiaoliang Qi (Tsinghua and Stanford)• Congjun Wu (Stanford and KITP/Santa Barbara)• Yongshi Wu (Utah)

The spin Hall effect

Can Moore’s law keep going?Power dissipation=greatest obstacle for Moore’s law! Modern processor chips consume ~100W of power of which

about 20% is wasted in leakage through the transistor gates.

The traditional means of coping with increased power per generation has been to scale down the operating voltage of the chip but voltages are reaching limits due to thermal fluctuation effects.

0

100

200

300

400

500

0.5 0.35 0.25 0.18 0.13 0.1 0.07 0.05

Active Power

Passive Power (Device Leakage)

350 250 180 130 100 70 50

500

500

400

300

200

100

0

Technology node (nm)

Po

we

r d

ensi

ty (

W/c

m)2

Spintronics

• The electron has both charge and spin.

• Electronic logic devices today only used the charge property of the electron.

• Energy scale for the charge interaction is high, of the order of eV, while the energy scale for the spin interaction is low, of the order of 10-100 meV.

• Spin-based electronic promises a radical alternative, namely the possibility of logic operations with much lower power consumption than equivalent charge based logic operations.

• New physical principle but same materials! In contrast to nanotubes and molecular electronics.

Manipulating the spin using the Stern-Gerlach experiment

• Problem of using the magnetic field:

• hard for miniaturization on a chip.

• spin current is even while the magnetic field is odd under time reversal => dissipation just as in Ohm’slaw.

E

v

EBeff

v~

effBSH

~

Relativistic Spin-Orbit Coupling • Relativistic effect: a particle

in an electric field experiences an internal effective magnetic field in its moving frame

• Spin-Orbit coupling is the coupling of spin with the internal effective magnetic field

E

Using SO: spin FET

V

- vBeff

- vBeff

-v

Beff

V/2

•Das-Datta proposal.

•Animation by Bernevig and Sinova.

Generalization of the quantum Hall effect

Fspinkijkspini

j ekEJ

h

e

q

pEJ HjijHi

2

• Quantum Hall effect exists in D=2, due to Lorentz force.

• Natural generalization to D=3, due to spin-orbit force:

• 3D hole systems (Murakami, Nagaosa and Zhang, Science 2003)

• 2D electron systems (Sinova et al, PRL 2004)

GaAsy

z

x

B

EJ

x: current direction y: spin directionz: electric field

GaAs

E

x

y

z

Valence band of GaAs

Luttinger Hamiltonian

( : spin-3/2 matrix, describing the P3/2 band)S

2

22

21 22

5

2

1Skk

mH

2/3000

02/100

002/10

0002/3

02/300

2/3010

0102/3

002/30

02/300

2/300

002/3

002/30

zyx SS

i

ii

ii

i

S

S

P

S

P3/2

P1/2

)(22

5

2

1 2

22

21 xVSkkm

H

Unitary transformation

)()()(22

5

2

1)()( 22

22

21 kUxVkUSkkm

kHUkUH z

Diagonalize the first term with a local unitary transformation

HH

LH

LH

HH

m

k

:

:

:

:

2

2

2

2

2

23

21

21

23

21

21

21

21

2

)()()()( DVkUiVkU k

ii

i Ak

iD

)()( kUk

kiUAi

i

: gauge field in k!

zy SiSiz eekUkSkUSkkU )(,)()(

Helicity basis Sk

ˆ

)ˆ(kU

)'ˆ(kU

Local gauge field in k space

HH

LH

LH

HH

didd

idddidd

idddidd

iddd

dkA ii

:

:

:

:

cos)(sin

)(sincossin

sincos)(sin

)(sincos

23

21

21

23

23

23

23

21

21

23

23

23

Adiabatic transport = potential V does not cause inter-band transitions only retain the intra-band matrix elements

Abelian approximation = retain only the intra-helicity matrix elements

HH

LH

LH

HH

didd

idddidd

idddidd

iddd

dkA ii

:

:

:

:

cos)(sin

)(sincossin

sincos)(sin

)(sincos

23

21

21

23

23

23

23

21

21

23

23

23

)(2

2eff xV

m

kH

)(~

kAk

iDx ii

ii

Effective Hamiltonian for adiabatic transport

kjijki

iii kEkm

kxEk

3

,

ijjiijjiji iFxxikxkk ],[,],[,0],[

Eq. of motion

3k

kF k

ijkij

(Dirac monopole)

ik

E

Drift velocity Topological term ijj F

eE

Nontrivial spin dynamics comes from the Dirac monopole at the center of k space, witheg=:

Dissipationless spin current induced by the electric field

The intrinsic spin Hall effect

• Key advantage:• electric field manipulation, rather than

magnetic field.• dissipationless response, since both

spin current and the electric field are even under time reversal.

• Topological origin, due to Berry’s phase in momentum space similar to the QHE.

• Contrast between the spin current and the Ohm’s law:

lkh

ewhereEJorRVI Fjj

22

/

)(6

,2

LF

HFspinkijkspin

ij kk

eEJ

Bulk GaAs

Ene

rgy

(eV

)

- vT

-v

- v-v

- v

-vT

Time reversal and the dissipationless spin current

Effect due to disorder

Rashba model: Intrinsic spin Hall conductivity (Sinova et al.(2004))

+ Vertex correction in the clean limit (Inoue, Bauer, Molenkamp(2003))

0S

8

eS

+ spinless impurities ( -function pot.)

8vertex e

S

xyyx kkm

kH

2

2

xJzyJ

xJ

zyJ

Luttinger model: Intrinsic spin Hall conductivity (Murakami et al.(2003)) )(

6 2

LF

HFS kk

e

+ spinless impurities ( -function pot.)

0vertex S

yxxy SkSkSkm

kH 2

21

2

2

xJzyJ

xJ

zyJ

Vertex correction vanishes identically!2DHG Bernevig+Zhang (PRL 2004)

Mott scattering or the extrinsic Spin Hall effectE

Electric field induces a transverse spin current.

• Extrinsic spin Hall effect

Spin-orbit couping

Mott (1929), D’yakonov and Perel’ (1971) Hirsch (1999), Zhang (2000)

up-spin down-spinimpurity

• Intrinsic spin Hall effect Berry phase in momentum space

impurity scattering = spin dependent (skew-scattering)

Independent of impurities !

Cf. Mott scattering

Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom, Science 306, 1910 (2004)

Experiment -- Spin Hall Effect in a 3D Electron Film

(i) Unstrained n-GaAs(ii) Strained n-In0.07Ga0.93As

-316 cm103T=30K, Hole density:

: measured by Kerr rotationzS

Y.K.Kato et al., Science (2004)

• unstrained GaAs -- no strain spin-orbit coupling• strained InGaAs -- no crystal orientation dependence• extrinsic quantum spin hall calculation (Engel, Rashba, Halperin)• sign mismatch? but right ballpark value

It should be extrinsic!

• Dresselhaus term is relevant, opposite sign.

• Dresselhaus term is small, but induced SHE is not small.• For Dresselhaus term the vertex correction does not cancel the intrinsic SHE.

• Dirty limit : SHE suppressed by some factor, which is roughly

It could be intrinsic!

Bernevig, Zhang, cond-mat (2004)

meVmeV 6.1/,025.0 4

2

10/

Experiment -- Spin Hall Effect in a 3D Electron Film

• Circular polarization %1

meV2.1/

• Clean limit :

much smaller than spin splitting

• vertex correction =0 (Bernevig, Zhang (2004))

• should be intrinsic

J. Wunderlich, B. Kästner, J. Sinova, T. Jungwirth, PRL (2005)

• LED geometry

Experiment -- Spin Hall Effect in a 2D Hole Gas

Direct measurement of the spin current?

E

A modified version of the standard drift-diffusion experiment in semiconductor physics. Optically inject up or down spin carriers, and observe the longitudinal charge drift and the spin-dependent transverse drift.

x

y

z

Spin – Orbit Coupling in Two Dimensions

Rashba Hamiltonian

Strong out-of plane junction electric field

E GaAs

General Hamiltonian for spin ½ systems:

• Upon momentum integration continuity equations:

• Rashba coupling (2D Asymmetric Quantum Wells):

Burkov Nunez and MacDonald; Mishchenko, Shytov and Halperin

Transport In Spin ½ Systems: Two Dimensions

Maxwell’s EquationsSpin-Orbit Coupling

Rashba SO Coupling: 2D PhotonBernevig, Yu and Zhang, PRL 95, 076602 (2005)

VS C

+-

+ -

R

Spintronics without spin injection and spin detection

In conventional charge dynamics, injected charge packets simply diffuses.

E

With a E field, the charge packets also drifts.Drift-diffusion is the fundamental process underlying all conventional electronics.

With strong spin-orbit coupling, injected charge packet spontaneously splits into two spin packets, propagating in opposite directions at the Rashba speed, without any applied E field. This effect can be used to construct a spin bus.

• Spin Hall effect is a profoundly deep effect in solid state physics,Natural generalization of the Hall effect and quantum Hall effect.

• Natural extensions of the spin Hall effect: orbitronics, spintronics withoutSpin injection and spin detection, quantum spin Hall effect.

• Need close interaction among theory, experiments and materials science. • Frontier of science and technology.

Conclusions