The Persistent Spin Helix

Shou-Cheng Zhang, Stanford University

Les Houches, June 2006

Credits

Collaborators:

• B. Andrei Bernevig (Stanford)

• Joe Orenstein (Lawrence Berkeley Lab)

• Chris Weber (Lawrence Berkeley Lab)

Outline

• Mechanisms of spin relaxation in solids

• Exact SU(2) symmetry of spin-orbit coupling models

• The Persistent Spin Helix (PSH)

• Boltzmann equations

• Optical spin grating experiments

Spin Relaxation in Solids• Without SO coupling, particle diffusion is the only mechanism to relax the spin.

21 Dq

Spin Relaxation in Solids

• With SO coupling, the dominant mechanism is the DP relaxation.

The spin-orbit field

: p

: Momentum relaxation time

zS

S

2

2 2,t t

t

The 2D random walk problem:

211 cos

2zS tS

1z

S

dSS

dt 21

S

The effective reduction of Sz:

The Rashba+Dresselhaus Model

2

2 y x x y x x y y

kH k k k k

m

The Rashba spin-orbit coupling.Can be experimentally tuned via proper grating.

The Dresselhauss spin-orbit coupling.

Increase Dresselhauss

The Rashba+Dresselhaus Model

The Dresselhaus [110] Model

For α=β 2

2 x y x y

kH k k

m

1

2x yk k k

Coordinate change 11

2 2x y

iU

Global spinrotation

2 2

ReD 22 z

k kH U HU k

m

2 2

110 22x y

x z

k kH k

m

Symmetric Quantum wells grown along the [110] direction:

Fermi Surface and the Shifting Property

k k Q

• The shifting property:

For the

ReDH model

4 , 0Q m Q

For the

110H model

4 , 0x yQ m Q

ReDH

110H

The Exact SU(2) Symmetry

0, , zQ Qk k Q k Q k k k k kk k kS 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 wavevector spin components

• Shifting property essential

The Exact SU(2) Symmetry

• The SU(2) symmetry is robust against spin-independent disorder and Coulomb (or other many-body) interactions.

q k q kkc c

0, , 0z

q Q qS S

0

0

, , 0,

, , 0

zq q Q q qq q

zq q q Q q q qq q

V S V S

V S V S

• A spin helix with wave vector has infinite life time,x yS S Q

Persistent Spin Helix

Physical Picture: Persistent Spin Helix

• Spin configurations do not depend on the particle initial momenta.

• For the same distance traveled, the spin precesses by exactly the same angle.

• After a length the spins all return exactly to the original configuration.

x

2L Q

(a) PSH for the model. The spin-orbit magnetic field is in-plane (blue), where as the spin helix is in the plane. (b) PSH for the model. The spin-orbit magnetic field , in blue, is out of plane, whereas the spin helix, in red, is in-plane.

ReDH

[110]H ,x zspinorbitB

PSH for the Model and the Model

ReDH 110H

The Non-Abelian Gauge Transformation

ReDH in the form of a background non-abelian gauge potential

22

ReD

12 .

2 2 z

kH k m const

m m

• Field strength vanishes; eliminate the vector potential by non-abelian gauge transf

, exp 2 , , , exp 2 ,x x i m x x x x x i m x x x

exp 4S x x x i m x S x

exp 4S x x x i m x S x

•Mathematically, the PSH is a direct manifestation of a non-abelian flux in the ground state of the models.

P. Q. Jin, Y. Q. Li, and F. C. Zhang, J. Phys. A 39, 7115 (2006)

2

Re 2D

kH H

m

0

The Boltzmann Transport Equations

21 2t i x x x xn D n B S B S

21 2 2t x i x x x z xS D S B n C S T S

22 1 1t x i x x x z xS D S B n C S T S

22 1 1 2t z i z x x x x zS D S C S C S T T S

2 2 21

2 2 22

2 ,

2

F

F

B k

B k

21

22

2 / ,

2 /

F

F

C k m

C k m

2 21

2 22

2 ,

2

F

F

T k

T k

For arbitrary α,β spin-charge transport equation is obtained for diffusive regime

For propagation on [110], the equations decouple two by two

.xS const

The Boltzmann Transport Equations

For α=β :

Gauge transformation

, cos 4 ,sin 4x yS S m x m x

2

2

kH

m (Free Fermi

gas)

Simple diffusion equation

2t i i iS D S

cos sin

sin cosx x

yy

S Sqx qx

Sqx qxS

2

2 x y x y

kH k k

m

x yS x S iS x x

x yS x S iS x x

2 2 2

2 2 2

2

2

t x x y x x y x

t y x y y x x y

S D S qD S Dq S

S D S qD S Dq S

Propagation on [1ῑ0] Propagation on [110]

Along special directions the four equations decoupled to two by two blocks

, 0x xq q q

,x z xn S S S

2 2 2 21,2 2 1 1 2

12 4

2i Dq T T T q C

At α=β 2 21 1 2,i Dq T i Dq

, 0x xq q q

,x zn S S S

2 2 2 21,2 1 2 2 1

12 4

2i Dq T T T q C

At α=β 21,2 1 1i Dq T C q

The behavior of Sz is diffusive and exponentially decaying; this is the passive direction

An infinite spin life-time of the Persistent Spin Helix; this is the active direction

2 4 0i q m Q At the shifting wave-vector Q

The Boltzmann Transport Equations

The Optical Spin Grating Experiment

Interference of two orthogonally polarized beams

An optical helicity wave generates an electron spin polarization wave

The pump-probe technique:•The spatially modulation of spin or charge is first introduced by the ‘pump’ laser pulse.•The time evolution of the modulation is measured by the diffraction of a probe beam.•Spin transport and relaxation properties are probed.

C. P. Weber et. al., Nature 437, 1330 (2005)

The Optical Spin Grating Experiment

Measurements of the decay, at q close to the ‘magic’ shifting vector, at Rashba close, but not equal to Dresselhauss Black is the active direction, red the passive.

The Optical Spin Grating Experiment

Fitting of experimental data to Boltzman transport equations, for Rashba/Dresselhauss ~ 0.2 - 0.3. Even though the Rashba and Dresselhauss are not yet equal, large enhancement of spin-lifetime for the spin helix is observed

• Minimize spin-decoherence while keeping strong spin-orbit coupling

• Shifted Fermi Surfaces: Fundamental property of some cond-mat systems, similar to nesting

• Exact SU(2) symmetry of systems with Rashba equal to Dresselhauss or Dresselhauss [110]; finite wave-vector generators

• Persistent Spin Helix

• Experimental discovery

Conclusions

Device

[110] GaAs

FM1 FM2