Date post: | 03-Jan-2016 |
Category: |
Documents |
Upload: | kitra-jordan |
View: | 30 times |
Download: | 0 times |
The Persistent Spin Helix
Shou-Cheng Zhang, Stanford University
Banff, Aug 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 gating.
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
For Dresselhauss = 0, the equations reduce to Burkov, Nunez and MacDonald, PRB 70, 155308 (2004);
Mishchenko, Shytov, Halperin, PRL 93, 226602 (2004)
.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
21
22
2 / ,
2 /
F
F
C k m
C k m
2 21
2 22
2 ,
2
F
F
T k
T k
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
Generation of the PSH Current
[110] GaAs
FM1 FM2
PSH associated with SU(2) charge – PSH current
FM2 pulse delayed from FM1 pulse
Two consecutive FM1 pulses delayed by
Ev
Generation of the PSH Current
Optical detection of oscillating spin at given spatial point. Dresselhauss [110]
ReD GaAs
FM1 FM2
Optical detection
For Rashba equal Dresselhauss:
Decay component:
• 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