the persistent spin helix shou-cheng zhang, stanford university les houches, june 2006
Post on 21-Dec-2015
219 views
TRANSCRIPT
![Page 1: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/1.jpg)
The Persistent Spin Helix
Shou-Cheng Zhang, Stanford University
Les Houches, June 2006
![Page 2: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/2.jpg)
Credits
Collaborators:
• B. Andrei Bernevig (Stanford)
• Joe Orenstein (Lawrence Berkeley Lab)
• Chris Weber (Lawrence Berkeley Lab)
![Page 3: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/3.jpg)
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
![Page 4: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/4.jpg)
Spin Relaxation in Solids• Without SO coupling, particle diffusion is the only mechanism to relax the spin.
21 Dq
![Page 5: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/5.jpg)
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:
![Page 6: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/6.jpg)
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
![Page 7: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/7.jpg)
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:
![Page 8: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/8.jpg)
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
![Page 9: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/9.jpg)
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
![Page 10: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/10.jpg)
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
![Page 11: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/11.jpg)
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
![Page 12: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/12.jpg)
(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
![Page 13: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/13.jpg)
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
![Page 14: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/14.jpg)
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
![Page 15: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/15.jpg)
.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
![Page 16: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/16.jpg)
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
![Page 17: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/17.jpg)
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)
![Page 18: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/18.jpg)
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.
![Page 19: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/19.jpg)
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
![Page 20: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/20.jpg)
• 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
![Page 21: The Persistent Spin Helix Shou-Cheng Zhang, Stanford University Les Houches, June 2006](https://reader035.vdocuments.net/reader035/viewer/2022062407/56649d545503460f94a3098a/html5/thumbnails/21.jpg)
Device
[110] GaAs
FM1 FM2