Transcript The Persistent Spin Helix

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.
  1 Dq2
Spin Relaxation in Solids
• With SO coupling, the dominant mechanism is the DP
relaxation.
S
z
S
 p  :
:

The spin-orbit field
Momentum relaxation time
The 2D random
walk problem:
   t  


  
2

t

,  2  2t
dS z
1
1 2
 S z 
The effective


S

1

cos






t




dt

2
reduction of Sz:  S 
S
1
S
 2
The Rashba+Dresselhaus Model
k2
H
   k y x  k x y     k x x  k y y 
2m
The Rashba spin-orbit coupling.
Can be experimentally tuned via
proper gating.
The Dresselhauss spin-orbit
coupling.
Increase Dresselhauss
The Rashba+Dresselhaus Model
For α=β
k2
H
   k x  k y  x   y 
2m
Global spin U  1 1  i     
x
y 

2
2
rotation

Coordinate change
k 
1
kx  k y 

2
2
2
k

k
H ReD  U  HU     2 k z
2m
The Dresselhaus [110] Model
Symmetric Quantum wells grown along the [110]
direction:
H110 
k x2  k y2
2m
 2 k x z
Fermi Surface and the Shifting Property
• The shifting property:
 

 k   k  Q

For the HReD model
H ReD
Q  4m , Q  0
For the H110 model
Qx  4m , Qy  0
H110
The Exact SU(2) Symmetry
• Finite wavevector spin components
SQ  k ckck Q ,
SQ  k ckQck  ,
 S0z , SQ   2 SQ ,
S0z  k ckck   ckck 
 SQ , SQ   S 0z
• Shifting property essential
 

 
 H ReD , c  c     k  Q    k c  c  0
k Q k  
k Q k 

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).
The Exact SU(2) Symmetry
• The SU(2) symmetry is robust against spin-independent disorder and
Coulomb (or other many-body) interactions.
q   k ckq ck
  q , SQ     q , S0z   0
 Vq q , SQ     Vq q , S0z   0,
 q
  q

 Vq q  q , SQ    Vq  q  q , S0z   0
 q
  q

x
y
• A S , S spin helix with wave vector Q has infinite life time
Persistent Spin Helix
Physical Picture: Persistent Spin Helix
• Spin configurations do not depend on the particle initial momenta.
• For the same x distance traveled, the spin precesses
by exactly the same angle.
• After a length L  2 Q the spins all
return exactly to the original
configuration.
PSH for the
H ReD Model and the H110 Model
(a) PSH for the H ReD model. The spin-orbit magnetic field is in-plane (blue),
where as the spin helix is in the  x , z  plane. (b) PSH for the H[110] model.
spin
The spin-orbit magnetic field Borbit , in blue, is out of plane, whereas the spin
helix, in red, is in-plane.
The Non-Abelian Gauge Transformation
H ReD in the form of a background non-abelian gauge potential
k2
1
2
H ReD 

 k  2m z   const.
2m 2m
P. Q. Jin, Y. Q. Li, and
F. C. Zhang, J. Phys.
A 39, 7115 (2006)
• Field strength vanishes; eliminate the vector potential by non-abelian gauge transf
  x , x   exp i2m x    x , x  ,   x , x   exp  i2m x    x , x 
H Re D
k2
H 
2m
S   x     x    x   exp  i4m x  S   x 
S   x     x    x   exp i4m x  S   x 
•Mathematically, the PSH is a direct manifestation of
a non-abelian flux in the ground state of the models.
   0
The Boltzmann Transport Equations
For arbitrary α,β spin-charge transport equation is obtained for diffusive regime
t n  D n  B1 x Sx  B2 x Sx
2
i
B1  2         kF2 2 ,
2
B2  2         kF2 2
2
t Sx  Di2Sx  B1 x n  C2 x Sz  T2Sx
C1  2     k F2 / m,
t Sx  Di2Sx  B2 x n  C1 x Sz  T1Sx
T1  2     k F2 ,
C2  2     k F2 / m
2
T2  2     k F2
2
t Sz  Di2Sz  C2 x Sx  C1 xSx  T1  T2  Sz
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)
The Boltzmann Transport Equations
For α=β :
k2
H
   k x  k y  x   y 
2m
Gauge
transformation
S   x   S x  iS y     x    x 
S   x   S x  iS y     x    x 
k2
H
2m
(Free Fermi gas)
 S x   cos qx  sin qx   S x 
 
 S   sin qx cos qx   S y 
 y
 t S x  D   2x   2y  S x  2qD x S y  Dq 2 S x


2
2
2
 t S y  D   x   y  S y  2qD x S x  Dq S y
Simple diffusion equation
 S , S    cos  4m x  ,sin  4m x 
Sx  const.
x
y


t Si  Di2 Si
The Boltzmann Transport Equations
Along special directions the four equations decoupled to two by two blocks
Propagation on [110]
Propagation on [1ῑ0]
qx  q, qx  0
qx  q, qx  0
n  Sx , Sz  Sx
i1,2   Dq 2 
n  Sx , Sz  S

1
2T2  T1  T12  4q 2C22
2

At α=β i1  Dq2  T1, i2  Dq2
The behavior of Sz is diffusive
and exponentially decaying; this
is the passive direction
C1  2     k F2 / m,
T1  2     k F2 ,
C2  2     k F2 / m
T2  2     k F2
2
2
i1,2   Dq 2 
At α=β

1
2T1  T2  T22  4q 2C12
2
i1,2  Dq2  T1 C1q
At the shifting wave-vector Q
i2  q  4m  Q  0
An infinite spin life-time of the
Persistent Spin Helix; this is the
active direction

The Optical Spin Grating Experiment
C. P. Weber et. al., Nature 437, 1330 (2005)
Interference of two orthogonally
polarized beams
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.
An optical helicity wave generates
an electron spin polarization wave
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
PSH associated with SU(2) charge – PSH current
FM2 pulse delayed from FM1
pulse
FM1
Two consecutive FM1 pulses
delayed by
FM2
[110] GaAs


v  E
Generation of the PSH Current
Optical detection of oscillating spin at given spatial point. Dresselhauss [110]
For Rashba equal Dresselhauss:
Optical
detection
Decay component:
FM1
FM2
ReD GaAs
Conclusions
• 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