Rashba and other spin-orbit interactions
Yuli Lyanda-Geller
Department of Physics and Birck
Nanotechnology Center, Purdue
University
Work in progress in collaboration with M.I. Dykman
Workshop on Quantum Engineering with Electrons on Helium,
May 21 2010, Princeton NJ.
Central Themes
• Spin Qubit – spin-dependent interactions
•Spin-Orbit – free electrons
•Spin-Orbit – Confined Electrons (Rashba Interaction)
• Condensed Matter Systems
• Electrons on He – Rashba Interaction
•Spin-Orbit Coupling – ripplons
•Electron-Electron Interactions
Electron spin qubits on the He-4/vacuum interface
He
m=2
H
m=1
Decoherence ?
1D hydrogenic spectrum
Spin-orbit effects ?
Spin-orbit interactions
mc2 I+U(r)
cp
Free relativistic electron
-mc2 I+U(r)
cp
Expansion in p/mc




p
e     [ p  U ]
H
 U (r ) 
H 
2m
2mc
4m  mc2
2
2mc2
Spin-orbit interactions
Confined electrons
mc2 I+U(z)
cp
Dirac Hamiltonian
cp
-mc2 I+U(z)
The same U acts on electrons and positrons
  



p
e
  [ p  U ]
H
 U (r ) 
H   * ( z )
( z )dz
2
2m
2mc
4m  mc
2
Spin-orbit interactions
Confined electrons
 

[ p  ] 
*
  ( z ) U  ( z ) dz
2 
4m  mc


U  iH , pz   i U ,i   iHpz  pz H 
z 


*

(
z
)

U

(
z
)
dz


i


 ( z)(Hpz  pz H )( z)dz
*
 H  E ; H  E
*
Ehrenfest theorem
*
 
[ p   ] * 
 ( z)U( z )dz  0
2 
4m  mc
!!
Spin-orbit interactions
Condensed Matter systems
ms2 I+U1(r)+
eEr
sp
sp
Eg
-ms2 I+U2(r) +
eEr
Different offsets, different barriers
For conduction and valence electrons or ground s and excited p states
Spin-orbit interactions
Condensed Matter systems

Uc
Uv
H so

U c  U v   [ p  eE ]

Uc
4m  ms 2
Different offsets in conduction
and valence bands
Lassnig (1985)
Rashba type spin-orbit interaction He
He
m=2
• Penetration into He
• Excited p-state
• Different parameters of quantized state
in s and p-state
• Eg =50 eV , s= 2.7x108 cm/s
• m – free electron mass

m=1
H so

  [ p  eE ]

4m  ms 2
Order of magnitude stronger than relativistic spin-orbit term.
T1= 1s (=100ns)
Lateral confinement
All discrete levels; no k
Capillary waves - ripplons
Thermally excited height variations of the helium surface

 (r , t )

r
2D in-plane coordinate


U (r )  eE (r , t )
Spectrum
 2  ( /  )k 3
Spin relaxation in a He surface quantum dot
H so  
 ( x p y   y p x ) eE z
4m  ms 2
Up and down spins are described by the same (blue) orbital wavefunction
Admixture by Rashba term to red state (different lateral orbital wavefunction
Ripplon
T1= 106s
For inelastic [processes, acoustic phonons are more important
Other spin-orbit interactions
Capillary waves - ripplons
Thermally excited height variations of the helium surface

 (r , t )

r
2D in-plane coordinate


U (r )  eE (r , t )
Spectrum
 
  [ p  U ]

4m  ms 2

H so
E – total electric field in z=direction
 2  ( /  )k 3
Spin-orbit interactions with ripplons
 
  [ p  k ]
 eE
4m  ms 2

ms2 I+U(r,t)
sp
sp
H so
-ms2 I+U(r,t)
Magnitude:  is the average mean displacement
k is the transferred wavevector
Ripplon coupling limits mobility (= 108 cm2 /V s )
H so
 
   z [ p  k ]z
 eE

2
4m  ms
Only in-plane wavevectors
z
Spin-orbit interactions with ripplons
H so
 
   z [ p  k ]z
 eE
4m  ms 2
Sz is not affected , no spin relaxation
Sx , Sy are affected , hence spin dephasing (decoherence)
Spin dephasing times 0.1-2 s
Corresponds to momentum relaxation =100ns
Lateral confinement
 
  [ p  U ]

4m  ms 2

H so
Direct spin-ripplon coupling to
Each of the excited level
T2= 102s
Virtual escape of electron from the dot
Other spin-orbit interactions
Solids : interaction of spin of one electron
with an orbital motion of another electron
Eg
Breit –Landau term in two-electron Dirac Model

Electron-electron Interactions contribute to relaxation and dephasing
Conclusions
• In 2D, Rashba interaction is present
• Coupling with ripplons is more important
• Spin dephasing times shorter than
spin relaxation times
•Spin-orbit terms in electron-electron
interactions