Origin of the spin-orbit interaction
B
E
v
s
+
E
Mott 1927
1
c
1
E×p
mc
µB 
i 2  
Ĥ =
σ i (E × p̂) = −
σ i (∇V × ∇)
2 2
mc
2m c
In a frame associated with the electron: B = E × v =
Zeeman energy in the SO field:
1/c expansion of the Dirac equation
p̂ 2
p̂ 4
2
  
2
Ĥ =
+ eV +
+
∇V+
σ i(∇V × p̂)
2 2
2 2
2 2
2m
8m c
8mc 
4m
c 


 


non-relativistic
K.E. correction
Darwin term
SOI
All three corection terms are of the same order
Smaller by a factor of 2 (“Thomson factor of two”, 1926)
Reason: non-inertial frame
Landau& Lifshits IV: Berestetskii, Lifshits, Pitaevskii
Quantum Electrodynamics, Ch. 33
( Ze
2
/ c
)
SO coupling
2
( Ze
2
/ c
)
2
Si14
Ga 31
Ge 32
As 33
Bi83
0.01 0.051 0.055 0.058 0.38
Ga
32
Ga
Si
14
Ge
32
33
Bi
83
Fine structure of atomic levels
Hydrogen
Runge-Lentz
(non-relativistic Coulomb)
+spin degeneracy
2s
2s1/2
Johnson-Lippman
(relativistic Coulomb)
degeneracy
lifted by radiation corrections
(Lamb shift)
Two types of SOI in solids
1)  Symmetry-independent:
exists in all types of crystals
stem from SOI in atomic orbitals
2) Symmetry-dependent:
exists only in crystals without inversion symmetry
a) Dresselhaus interaction (bulk): Bulk-Induced-Assymetry (BIA)
b) Bychkov-Rashba (surface): Surface-Induced-Asymmetry (SIA)
Example of symmetry-independent SOI:
SO-split-off valence bands
I
E: j = 1 / 2; jz = ±1 / 2
no I
2
2
⎡ (ZGa + Z As ) / 2 ⎤ ⎛ ZGe ⎞
⎛ 32 ⎞
=
=
⎜⎝ ⎟⎠ = 5.2
⎢
⎥ ⎜
⎟
Z Si
14
⎣
⎦ ⎝ Z Si ⎠
Δ GaAs / Δ Si = .34/0.044 ≈ 7.7
Δ GaAs / Δ Ge = .34/.029 ≈ 1.2
2
Winkler, Ch. 3
Non-centrosymmetric crystals
space group: O h7 (non-symmorphic)
factor group Oh = Td × I
S. Sque (Exter)
Diamond (C): Si, Ge
[111]
3a
3a / 4
[111]
3a
3a / 4
space group: T 2d (symmorphic)
point group: Td (methane CH 4 )
Zincblende (ZnS): GaAs, GaP, InAs, InSb, ZnSe, CdTe …
Additional band splitting in non-centrosymmetric crystals
Kramers theorem: if time-reversal symmetry is not broken,
all eigenstates are at least doubly degenerate
if ψ is a solution, ψ * is also solution
Kramers doublets
ε s ( k ) , s = ±1 (not necessary spin projection!)
Time reversal symmetry: k-k,t-t
ε s ( k ) = ε − s ( −k )
No SOI: ε ( k ) = ε ( −k ) regardless of the inversion symmetry
With SOI:
i) If a crystal is centrosymmetric
εs (k) =
 ε − s ( −k ) =
 ε−s (k) ⇒ εs (k) = ε−s (k)
t →−t
k→− k
ii) If a crystal is non-centrosymmetric,
εs (k) ≠ ε−s (k)
B=0 “spin” splitting
Symmetry-dependent SOI:
Dresselhaus band splitting in
non-centrosymmetric crystals
Surface-induced asymmetry: Rashba interaction
z>0 and z<0 half-spaces
are not equivalent:
n̂ has a specified direction

k,
n̂,
σ
Three vectors:
n̂
k
E. I. Rashba and V. I. Sheka: Fiz. Tverd. Tela 3 (1961) 1735; ibid. 1863;
Sov. Phys. Solid State 3 (1961) 1257; ibid. 1357.
How to form a scalar?
Yu. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984);
Yu. A. Bychkov and E. I. Rashba, J. Phys. C: Solid State Phys. 17 (1984) 6039.
(

k2
k2
Ĥ R =
+ α n̂i(σ × k ) =
+ α σ x ky − σ y kx
2m
2m
t → −t : k → −k, σ → −σ
⎛ k 2 / 2m
HR = ⎜
⎜ α k − ik
y
x
⎝
(
)
(
)
)
2
α ky + ikx ⎞
k
⎟ ⇒ ε =
± αk
±
2
⎟
k / 2m
2m
⎠
Rashba states
ε
ΔR
εF
k
k2
ε± =
± αk
2m
Sz, ±
Sx, ±
1 ⎞
1 ⎛
ψ± =
⎜
⎟
iφ
2 ⎝ ie k ⎠
1
= ψ ±σ zψ ± = 0
2
1
1
1
1
x
y
= ψ ±σ ψ ± = ± sin φ k ; Sy, ± = ψ ±σ ψ ± =  cos φ k
2
2
2
2
kiS± = 0
Conductance of Rashba wires
BSO
(
k2
2D : Ĥ R =
+ α σ x ky − σ y kx
2m
kx2
1D : Ĥ R =
− ασ y kx
2m
)
Quay et al. Nature Physics 6, 336 - 339 (2010)
Hole wire
Dresselhaus+Rashba
Dresselhaus Hamiltonian
G. Dresselhaus: Phys. Rev. 100(2), 580–586 (1955)
zincblende (AIIIBV) lattice


H D = DΩ ( k )iσ

Ω = kx ky2 − kz2 , ky ( kz2 − kx2 ) , kz kx2 − ky2
( (
)
(
))
(001) surface (interface) of a AIIIBV semiconductor
kz = 0, k 2z ~ 1 / d 2
⎛
⎞

2
2
2
2
2
Ω = ⎜ kx k y − kx k z , ky kz − ky k x ,0 ⎟ = kz −kx , ky ,0
 ⎠
⎝
(
cubic→0
)
cubic→0
x
x
(
H DS = β σ y ky − σ k
)
in general: Rashba+Dresselhaus
(
) (
)
(
)
(
2
k2
k
H=
+ α σ x ky − σ y kx + β σ y ky − σ x kx =
+ σ x α ky − β kx + σ y α kx + β ky
2m
2m
)
Datta-Das Spin Transistor
 k 2 gµB 
2α
H=
+
σ iB R (k); B R k =
k × n ≡ Rashba field
2m
2
gµB
()
even number of π rotations: ON
odd number of π rotations: OFF
Loss-DiVincenzo Quantum Computer
Phys. Rev. A 57, 120 (1998)
GaAs: strong spin-orbit coupling; strong hyperfine interaction
(Extrinsic) Spin-Hall Effect
FSO
J
FSO
Js
Spin Hall Effect: the regular current (J) drives a spin
current (Js) across the bar resulting in a spin
accumulation at the edges.
e
Skew scattering
More spin up electrons are
deflected to the right than to the
left (and viceversa for spin down)
impurity
Side Jump
e
For a given deflection, spin up and
spin down electrons make a sidejump in opposite directions.
Intrinsic Spin-Hall Effect
No observations as of yet
k
I
unbounded 2D: magnetoelectric effect
[V. M. Edelstein, Solid State Comm. 73, 233 (1990). "
Can an electric field produce
n
magnetization?
Drift momentum
BR
kx = eEτ
M
2
 k
gµ 
2α
H=
+ B σ iB R (k); B R k =
k × n ≡ Rashba field
2m
2
gµB
()
2α
2α eEτ
M y = µB B =
kx =
g
g
Impurities are only necessary to maintain steady state
Current induces
steady Rashba field
y
R
(but forgetting about them leads to incorrect results--”universal spin-Hall conductivity”
Murakami et al. Science 301, 1348 (2003)
Sinova et al. Phys. Rev. Lett. 92, 126603 (2004)
References
1.
M. I. Dyakonov, ed. Spin Physics in Semiiconductors, Spinger 2008
http://www.springerlink.com.lp.hscl.ufl.edu/content/g7x071/#section=215878&page=1
available through UF Library
Ch. 1 M.I. Dyakonov, Basics of Semiconductor and
Spin Physics.
Ch. 8 M. I. Dyakonov and A. V. Khaetskii,
Spin Hall Effect
2. R. Winkler, Spin-Orbit Coupling Effects in 2D Electron
and Hole System, Springer 2003.
3. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, Springer 1999
4. J. Fabian et al. Semiconductor Spintronics, Acta Physica Slovaca 57, 565 (2007)
arXiv:0711.1461
5. I. Zutic, J. Fabian, and S. Das Sarma, Spintronics: Fundamentals and
Applications, Rev. Mod. Phys. 76, 323 (2004)
6. M. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Applications
to the Physics of Condensed Matter, Springer 2008.