Download Spin Hall Effect in

Spin Hall Effect in
1. Rashba Electron Systems in Quantum Hall Regime
2. p-type GaAs Quantum Well with Rashba Coupling
Fu-Chun Zhang, The Univ. of Hong Kong
Collaborators:
Topic 1: Shun-Qing Shen , and Y. Bao (Univ. of Hong Kong),
Mike Ma (Cincinnati), X.C. Xie (Oklahoma and IOP, Beijing)
Topic 2: Xi Dai (HKU), Zhong Fang , Yu-Gui Yao (IOP, Beijing)
PRL 92, 2004, PRB 71, 155316 (2005), cond-mat/0503592;
cond-mat/0507603
1. Spin Hall effect in 2D Rashba electron systems
in quantum Hall regime
Motivation: Why magnetic field?
• Rich physics of quantum Hall effect, dissipation-less;
Key points
• Relation between Spin polarization and spin Hall current
• Zeeman and Rashba terms compete to induce level crossings
• Resonant spin Hall effect if Fermi energy at level crossing
• Experimentally measurable
In collaboration with S. Q. Shen, M. Ma, X. C. Xie, Y. J. Bao
System and model hamiltonian
2D electron gas with spin-orbit
coupling in a magnetic field
e → 2
( p+
A)
→
→
→
→
r r
e
c
A) × σ ] + g S µ BS • B
+ λ z • [( p +
H =
2m
c
Kinetic
Rashba
Zeeman
→
Spin Hall current and spin polarization
λ
j
z
s,x
hg sµ B B
= −
4m
σ
y
At g_s =0, or Zeeman energy = 0,
we have spin Hall current =0.
j
z
s,x
= 1 / 2{s z , v x }
dsx / dt = (1 / ih)[s x , H ],
< dsx / dt >= 0,
average − in − an − eigenstate
→
Single electron ( E = 0 case) with Rashba coupling
Rashba, and also Loss et al.
→ →
→
→
1 → e→ 2
e→ →
H =
( p + A) + g s µ B S • B + λ Z • [( p + A) × σ ]
*
2 me
c
c
Kinetic
Zeeman
Rashba coupling (spin-orbit)
Single electron energy
~
1
2
E ~ = hω (n ±
(1 − g ) + 8 n η 2 )
n,±
2
g sm
λ2m 2c
2
η =
,g =
3
eB h
2me
~
B
Energy levels
~
n = 2,+
2D electron with Rashba
coupling in B-field
~
n = 2,−
~
n =1,−
~
n = 0,+
Mixed spin up and
spin down in different
Landau level
(state entangled: spin and
orbit w.f. not decoupled)
λ =η = 0
η2 ≠ 0
Spectrum of Rashba system in B-field
Landau levels of an electron as functions ηof = λ ml b / h
for g = g s m / 2 m e = 0 . 1 (In0.52Al0.48As/In0.53Ga0.47As).
Arrows indicate those level crossings giving rise to
resonant spin Hall conductance.
2
Magnetization
Level crossing
Average spin σ Z (unit h / 2 ) per electron as
− 11
a function of 1/B. The parameters used λ = 0 .9 × 10 eVm
16
2
n
=
1
.
9
×
10
/
m
are e
, g s = 4 , m = 0.05 me taken for the
inversion heterostructure In0.53Ga0.47As/In0.52Al0.48As.
j xz
h
= S z vx
2
Fermi level at the level crossing,
spin Hall current resonant (peaked)
a + λ = λc
λ
λ =0
λc
For any λ ≠ 0 , there is one (B,ne)
for resonance
Spin Hall conductance v.s. 1/B
λ = 0.9 ×10 −11 eVm
ne = 1.9 × 1016 / m 2
g s = 4.0; m = 0.05me
Resonant Spin Hall Current Density
Formalism for
charge and spin
currents,
perturbation
H = H 0 (E ) + H ′
H ′ = − (η el bσ y + p x c / B ) E
 cos θ ns n , p x
~
n , p x , s = 
 i sin θ ns n − 1, p x
σ y = Pauli
matrix
p x = const .
( jc , s ) n , p x , s = ( jc(,0s) ) n , p x , s + ( jc(1, s) ) n , p x , s
( jc(,0s) ) n , p x , s =< n, p x , s | jc , s | n, p x , s >
(1)
c,s n, px ,s
(j )
=∑
n ', s '
< n′p x s′ | H ′ | np x s >< np x s | jc , s | n′p x s′ >
(ε ns − ε n′s′ )
+ h.c.




The effect of E-field
1, ↑
1, ↑
0, ↓
0, ↓
0, ↑
0, ↑
E = 0
Carries no current
0, ↑
E=0
+
1
[ 0, ↑ + (iβ 1, ↑ + α 0, ↓ )]
2
State carries a final spin current
Discussions on the resonance
About anti level crossing. For non-magnetic
impurity, u^2/\hbar omega, small.
E-field must be larger than all energy scales
to see the resonance:
temperature, level separation,
deviation of B-field from the resonant field
Edge spin current and spin
polarization in quantum Hall regime
Edge state: Rashba coupling=0
MacDonald and Streda (1980’s)
Energy spectrum of edge state in quantum
Hall system with Rashba coupling
(at a distance 4 times of magnetic length)
Edge spin Current and spin polarization
E=0
V ( y ) = 0 if
y ∈ (− L / 2, L / 2);
otherwise
+∞
Spin polarization and spin current
at a small E-field =0. 01 V/m
blue: energy separation in bulk > eE l_b,
black: similar; red: < eB l_b
In calculation, we Assume voltage drops only at the
edges, and bulk states contributes no spin current
Summary of SHE in quantum Hall region
Resonance condition:
– Rashba 2DEG with g >0,
– Dresselhaus with g<0
Expected to see the resonance near the critical
magnetic field at low T
At the resonant field, a small E-field changes
spin polarization from z to y to -z.
Expt.: measure spin polarization
Resonant intrinsic spin Hall effect
in p-type GaAs quantum well structure
Luttinger Hamiltonian with a Rashba
spin-orbit coupling arising from the structural
inversion symmetry breaking.
Rashba term induces an energy level crossing
in the lowest heavy hole
sub-band, which gives rise to a resonant spin
Hall conductance.
The resonance may be used to identify the
intrinsic spin Hall effect.
In collaboration with X. Dai, Z. Fang, Y. G. Yao
Hamiltonian (Luttinger + Rushba)
H = HL - λ(z × p) · S + V (z)
For a given k, in plane wave vector,
V(z): infinity potential walls.
Limiting cases of the model
\lambda =0 , Luttinger hamiltonian. Band
structure studied by Yu Cardona and others
with a more realistic potential.
SHE studied by Bernevig and S. C. Zhang.
kL << 1 limit, heavy hole sub-bands studied by
Schielmann and Loss by using perturbation
theory to the lowest order. Rashba effect: k^3
Here we consider interplay of Rashba and Luttinger
terms.
Method: using basis of k=0 wavefunctions and
truncated numerical method
Sub-band dispersion: level crossings
Left: No Rashba, similar to Yu & Cardona, Huang et al.
\lambda=hbar^2/mL=1; Right: \lambda= 3.
K: in-plane momentum, 2L =well width.
\gamma_1=7, \gamma_2 =1.9, m=m_e.
Mid:
Basis function: k=0 eigen states.
Spin Hall conductance in GaAs (Kubo formula)
hole density = 5x 10^{11} /cm^2, well thickness = 87 A,
lifetime = 2 x10^{-11} s, or mobility = 10000 cm^2/sV.
Heavy hole sub-band level crossing
\lambda_0 = \hbra^2/mL
Resonant Rashba coupling
Spin Hall conductance for small \lambda
(level crossing only in light hole subbands)
well thickness= 83A, \lambda = \hbar^2/mL,
lifetime= 2x10^{-11}s. Red curve: \lambda=0
Vertex correlations
Have not done calculations.
Resonance related to heavy hole subband expected to survive with vertex
correction.
Resonance related to light subband
crossing needs more cautious.