Spin Hall Effect in1. Rashba Electron Systems in Quantum Hall Regime
2. p-type GaAs Quantum Well with Rashba Coupling
Fu-Chun Zhang, The Univ. of Hong Kong
PRL 92, 2004, PRB 71, 155316 (2005), cond-mat/0503592;cond-mat/0507603
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)
1. Spin Hall effect in 2D Rashba electron systems
in quantum Hall regimeMotivation: 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 hamiltonian2D electron gas with spin-orbit coupling in a magnetic field
BSgAcepz
m
Acep
H BS
rr•+×+•+
+=
→→→→
→→
µσλ ])[(2
)( 2
Kinetic Rashba Zeeman
Spin Hall current and spin polarizationSpin Hall current and spin polarization
},{2/1, xzzxs vsj =
yBsz
xs mBgj σµλ
4,h
−=
At g_s =0, or Zeeman energy = 0, we have spin Hall current =0.
eigenstateaninaveragedtds
Hsidtds
x
xx
−−−>=<
=,0/
],,)[/1(/ h
])[()(21 2*
→→→→→→→→
×+•+•++= σλµ AcepZBSgA
cep
mH Bs
e
Single electron 0=→
E( case) with Rashba coupling
Rashba, and also Loss et al.
Kinetic Zeeman Rashba coupling (spin-orbit)
B
Single electron energy
e
s
n
mmgg
eBcm
ngnE
2,
)8)1(21(
3
222
2~
2~
,~
==
+−±=±
h
h
λη
ηω
Energy levels2D electron with Rashbacoupling in B-field
Mixed spin up and spin down in different Landau level(state entangled: spin and orbit w.f. not decoupled)
−= ,1~n
0==ηλ 02 ≠η
+= ,0~n
−= ,2~n
+= ,2~n
Landau levels of an electron as functions offor (In0.52Al0.48As/In0.53Ga0.47As).Arrows indicate those level crossings giving rise to resonant spin Hall conductance.
2/ hbmlλη =
1.02/ == es mmgg
Spectrum of Rashba system in B-field
Average spin (unit ) per electron asa function of 1/B. The parameters used are , , taken for the inversion heterostructure In0.53Ga0.47As/In0.52Al0.48As.
eVm11109.0 −×=λ216 /109.1 mne ×= 4=sg emm 05.0=
Zσ 2/h
Level crossing
Magnetization
0≠λFor any , there is one (B,ne) for resonance
xzz
x vSj2h
=
0=λ
ca λλ =+
cλ
λ
Fermi level at the level crossing, spin Hall current resonant (peaked)
Spin Hall conductance Spin Hall conductance v.sv.s. 1/B. 1/B
es
e
mmgmneVm
05.0;0.4/109.1
109.0216
11
==×=
×= −λ
Resonant Spin Hall Current DensityResonant Spin Hall Current Density
Formalism for charge and spin currents,
perturbation
..)(
||||)(
,,||,,)(
)()()(
','
,,,
)1(,
,,,)0(,
,,)1(,,,
)0(,,,,
chspnjsnpsnpHspn
j
spnjspnj
jjj
sn snns
xscxxxspnsc
xscxspnsc
spnscspnscspnsc
x
x
xxx
+−
>′′><′′′<=
>=<
+=
∑′′εε
.
,1sin,cos
,,~
)/()(0
constp
matrixPauli
pnipn
spn
EBcpelHHEHH
x
y
xns
xnsx
xyb
=
=
−
=
+−=′′+=
σ
θθ
ση
The effect of E-field
)],0,1(,0[21
↓+↑+↑ αβi+= 0E
↑,1
↑,0
↓,0
↑,1
↑,0
↓,0
↑,00=E
Carries no current State carries a final spin current
Discussions on the resonanceDiscussions on the resonance
About anti level crossing. For nonAbout anti level crossing. For non--magnetic magnetic impurity, u^2/impurity, u^2/\\hbar omega, small.hbar omega, small.EE--field must be larger than all energy scalesfield must be larger than all energy scalesto see the resonance:to see the resonance:temperature, level separation, temperature, level separation, deviation of Bdeviation of B--field from the resonant fieldfield from the resonant field
Edge spin current and spin Edge spin current and spin polarization in quantum Hall regimepolarization in quantum Hall regime
Edge state: Edge state: RashbaRashba coupling=0 coupling=0 MacDonald and MacDonald and StredaStreda (1980(1980’’s)s)
Energy spectrum of edge state in quantum Energy spectrum of edge state in quantum Hall system with Hall system with RashbaRashba couplingcoupling
(at a distance 4 times of magnetic length)(at a distance 4 times of magnetic length)
Edge spin Current and spin polarizationEdge spin Current and spin polarization0=E
∞+−∈= otherwiseLLyifyV );2/,2/(0)(
blue: energy separation in bulk > eE l_b,black: similar; red: < eB l_b
Spin polarization and spin current at a small E-field =0. 01 V/m
In calculation, we Assume voltage drops only at the edges, and bulk states contributes no spin current
Summary of SHE in quantum Hall regionSummary of SHE in quantum Hall region
Resonance condition:Resonance condition:–– RashbaRashba 2DEG with g >0, 2DEG with g >0, –– DresselhausDresselhaus with g<0with g<0Expected to see the resonance near the critical Expected to see the resonance near the critical
magnetic field at low Tmagnetic field at low TAt the resonant field, a small EAt the resonant field, a small E--field changes field changes
spin polarization from z to y to spin polarization from z to y to --z.z.ExptExpt.: measure spin polarization.: measure spin polarization
Resonant intrinsic spin Hall effect in p-type GaAs quantum well structure
Luttinger Hamiltonian with a Rashbaspin-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 (Hamiltonian (LuttingerLuttinger + + RushbaRushba))H = HL - λ(z × p) · S + V (z)
V(z): infinity potential walls.
For a given k, in plane wave vector,
Limiting cases of the modelLimiting cases of the model
\\lambda =0 , lambda =0 , LuttingerLuttinger hamiltonianhamiltonian. Band . Band structure studied by Yu Cardona and others structure studied by Yu Cardona and others with a more realistic potential.with a more realistic potential.SHE studied by SHE studied by BernevigBernevig and S. C. Zhang.and S. C. Zhang.kLkL << 1 limit, heavy hole sub<< 1 limit, heavy hole sub--bands studied by bands studied by SchielmannSchielmann and Loss by using perturbation and Loss by using perturbation theory to the lowest order. theory to the lowest order. RashbaRashba effect: k^3effect: k^3
Here we consider interplay of Here we consider interplay of RashbaRashba and and LuttingerLuttingerterms. terms.
Method: using basis of k=0 Method: using basis of k=0 wavefunctionswavefunctions and and truncated numerical methodtruncated numerical method
SubSub--band dispersion: level crossingsband dispersion: level crossings
Left: No Rashba, similar to Yu & Cardona, Huang et al. Mid: \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.
Basis function: k=0 eigen states.
Spin Hall conductance in Spin Hall conductance in GaAsGaAs (Kubo formula)(Kubo formula)hole density = 5x 10^{11} /cm^2, well thickness = 87 A, hole density = 5x 10^{11} /cm^2, well thickness = 87 A, lifetime = 2 x10^{lifetime = 2 x10^{--11} s, or mobility = 10000 cm^2/sV.11} s, or mobility = 10000 cm^2/sV.
\lambda_0 = \hbra^2/mLHeavy hole sub-band level crossing
Resonant Resonant RashbaRashba coupling coupling
Spin Hall conductance for small Spin Hall conductance for small \\lambdalambda(level crossing only in light hole (level crossing only in light hole subbandssubbands) )
well thickness= 83A, well thickness= 83A, \\lambda = lambda = \\hbar^2/mL, hbar^2/mL, lifetime= 2x10^{lifetime= 2x10^{--11}s. Red curve: 11}s. Red curve: \\lambda=0lambda=0
Vertex correlationsVertex correlations
Have not done calculations.Have not done calculations.Resonance related to heavy hole subResonance related to heavy hole sub--band expected to survive with vertexband expected to survive with vertexcorrection.correction.Resonance related to light Resonance related to light subbandsubbandcrossing needs more cautious.crossing needs more cautious.
