Spin Hall Effect inpeople.physics.tamu.edu/sinova/Workshop_talks/FuChun... · 2006-03-01 · Spin...
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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)
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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
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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
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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
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])[()(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
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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
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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
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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
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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)
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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
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Resonant Spin Hall Current DensityResonant Spin Hall Current Density
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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
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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
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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
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Edge spin current and spin Edge spin current and spin polarization in quantum Hall regimepolarization in quantum Hall regime
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Edge state: Edge state: RashbaRashba coupling=0 coupling=0 MacDonald and MacDonald and StredaStreda (1980(1980’’s)s)
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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)
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Edge spin Current and spin polarizationEdge spin Current and spin polarization0=E
∞+−∈= otherwiseLLyifyV );2/,2/(0)(
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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
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In calculation, we Assume voltage drops only at the edges, and bulk states contributes no spin current
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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
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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
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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,
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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
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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.
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Basis function: k=0 eigen states.
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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
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Resonant Resonant RashbaRashba coupling coupling
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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
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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.