The spin Hall effect - Physics and Astronomy at...
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The spin Hall effect
Shoucheng Zhang (Stanford University)
Collaborators:Shuichi Murakami, Naoto Nagaosa (University of Tokyo)Andrei Bernevig, Congjun Wu, Taylor Hughes (Stanford University)Xiaoliang Qi (Tsinghua), Yongshi Wu (Utah)
APCTP 2005/08
Science 301, 1348 (2003)PRB 69, 235206 (2004), PRL93, 156804 (2004)cond-mat/0504147, cond-mat/0505308,…
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My view on the status of the SHE
Fspinkijkspini
j ekEJ ∝= σεσ
he
qpEJ HjijHi
2
== σεσ
• Quantum Hall effect exists in D=2, due to Lorentz force.
• Natural generalization to D=3, due to spin-orbit force:
• 3D hole systems (Murakami, Nagaosa and Zhang, Science 2003)• 2D electron systems (Sinova et al, PRL 2004)
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My view on the status of the SHE• The intrinsic spin Hall conductivity vanishes in the n-type Rashba
model due to vertex corrections. But an uniform magnetization isinduced by the electric field.
• The intrinsic spin Hall conductivity is finite in the p-type Luttingermodel in 3D and the p-type Rashba model in 2D. In fact, the impurity vertex correction vanishes identically for delta-function scatters. Interestingly, no uniform magnetization is induced by the electric field. =>An interesting complimentarity principle?
• The UCSB effect could either be intrinsic or extrinsic. Maybe they can be distinguished by the sign of the effect.
• The Hitachi-Cambridge effect could be intrinsic. The best opportunity to compare theory with experiment.
• The non-conservation of spin current, and relationship between the spin current and spin accumulation are still outstanding theoretical issues, but maybe the naïve expectations are basically correct.
• The new frontier in SHE theory is the quantum SHE.
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Rashba model:
Intrinsic spin Hall conductivity (Sinova et al.(2004))
+ Vertex correction in the clean limit (Inoue et al (2003), Mishchenko et al,Sheng et al (2005))
Effect due to disorder
0=Sσ
πσ
8e
S =
+ spinless impurities ( -function pot.)
πσ
8vertex e
S −=
( )xyyx kkm
kH σσλ −+=2
2
Green’s function method
xJzyJ
+ ⋅⋅⋅+xJ
zyJ
δ
Luttinger model:
Intrinsic spin Hall conductivity (Murakami et al.(2003)) )(6 2
LF
HFS kke
−=π
σ
+ spinless impurities ( -function pot.)
0vertex =Sσ
( ) ( )yxxy SkSkSkm
kH ⋅−⋅+⋅+= 22
1
2
2λλ
xJzyJ
+ ⋅⋅⋅+xJ
zyJ
δ
Vertex correction vanishes identically!(Murakami (2004), Bernevig+Zhang (2004)
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Quantum Spin Hall
• Can one have a quantum spin Hall effect without any external magnetic field and T breaking?
• Landau level problem:
raE =
raE =chargeρ
raE =chargeρ
GaAs
E
• 2D momenta and E field, σz only:
raE =chargeρ
• Hamiltonian for spin-orbit coupling:
( ) σµ⋅×++= Ep
mcgear
mpH B2
2
2
BrAm
AepHrrr
rr
×=−
=21
2)( 2
• Example of such a field: inside a uniformly charged cylinder
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Quantum Spin Hall • In semiconductors without inversion symmetry, shear strain is like an
electric field in terms of the SO coupling term
dh TO ⎯⎯ →⎯breakingsymmetry inversion
cubic gp symm gp: Ixyz ≡ (rotation part only, inversion not a symmetry)
⎪⎩
⎪⎨
⎧
⇒⎪⎩
⎪⎨
⎧
+++
⇒≡
y
x
z
EEE
yzxxzxyzyzyxxy
Ixyz~~~
~~z~
xz
yz
xy
εεε
ayaxraE
xz
yz
xy
===
↔=εεε 0
zyyzxxz ppCDrm
pH σεε )(22
322
−++=h
(shear strain gradient creates the same SO coupling situation as a radialyincreasing electric field)
zxyyx ypxpRyxppH σ)(2222 −++++= (up to a coordinate re--scaling)
aDmCR 2
23
h=
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Quantum Spin Hall
GaAs
E• Hamiltonian for electrons:
zxyyx ypxpRyxppH σ)(2222 −++++=
• Tune to R=2
( )( ) ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−= 2
2
00
ApApH
)0,,( xyA −=
• Spin up
( )21 ApH −=
effectiveB
• Spin down
( )21 ApH +=
effectiveB
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Quantum Spin Hall
• P,T-invariant system
0arg =xyechσ
πσ
42
22
2 eeh
espin ==
h
• Spin up
effectiveB
∗−=
zzn
n en
z 21
!πφ
• Spin down
effectiveB
( ) ∗−∗
=zz
m
m em
z 21
!πφ
( ) ( ) ( )njiji
jim
jijiji
jiji
mjii zzzzzzz ∏∏∏
=↓=↑<
∗
=↓<
∗∗
=↑<
−−−=↓↑Ψ,;,;,;
),,(
• Halperin-like wavefunction
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Quantum Spin Hall
• Purely electrical detection measurement, measure xx
echargρ
ν
xyσxxσ
xxρ
• More effort to directly measure , open question. spinσ
• Landau Gap and Strain Gradient
aCELandau 3=∆ m/s108 53 ×=h
Cstrain gradient=a
m10over %1for µ=a mKELandau 10=∆
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Spin-Hall insulator: dissipationless spin transport without charge transport (PRL 93, 156804, 2004)
• In zero-gap semiconductors, such as HgTe, PbTe and α-Sn, the HH band is fully occupied while the LH band is completely empty.
• A charge gap can be induced by pressure. In this case, charge conductivity vanishes, but the spin Hall conductivity is maximal.
ae
s 1.0−≈σ
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Topological Quantization of the AHEMagnetic semiconductor with SO coupling (no Landau levels):
charge Hall conductance topological quantized to be n/2π
Charge Hall effect of a filled band:
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Topological Quantization of SHE
LH
HH
SHE is topological quantized to be n/2π
Paramagnetic semiconductors such as HgTe and α-Sn:
In the presence of mirror symmetry z->-z, d1=d2=0! In this case, the H becomes block-diagonal:
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Topological Quantization of Spin Hall • Physical Understanding: Edge states
In a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states.
Energy spectrum on stripe geometry.
Laughlin’s Gauge Argument:
When turning on a flux threading a cylinder system, the edge states will transfer from one edge to another
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Quantum spin Hall effect in graphene (Haldane, Kane&Mele)
• SO coupling opens up a gap at the Dirac point.• One pair of TR edge state on each edge.• Numerical calculation indicate stability (Sheng et al)
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Topological Quantization of Spin Hall • Physical Understanding: Edge states
When an electric field is applied, n edge states with Γ12=+1(−1) transfer from left (right) to right (left).
Γ12 accumulation Spin accumulation
Conserved Non-conserved
+=
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Stability at the edge• The edge states of the QSHE is the
1D helical liquid. Opposite spins have the opposite chirality at the same edge.
• It is different from the 1D chiralliquid (T breaking), and the 1D spinless fermions. T2=1 for spinlessfermions and T2=-1 for helical liquids.
• One particle backscattering is forbidden by the T symmetry. (Kane&Mele), however, two particle backscattering is allowed.
• An new kind of stability!
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Conclusion & Discussion
• A new type of dissipationless quantum spin transport.• Natural generalization of the quantum Hall effect.• Lorentz force and spin-orbit forces are both velocity
dependent.• U(1) to SU(2), 2D to 3D.
• Quantum SHE.• A new type of 1D metal: the helical liquid.• Standard semiconductors with a strain gradient, narrow gap
semiconductors and monolayers of graphene.• More experiments!