The spin Hall effect

1
The spin Hall effect
Shoucheng Zhang (Stanford University) Collaborato
rs Andrei Bernevig, Congjun Wu
(Stanford) Xiaoliang Qi (Tsinghua), Yongshi Wu
(Utah) Murakami, Nagaosa (Tokyo)
Science 301, 1348 (2003) PRB 69, 235206 (2004),
PRL93, 156804 (2004) cond-mat/0504147,
cond-mat/0505308
Australia meeting, 2006/01
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Generalization of the quantum Hall effect

• Quantum Hall effect exists in D2, due to Lorentz
  force.

• Natural generalization to D3, 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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The intrinsic spin Hall effect

• Key advantage
• electric field manipulation, rather than magnetic
  field.
• dissipationless response, since both spin current
  and the electric field are even under time
  reversal.
• Topological origin, due to Berrys phase in
  momentum space similar to the QHE.
• Contrast between the spin current and the Ohms
  law

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Experiment -- Spin Hall effect in a 3D electron
film
Y.K.Kato, R.C.Myers, A.C.Gossard, D.D. Awschalom,
Science 306, 1910 (2004)
(i) Unstrained n-GaAs (ii) Strained
n-In0.07Ga0.93As
T30K, Hole density
measured by Kerr rotation
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Experiment -- Spin Hall effect in a 2D hole gas
--
J. Wunderlich, B. Kästner, J. Sinova, T.
Jungwirth, PRL (2005)

• LED geometry

• Circular polarization

• Clean limit

much smaller than spin splitting

• vertex correction 0
• (Bernevig, Zhang (2004))

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• What about the quantum anomalous Hall effect and
  the quantum spin Hall effect?

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Key ingredients of the quantum Hall effect

• Time reversal symmetry breaking.
• Bulk gap.
• Gapless chiral edge states.

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Topological Quantization of the AHE
(cond-mat/0505308)
Magnetic semiconductor with SO coupling (no
Landau levels)
General 22 Hamiltonian
Example
Rashbar Spin-orbital Coupling
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Topological Quantization of the AHE
(cond-mat/0505308)
Hall Conductivity
Insulator Condition
Quantization Rule
The Example
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Origin of Quantization Skyrmion in momentum space
Skyrmion number1
Skyrmion in lattice momentum space (torus) Edge
state due to monopole singularity
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band structure on stripe geometry and topological
edge state
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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 a-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.

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Luttinger Model for spin Hall insulator
Bulk Material zero gap
l1/2,-1/2
l3/2,-3/2
Symmetric Quantum Well, z?-z mirror
symmetry Decoupled between (-1/2, 3/2) and (1/2,
-3/2)
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Topological Quantization of SHE
Luttinger Hamiltonian rewritten as
In the presence of mirror symmetry z-gt-z,
ltkzgt0?d1d20! In this case, the H becomes
block-diagonal
LH
HH
SHE is topological quantized to be n/2p
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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.
Laughlins Gauge Argument When turning on a flux
threading a cylinder system, the edge states will
transfer from one edge to another
Energy spectrum on stripe geometry.
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Topological Quantization of Spin Hall

• Physical Understanding Edge states

When an electric field is applied, n edge states
with G121(-1) transfer from left (right) to
right (left).
G12 accumulation ? Spin accumulation
Conserved
Non-conserved


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Quantum spin Hall effect in graphene (Haldane,
KaneMele)

• SO coupling opens up a gap at the Dirac point.
• One pair of TR edge state on each edge.

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Edge state contribution to Charge transport

• Edge state carry chiral spin current but
  non-chiral charge current.
• Quantized residual conductance in topological
  insulator

Schematic Picture of Conductance
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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 chiral liquid (T
  breaking), and the 1D spinless fermions.

• T21 for spinless fermions and T2-1 for helical
  liquids.

• Single particle backscattering is not possible
  for helical liquids!

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Stability at the edge

• Kane Mele
• Wu, Bernevig and Zhang
• Xu and Moore
• Sheng et al

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Conclusion Discussion

• Quantum AHE.
• Ferromagnetic insulators with spin-orbit
  coupling.
• Topologically non-trivial band gap.
• Hall conductanceSkyrmion number in momentum
  space.
• Number of chiral edge modesSkyrmion number in
  momentum space.
• Quantum SHE
• Standard semiconductor with a strain gradient,
  narrow gap semiconductors, and monolayers of
  graphene.
• A new type of 1D metal the helical liquid.
• Stability ensured by the time reversal symmetry
  of the spin current.