Quantum spin Hall effect

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Quantum spin Hall effect
Shoucheng Zhang (Stanford University) Collaborato
rs Andrei Bernevig, Congjun Wu
(Stanford) Xiaoliang Qi (Tsinghua), Yongshi Wu
(Utah) Murakami, Nagaosa (Tokyo)
cond-mat/0504147 cond-mat/0505308 cond-mat/0508273
HK meeting, 2005/12
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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.