Topological Orbital Angular Momentum Hall Current

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Topological Orbital Angular Momentum Hall Current
Jiangping Hu Department
of Physics Purdue University

CCAST, 05/31/2005
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Collaborators

• Chang Liu

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Quantum Hall Effect
Quantum Hall Conductance
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Outline

• Introduction of Spin-Orbital Coupling
• Spin Hall effect
• Orbital Angular Momentum Hall Current
• Topological Nature of Angular Momentum Hall
  Current
• Detection of Spin Hall Physics from ARPES

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Spintronics

• Can we control spin and create spin current
  electrically?
• Advantage Integrate perfectly with conventional
  device and technology.
• Challenge No direct coupling between electric
  field and spins
• Key Idea the spin degree of freedom can be
  driven by electric field through spin-orbit
  coupling.

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Motivation

• New topological effects
• Orbit-spin coupling What are the orbital part
  doing?
• How to detect spin Hall physics

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Spin Orbit Coupling

• spin orbit coupling

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Rasha-Dresselhaus S-O coupling

• 2 dimension
• Spin-1/2

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Luttinger Spin-Orbit coupling
p-orbit (x,y,z) spin ½ 6 states
spin-orbit coupling
3 bands
Luttinger Hamiltonian
( spin-3/2 matrix, describing the P3/2 band)

• 3 dimension
• Spin 3/2

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Spin Hall Current
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Adiabatic approximation

• Validity of the adiabatic approximation the
  characteristic time for momentum charge are much
  longer than the relaxation time of spins due to
  the spin orbital coupling.

• Single particle Hamiltonian

• Using a time dependent gauge

E

• Instantaneous Eigenstate

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A simple derivation of spin current

• Final formula

• In two band systems, for each momentum P, the
  spin current induced by the electric field are
  exactly opposite in the two bands. Therefore, the
  spin current comes from the difference between
  the two Fermi surface of the two bands.

• Important for the valid of
• adiabatic approximation

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Spin Hall Current in R-D Systems
Kubo Formula
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• Only magnetization can be detected !

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Orbit vs Spin

• Spin orbit coupling
• Magnetic moment

A real magnetization is contributed from both
orbit and spin.
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Orbit Angular Momentum Hall Current
Definition
b0 Lz Sz, H0 a0
Lz-Sz , H 0
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Gauge in Momentum Space
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Gauge field an angular momentum flux
v

• Gauge field

y
x
Force
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Topological Orbit Hall Current
Derivation of orbit hall current
Orbit Hall conductance
Spin Hall conductance from this derivation is
zero! No topological Spin Hall conductance!
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Conclusion

• Spin Hall effect in R-D case is not topological
  effect, However, orbit Hall current is purely
  topological effect
• Orbit Hall current is larger than spin Hall
  current in general in R-D case
• Orbit Hall current does not change sign in all
  coupling parameter region.
• When Rashba coupling is larger than Dresselhaus
  coupling, spin Hall current and orbit Hall
  current run in opposite direction. The
  magnetization direction near edge due to the
  total angular momentum Hall current is opposite
  to the prediction from the spin Hall current
• Due to the topological nature, the orbit Hall
  current is expected to survive in the weak
  disorder limit.

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Detection of Spin Hall Effect By APRES
Energy conservation EB hn - Ekin - F Momentum
conservation K k G
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ARPES Theory
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ARPES Spin Orbit Coupling
Rashba
Luttinger

• Spin orbit coupling induces an additional term in
  ARPES
• It is directly proportional to electric field and
  has specific
• momentum dependence determined by spin
  orbit coupling
• The effect can be strong

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Measurement
Determine the band
Spin orbit coupling effect
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Polarization and Dichroism
Mirror plane
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Conclusion

• Orbit Angular Momentum Hall Current is
• also important in spin orbit coupling system
• There is a topological nature for orbit angular
  momentum Hall current
• The origin of spin Hall effect can be detected by
  ARPES
• New Dichorism effect can be induced by electric
  field in spin orbit coupling systems.