Spin Hall Effect in Quantum Hall Regime

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Spin Hall Effect in Quantum Hall Regime

• Dr. Shun-Qing Shen
• ?????
• Department of Physics
• The University of Hong Kong
• June 3, 2005

CCAST 05
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Collaborators

• Professor Michael Ma (University of Cincinatti)
• Professor Xin-Cheng Xie (Oklahoma State
  University)
• Professor Fu-Chun Zhang (The University of Hong
  Kong )
• Professor Liang-Bin Hu (South China Normal
  University)
• Mr. Jian Li (The University of Hong Kong)
• Miss Yun-Juan Bao (The University of Hong Kong)

• References
• S.Q. Shen, Phys. Rev. B 70, 081311 (R)(2004)
• S.Q. Shen, M. Ma, X.C. Xie, and F.C. Zhang, Phys.
  Rev. Lett. 92, 256603 (2004)
• S.Q. Shen, Y.J. Bao, M. Ma, X.C. Xie, and F.C.
  Zhang, Phys. Rev. B 71, 155316(2005)
• J. Li, L.B. Hu, and S.Q. Shen, Phys. Rev. B 73,
  (R) (2005)
• Y.J. Bao, H.B. Zhuang, S.Q. Shen, and F. C.
  Zhang, cond-mat/0503592

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What is the spin current?

• Electrons carry both charge and spin. Spin
  looks like a tiny spinning ball, and may be
  represented with a vector. It has two components
  spin-up and -down.

If the currents of electrons for different spins
are opposite,
The charge current
The spin current
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Spin Current and Vector Potential
In many body systems if the vector potential for
electrons is spin dependent,
Spin-dependent current
Special case
Equivalent definition
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Spin-Orbit Interaction
The interaction describes the effect of an
electrons orbital motion on the orientation of
its spin. The B field due to the relative orbital
motion of nuclear charge is given by
The potential energy of the spin momentum
Relativistic quantum correction
This coupling can be derived from Dirac Equation.
The relativistic correction factor of ½, the
Thomas precession, has been included.
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Spin-orbit coupling in 2D quantum wells

• Rashba coupling (1960) from structure inversion
  asymmetry

• Dresselhaus coupling (1955) from bulk inversion
  asymmetry.

2DEG
Spin-dependent vector potential
The relative strength of two couplings can be
extracted from photoncurrent measurements.
Ganichev et al, PRL (2004)
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Spin Transformation and Symmetry
Spin gauge transformation
Spin current
Rigorous result
is even about the two couplings
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Berry phase
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Spin Hall Conductance and Berry Phase in 2DEG
J
EX
In a clean limit, the linear response theory gives
for Rashba coupling Sinova et al, PRL
(2004)
for Rashba and Dresselhaus coupling Shen,
PRB 70, 081311 (R) (2004)
Berry phase
Condition Two bands are filled. The conductance
is not a constant if only one band is filled.
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Spin force on a moving electron
Ehrenfest theorem
Li, Hu, and Shen, cond-mat/0502102
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Reversible Spin Hall Effect
In semiconductors, such as GaAs and InAs, Rashba
and Dresselhaus coupling are usually of the same
order of magnitudes. The Rashba coupling is
tunable by the gate voltage or electric field
perpendicular to the plain.
Nitta et al, PRL (1997) Grundler, PRL (2000)
Spin Hall conductance
Shen, PRB (2004) Sinitsyn et al, PRB (2004)
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Impurity Effect
Li, Hu, and Shen Cond-mat/0502102
D
R
Inoue et al, PRB (2004) Mischenko et al, PRL
(2005) Dimitova et al, (2004) Rashba,
(2004) Zhang, (2005)
Sheng et al, PRL (2005) Nomura et al, PRB
(2005) Nikonic et al, Li, Hu and Shen, PRB (2005)
Sugimoto, Onoda, Murakami, Nagaosa, 2005
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Resonant Spin Hall Effectin the presence of
magnetic field and Rashba spin-orbt coupling
The spin-orbit coupling competes with Zeeman
splitting to introduce additional degeneracy
between different Landau levels at certain
magnetic fields. This degeneracy, if occurring at
the Fermi level, gives rise a resonant spin Hall
conductance. The spin current as well as the
charge current is non-dissipative in the quantum
regime.
Shen, Ma, Xie, and Zhang, PRL 92, 256603 (2004)
Shen, Bao, Ma, Xie, and Zhang, PRB 71, 155316,
(2005)
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Integer Quantum Hall Effect
Charge Hall conductance
Spin Hall conductance without spin-orbit coupling
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We take the periodic boundary condition Along the
x-direction.
Dresselhaus
Rashba
Rashba Schliemann and Loss Wang and Vasilopoulos
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FIG. 1. Schematic layer structure of an inverted
In0.53Ga0.47As/In0.52Al0.48As heterostructure.
FIG. 2. Calculated conduction band diagram and
electron distribution.
Nitta et al, PRL 78, 1335 (1997)
FIG. 3. Schubnikovde Haas oscillations as a
function of the gate voltages.
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Analytical Solutions with Rashba coupling
In the weak coupling limit
In the strong coupling limit
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Spectrum with the Rashba coupling
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Perturbation Theory
To calculate the spin current, we study the
system in the presence of the E-field. The spin
current is calculated in the perturbation theory.
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Spin and charge Hall current
Charge Hall conductance
The charge Hall conductance is unaffected by the
spin-orbit coupling.
Spin Hall conductance
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Average spin per electron
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Spin Hall conductance v.s. 1/B
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Near the resonant point

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Transition from Sz to Sy
Sy
Sz
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Scaling Behaviors
At low temperature near the resonant point,
The remaining filling factor on the two levels
At the resonant point and T0,
At finite temperature,
The analysis is based on the truncation
approximation, which goes beyond the perturbation
theory. Thus the resonant spin Hall current is
non-linear with the E-field.
The weight of resonant peak
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Resonant Spin Hall Current Density
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Effect of Dresselhaus coupling
As the Dresselhaus coupling increases from zero,
the resonance is shifted to lower magnetic field,
the resonance occurs at higher Landau levels with
a weaker resonant strength.
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Edge State
Assume the electrons are confined in the sample
of width Ly, and take a periodic boundary
condition along the x-direction. The wave
function must vanish at the boundary. This
condition leads to the edge state and edge current
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Solution of the edge states
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Truncation Approximation
Inclusion of the Rashba coupling In the bulk
the two states n, -1gt and n1,1gt are mixed and
an analytical solution can be obtained. Near the
boundary an analytical solution seems to be
unlikely. We choose the energy eigenstates
without spin-orbit coupling as a base
wavefunctions. To calculate the eigen values of
lower energy and the corresponding eigenstates,
the Hamiltonian is diagonalized by ignoring the
sufficient higher Landau levels.
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Energy level anticrossing the edge effect
(Four magnetic lengths away from the edge)
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Edge spin Current and spin polarization
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Spin-y in a Small E-Field
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Non-Linear Effect of Electric Field
For the two degenerated levels in an E-field
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Spin current and spin polarization
In a Rashba system,
In an energy eigenstate,
This relation provides a way to extract the spin
current from the spin polarization in experiment.
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Charge and spin Hall conductance from the edge
states
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Total spin Hall current

• In a realistic sample with a finite size the bulk
  Hall current density will not be suppressed
  completely though it decays with the size of the
  sample even if the impurities are taken into
  account.
• The ratio of the edge charge current and the
  total current and the distribution of the applied
  electric field were determined by the impurity,
  and the size of samples. (Thouless, 93 Hirai and
  Komiyama, 94)

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Conclusion
The spin-orbit coupling competes with Zeeman
splitting to introduce additional degeneracy
between different Landau levels at certain
magnetic fields. This degeneracy, if occurring at
the Fermi level, gives rise a resonant spin Hall
conductance, and generate a spin phase
transition. The relation between the spin Hall
current and spin polarization provides an
explicit way to extract the spin current
distribution from the measurement of spin
polarization Sy.
Acknowledgment This work was supported by the
Research Grant Council of Hong Kong.