A1258689593cKYAU

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The spin Hall effect The quantum AHE and the
SHE The persistent spin helix
Shou-cheng Zhang, Stanford University
Les Houches, June 2006
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Credits

• Collaborators
• Andrei Bernevig (Stanford)
• Taylor Hughes (Stanford)
• Shuichi Murakami (Tokyo)
• Naoto Nagaosa (Tokyo)
• Xiaoliang Qi (Tsinghua and Stanford)
• Congjun Wu (Stanford and KITP/Santa Barbara)
• Yongshi Wu (Utah)

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The spin Hall effect
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Can Moores law keep going?
Power dissipationgreatest obstacle for Moores
law! Modern processor chips consume 100W of
power of which about 20 is wasted in leakage
through the transistor gates. The traditional
means of coping with increased power per
generation has been to scale down the operating
voltage of the chip but voltages are reaching
limits due to thermal fluctuation effects.

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Spintronics

• The electron has both charge and spin.
• Electronic logic devices today only used the
  charge property of the electron.
• Energy scale for the charge interaction is high,
  of the order of eV, while the energy scale for
  the spin interaction is low, of the order of
  10-100 meV.
• Spin-based electronic promises a radical
  alternative, namely the possibility of logic
  operations with much lower power consumption than
  equivalent charge based logic operations.
• New physical principle but same materials! In
  contrast to nanotubes and molecular electronics.

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Manipulating the spin using the Stern-Gerlach
experiment

• Problem of using the magnetic field
• hard for miniaturization on a chip.
• spin current is even while the magnetic field is
  odd under time reversal gt dissipation just as in
  Ohmslaw.

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Relativistic Spin-Orbit Coupling

• Relativistic effect a particle in an electric
  field experiences an internal effective magnetic
  field in its moving frame

• Spin-Orbit coupling is the coupling of spin with
  the internal effective magnetic field

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Using SO spin FET
V
V/2

• Das-Datta proposal.
• Animation by Bernevig and Sinova.

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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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Valence band of GaAs
S
S
P3/2
P
P1/2
Luttinger Hamiltonian
( spin-3/2 matrix, describing the P3/2 band)
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Unitary transformation
Diagonalize the first term with a local unitary
transformation
Helicity basis
gauge field in k!
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Local gauge field in k space
Adiabatic transport potential V does not cause
inter-band transitions
? only retain the intra-band matrix
elements
Abelian approximation retain only the
intra-helicity matrix elements
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Effective Hamiltonian for adiabatic transport
(Dirac monopole)
Nontrivial spin dynamics comes from the Dirac
monopole at the center of k space, with egl
Eq. of motion
Drift velocity
Topological term
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Dissipationless spin current induced by the
electric field
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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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Time reversal and the dissipationless spin current
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Effect due to disorder
spinless impurities ( -function pot.)
Luttinger model Intrinsic spin Hall
conductivity (Murakami et al.(2003))
Vertex correction vanishes identically! 2DHG
BernevigZhang (PRL 2004)
Rashba model Intrinsic spin Hall
conductivity (Sinova et al.(2004))
spinless impurities ( -function pot.)
Vertex correction in the clean limit
(Inoue, Bauer, Molenkamp(2003))
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Mott scattering or the extrinsic Spin Hall effect
Electric field induces a transverse spin current.

• Extrinsic spin Hall effect

Mott (1929), Dyakonov and Perel (1971) Hirsch
(1999), Zhang (2000)

• impurity scattering spin dependent
  (skew-scattering)

Spin-orbit couping
down-spin
up-spin
impurity
Cf. Mott scattering

• Intrinsic spin Hall effect Berry phase in
  momentum space
• 
  

Independent of impurities !
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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 3D Electron
Film
Y.K.Kato et al., Science (2004)

• unstrained GaAs -- no strain spin-orbit
  coupling
• strained InGaAs -- no crystal orientation
  dependence
• extrinsic quantum spin hall calculation (Engel,
  Rashba, Halperin)
• sign mismatch? but right ballpark value

It should be extrinsic!
Bernevig, Zhang, cond-mat (2004)

• Dresselhaus term is relevant, opposite sign.
• Dresselhaus term is small, but induced SHE
  is not small.
• For Dresselhaus term the vertex correction does
  not cancel the intrinsic SHE.
• Dirty limit
• ? SHE suppressed by some factor, which
  is roughly

It could be intrinsic!
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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))
• should be intrinsic

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Direct measurement of the spin current?
A modified version of the standard
drift-diffusion experiment in semiconductor
physics. Optically inject up or down spin
carriers, and observe the longitudinal charge
drift and the spin-dependent transverse drift.
z
y
E
x
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Spin Orbit Coupling in Two Dimensions
General Hamiltonian for spin ½ systems
Rashba Hamiltonian
Strong out-of plane junction electric field
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Transport In Spin ½ Systems Two Dimensions

• Upon momentum integration continuity equations

• Rashba coupling (2D Asymmetric Quantum Wells)
  

Burkov Nunez and MacDonald Mishchenko, Shytov
and Halperin
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Rashba SO Coupling 2D Photon
Bernevig, Yu and Zhang, PRL 95, 076602 (2005)
Maxwells Equations
Spin-Orbit Coupling
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Spintronics without spin injection and spin
detection

-
S
C
V
R

-
With strong spin-orbit coupling, injected charge
packet spontaneously splits into two spin
packets, propagating in opposite directions at
the Rashba speed, without any applied E field.
This effect can be used to construct a spin bus.

In conventional charge dynamics, injected charge
packets simply diffuses.
With a E field, the charge packets also
drifts. Drift-diffusion is the fundamental
process underlying all conventional electronics.
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Conclusions

• Spin Hall effect is a profoundly deep effect in
  solid state physics,
• Natural generalization of the Hall effect and
  quantum Hall effect.
• Natural extensions of the spin Hall effect
  orbitronics, spintronics without
• Spin injection and spin detection, quantum spin
  Hall effect.
• Need close interaction among theory, experiments
  and materials science.
• Frontier of science and technology.