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

2. 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)

3. The spin Hall effect

4. Can Moore’s law keep going?
Power dissipation=greatest obstacle for Moore’s 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.

5. 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 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.

6. 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 => dissipation just as in Ohm’slaw. 2 2

7. 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

8. Using SO: spin FET - - - V/2 V v v v Das-Datta proposal.
Beff
Beff
Beff
Das-Datta proposal.
Animation by Bernevig and Sinova.

9. Generalization of the quantum Hall effect
Quantum Hall effect exists in D=2, due to Lorentz force.
GaAs
Natural generalization to D=3, due to spin-orbit force:
x: current direction
y: spin direction
z: electric field
GaAs
3D hole systems (Murakami, Nagaosa and Zhang, Science 2003)
2D electron systems (Sinova et al, PRL 2004)

11. Valence band of GaAs Luttinger Hamiltonian S S P3/2 P P1/2
( : spin-3/2 matrix, describing the P3/2 band)

12. Unitary transformation
Diagonalize the first term with a local unitary transformation
Helicity basis
: gauge field in k!

13. 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

14. Effective Hamiltonian for adiabatic transport
(Dirac monopole)
Nontrivial spin dynamics comes from the
Dirac monopole at the center of k space, with
eg=l:
Eq. of motion
Drift velocity
Topological term

15. Dissipationless spin current induced by the electric field

16. The intrinsic spin Hall effect
Bulk GaAs
Energy (eV)
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 Berry’s phase in momentum space similar to the QHE.
Contrast between the spin current and the Ohm’s law:

17. - - - - - - T T Time reversal and the dissipationless spin current v v

18. Effect due to disorder Luttinger model:
Intrinsic spin Hall conductivity (Murakami et al.(2003))
+ spinless impurities ( -function pot.)
Vertex correction vanishes identically!
2DHG Bernevig+Zhang (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))

19. Mott scattering or the extrinsic Spin Hall effect
Electric field induces a transverse spin current.
Extrinsic spin Hall effect
Mott (1929), D’yakonov 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 !

20. 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
T=30K, Hole density:
: measured by Kerr rotation

21. 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!

22. 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

23. 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

24. Spin – Orbit Coupling in Two Dimensions
General Hamiltonian for spin ½ systems:
Rashba Hamiltonian E
GaAs
Strong out-of plane junction electric field

25. 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

26. Rashba SO Coupling: 2D Photon
Bernevig, Yu and Zhang, PRL 95, (2005)
Spin-Orbit Coupling
Maxwell’s Equations

27. Spintronics without spin injection and spin detection+ - S C R V + E -
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.

28. 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.