1. Anomalous Hall effects :
Merging The Semiclassical and Microscopic Theories
JAIRO SINOVA
Karlsruhe, Germany, July 16th 2008
Research fueled by:
NERC

2. OUTLINE

3. Anomalous Hall effect:
Spin-orbit coupling “force” deflects like-spin particles I _
FSO
majority
minority V
Simple electrical measurement
of magnetization
InMnAs
controversial theoretically: three contributions to the AHE (intrinsic deflection, skew scattering, side jump scattering)

4. A history of controversy
(thanks to P. Bruno– CESAM talk)

5. Anomalous Hall effect: what is necessary to see the effects?_
FSO
majority
minority V
Necessary condition for AHE: TIME REVERSAL SYMMETRY MUST BE BROKEN
Need a magnetic field and/or magnetic order
BUT IS IT SUFFICIENT?
(P. Bruno– CESAM 2005)

6. Local time reversal symmetry being broken does not always mean AHE present
Staggered flux with zero average flux:  -
Translational invariant so xy =0
Similar argument follows for antiferromagnetic ordering -  -  - 
Is xy zero or non-zero?
Does zero average flux necessary mean zero xy ? - 3
No!! (Haldane, PRL 88)
(P. Bruno– CESAM 2005)

7. Is non-zero collinear magnetization sufficient?
In the absence of spin-orbit coupling a spin rotation of  restores TR symmetry and xy=0
If spin-orbit coupling is present there is no invariance under spin rotation and xy≠0
(P. Bruno– CESAM 2005)

8. Collinear magnetization AND spin-orbit coupling → AHE
Does this mean that without spin-orbit coupling one cannot get AHE?
Even non-zero magnetization is not a necessary condition
No!! A non-trivial chiral magnetic structure WILL give AHE even without spin-orbit coupling
Mx=My=Mz=0
xy≠0
(P. Bruno– CESAM July 2005)
Bruno et al PRL 04

9. COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING vs
COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING vs. CHIRAL MAGNET STRUCTURES
AHE is present when SO coupling and/or non-trivial spatially varying magnetization (even if zero in average)
Spatial dependent magnetization: also can lead to AHE. A local transformation to the magnetization direction leads to a non-abelian gauge field, i.e. effective SO coupling (chiral magnets), which mimics the collinear+SO effective Hamiltonian in the adiabatic approximation
SO coupled chiral states: disorder and electric fields lead to AHE/SHE through both intrinsic and extrinsic contributions
So far one or the other have been considered but not both together, in the following we consider only collinear magnetization + SO coupling

10. Movie created by Mario Borunda
Intrinsic deflection
Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure)
Movie created by Mario Borunda
Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling.

11. Skew scattering Movie created by Mario Borunda
Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators.
Movie created by Mario Borunda

12. Side-jump scattering
Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step.
Related to the intrinsic effect: analogy to refraction from an imbedded medium
Movie created by Mario Borunda

13. Intrinsic deflection Side jump scattering Skew scattering E
Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure)
Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling.
Side jump scattering
Related to the intrinsic effect: analogy to refraction from an imbedded medium
Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step.
Skew scattering
Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. This is also known as Mott scattering used to polarize beams of particles in accelerators.

14. Microscopic vs. Semiclassical
Need to match the Kubo, Boltzmann, and Keldysh
Kubo: systematic formalism
Boltzmann: easy physical interpretation of different contributions (used to define them)
Keldysh approach: also a systematic kinetic equation approach (equivelnt to Kubo in the linear regime). In the quasiparticle limit it must yield Boltzmann eq.

15. THE THREE CONTRIBUTIONS TO THE AHE: MICROSCOPIC KUBO APPROACH
Skew
σHSkew  (skew)-1 2~σ0 S where
S = Q(k,p)/Q(p,k) – 1~
V0 Im[<k|q><q|p><p|k>]
Skew scattering
n, q
m, p
n’, k
Side-jump scattering
Vertex Corrections
 σIntrinsic
Intrinsic AHE: accelerating between scatterings
n, q
n’n, q
Intrinsic
σ0 /εF

16. FOCUS ON INTRINSIC AHE (early 2000’s): semiclassical and Kubo
STRATEGY: compute this contribution in strongly SO coupled ferromagnets and compare to experimental results, does it work?
n, q
n’n, q
Kubo:
Semiclassical approach in the “clean limit”
In ferromagnetic systems where SO coupling is intrinsic and strong the Kubo formalism in the weak scattering limit and a semiclassical treatment of Bloch wave-packet dynamics captures a nonzero anomalous Hall CONDUCTIVITY !! (The Berry’s phase in k-space approach)
K. Ohgushi, et al PRB 62, R6065 (2000); T. Jungwirth et al PRL 88, 7208 (2002);
T. Jungwirth et al. Appl. Phys. Lett. 83, 320 (2003); M. Onoda et al J. Phys. Soc. Jpn. 71, 19 (2002); Z. Fang, et al, Science 302, 92 (2003).

17. Success of intrinsic AHE approach in comparing to experiment: phenomenological “proof”
DMS systems (Jungwirth et al PRL 2002, APL 03)
Fe (Yao et al PRL 04)
layered 2D ferromagnets such as SrRuO3 and pyrochlore ferromagnets [Onoda and Nagaosa, J. Phys. Soc. Jap. 71, 19 (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003), Shindou and Nagaosa, Phys. Rev. Lett. 87, (2001)]
colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999).
CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004)
AHE in GaMnAs
AHE in Fe
Berry’s phase based AHE effect is quantitative-successful in many instances BUT still not a theory that treats systematically intrinsic and extrinsic contribution in an equal footing
Experiment
sAH  1000 (W cm)-1
Theroy
sAH  750 (W cm)-1

18. AHE in Rashba systems with weak disorder:
INTRINSIC+EXTRINSIC: REACHING THE END OF A DECADES LONG DEBATE
AHE in Rashba systems with weak disorder:
Dugaev et al (PRB 05)
Sinitsyn et al (PRB 05, PRB 07)
Inoue et al (PRL 06)
Onoda et al (PRL 06, PRB 08)
Borunda et al (PRL 07), Nuner et al (PRB 07, PRL 08)
Kovalev et al (PRB 08)
All are done using same or equivalent linear response formulation–different or not obviously equivalent answers!!!
The only way to create consensus is to show (IN DETAIL) agreement between ALL the different equivalent linear response theories both in AHE and SHE

19. Kubo-Streda formula summary Semiclassical Boltzmann equation
Golden rule:
In metallic regime:
J. Smit (1956):
Skew Scattering
Calculation done easiest in normal spin basis

20. Semiclassical approach II:
Golden Rule:
Coordinate shift:
Modified
Boltzmann
Equation:
Sinitsyn et al PRB 06
Berry curvature:
velocity:
current:

21. “AHE” in graphene EF
Armchair edge
Zigzag edge

22. Single K-band with spin up
Kubo-Streda
formula:
In metallic regime:
Sinitsyn et al PRB 07

23. Comparing Botlzmann to Kubo in the chiral basis

24. AHE in Rashba 2D system n’n, q n, q Inversion symmetry
(differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator)
To put this parsing of the contributions to the spin transport coefficients we turn now to a specific example of two dimensional Rashba systems where we are going to consider electron and hole doped type systems and see clear differences between them. The 2DEG+Rashba system, which was the one our team initially focused upon), originates from the broken inversion symmetry in the 2D trap and gives a SO linear in momentum (which we now understand clearly to be the originator of the now cleared up confusion).
Inversion symmetry
 no R-SO
Broken inversion symmetry
 R-SO
Bychkov and Rashba (1984)

25. Only when ONE both sub-band there is a significant contribution
AHE in Rashba 2D system
Kubo and semiclassical approach approach: (Nuner et al PRB08, Borunda et al PRL 07)
Only when ONE both sub-band there is a significant contribution
When both subbands are occupied there is additional vertex corrections that contribute

26. AHE in Rashba 2D system
Keldysh and Kubo match analytically in the metallic limit
When both subbands are occupied the skew scattering is only obtained at higher Born approximation order AND the extrinsic contribution is unique (a hybrid between skew and side-jump)
Kovalev et al PRB 08
Numerical Keldysh approach (Onoda et al PRL 07, PRB 08)
Solved within the self consistent T-matrix approximation for the self-energy

27. AHE in Rashba 2D system: “dirty” metal limit?
Is it real? Is it justified? Is it “selective” data chosing?
Can the kinetic metal theory be justified when disorder is larger than any other scale?

28. OTHER ANOMALOUS HALL EFFECTS

29. Spin Hall effect I V=0 Spin-current generation in non-magnetic systems
Take now a PARAMAGNET instead of a FERROMAGNET: Spin-orbit coupling “force” deflects like-spin particles I _
FSO
V=0
non-magnetic
Carriers with same charge but opposite spin are deflected by the spin-orbit coupling to opposite sides.
Spin-current generation in non-magnetic systems
without applying external magnetic fields
Spin accumulation without charge accumulation
excludes simple electrical detection

30. Spin Hall Effect Paramagnets (Dyaknov and Perel) Interband
Coherent Response
 (EF) 0
Occupation #
Response
`Skew Scattering‘
(e2/h) kF (EF )1
X `Skewness’
[Hirsch, S.F. Zhang]
Intrinsic
`Berry Phase’
(e2/h) kF
[Murakami et al, Sinova et al]
Influence of Disorder
`Side Jump’’
[Inoue et al, Misckenko et al, Chalaev et al…]
Paramagnets

31. INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/ ) Sinova et al PRL 2004 (cont-mat/ )
as there is an intrinsic AHE (e.g. Diluted magnetic semiconductors), there should be an intrinsic spin-Hall effect!!!
n, q
n’n, q
(differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator)
To put this parsing of the contributions to the spin transport coefficients we turn now to a specific example of two dimensional Rashba systems where we are going to consider electron and hole doped type systems and see clear differences between them. The 2DEG+Rashba system, which was the one our team initially focused upon), originates from the broken inversion symmetry in the 2D trap and gives a SO linear in momentum (which we now understand clearly to be the originator of the now cleared up confusion).
Inversion symmetry
 no R-SO
Broken inversion symmetry
 R-SO
Bychkov and Rashba (1984)

32. ‘Universal’ spin-Hall conductivity
n, q
n’n, q
Looking at the intrinsic bubble contribution to the transport coefficient this gives a contribution of e/8pi when both subbands are occupied and goes linearly to zero with density when one of the subbands are occupied (so, although not apparent, it is dependent on the SO coupling parameter lambda).
Color plot of spin-Hall conductivity:
yellow=e/8π and red=0

33. SHE conductivity: all contributions– Kubo formalism perturbation theory
Skew
σ0 S
n, q
n’n, q
Intrinsic
σ0 /εF
Vertex Corrections
 σIntrinsic
Within the diagrammatic perturbation treatment we have similar terms as before but now the bare vertex contain the electric field coupling term (j_v) and the response to a chose definition of the spin current which in this case we take it to be the one usually taken (the symmetric product of v and s)
= j = -e v
= jz = {v,sz}

34. Disorder effects: beyond the finite lifetime approximation for Rashba 2DEG
Question: Are there any other major effects beyond the finite life time broadening? Does side jump contribute significantly?
n, q
n’n, q +
+…=0
For the Rashba example the side jump contribution cancels the intrinsic contribution!!
Inoue et al PRB 04
Dimitrova et al PRB 05
Raimondi et al PRB 04
Mishchenko et al PRL 04
Loss et al, PRB 05
Once the vertex-side-jump corrections are taken into account it was noticed that the vertex ladder correction cancels this intrinsic bubble contribution in the limit of tau-> infinity. It was also shortly understood that in other systems (where the SO coupling was not linear in k) these vertex corrections vanished and did not influence the calculated spin-transport coefficient. This is the puzzle that has kept many of us up at night over the past year and which now it has been resolved—everyone agreeing that indeed the disorder scattering gives a zero SHE in the 2DEG-Rahsba system but that this consideration does not translate to 3D systems of 2DHG-Rashba systems.
Ladder partial sum vertex correction:
the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05)

35. First experimental observations at the end of 2004
Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/
PRL 05
Experimental observation of the spin-Hall effect in a two
dimensional spin-orbit coupled semiconductor system
Co-planar spin LED in GaAs 2D hole gas: ~1% polarization
CP [%]
1.505
1.52
Light frequency (eV)
Kato, Myars, Gossard, Awschalom, Science Nov 04
Observation of the spin Hall effect bulk in semiconductors
Local Kerr effect in n-type GaAs and InGaAs:
~0.03% polarization (weaker SO-coupling, stronger disorder)

36. OTHER RECENT EXPERIMENTS
Transport observation of the SHE by spin injection!!
Valenzuela and Tinkham cond-mat/ , Nature 06
Saitoh et al APL 06
Sih et al, Nature 05, PRL 05
“demonstrate that the observed spin accumulation is due to a transverse bulk electron spin current”

37. QUANTUM SPIN HALL EFFECT
(Physics Today, Feb 2008)

38. Anomalous Hall effect in cold atoms
Unitary transformation
State vector in the pseudospin basis:
Interaction of atoms with laser fields
Diagonalization with local unitary transformation:
Pure gauge:

39. U(m) adiabatic gauge field
We can then introduce
the adiabatic condition and reach the dynamical evolution of degenerate ground subspace
U(m) adiabatic
gauge field:
Berry phase, Spin-orbit coupling, etc.

40. Spin-orbit coupling and AHE in atomsx y z
Two dark-states consist of the degenerate subspace (pseudospin-1/2 states):
Mixing angle:
U(2) adiabatic gauge field:
(see also Staunesco, Galitskii, et al PRL 07)

41. AHE y x -y Effective trap potentials:
Employ Gaussian laser beams and set the trap:
The effective Hamiltonian ( ):
AHE

42. Mario Borunda
Texas A&M U.
Xin Liu
Texas A&M U.
Alexey Kovalev
Texas A&M U.
Nikolai Sinitsyn
LANL
Ewelina Hankiewicz
U. of Missouri
Texas A&M U.
Tomas Jungwirth
Inst. of Phys. ASCR
U. of Nottingham
Joerg Wunderlich
Cambridge-Hitachi
Laurens Molenkamp
Wuerzburg
Kentaro Nomura
U. Of Texas
Branislav Nikolic
U. of Delaware
Allan MacDonald
U of Texas
Other collaborators: Bernd Kästner, Satofumi Souma, Liviu Zarbo, Dimitri Culcer , Qian Niu, S-Q Shen, Brian Gallagher, Tom Fox, Richard Campton, Winfried Teizer, Artem Abanov

43. EXTRAS

45. Spin-orbit coupling interaction (one of the few echoes of relativistic physics in the solid state)
Ingredients: -“Impurity” potential V(r)
- Motion of an electron
Produces
an electric field
In the rest frame of an electron
the electric field generates and
effective magnetic field
This gives an effective interaction with the electron’s magnetic moment
Since I am the first talk and the topic of spin-orbit coupling will come up often let me introduce its basic notion. The spin orbit coupling interactions is nothing but an effective magnetic field interaction felt by a moving charge due to a changing electric field due to its relative motion in a potential generating such an electric field. Maxwell’s equations show us that this relative motion of the charge and the electric field induces an effective magnetic field proportional to the orbital momentum of the quasiparticle.
As illustrated in the equation shown the spin quantization axis for such a
CONSEQUENCES
If part of the full Hamiltonian quantization axis of the spin now depends on the momentum of the electron !!
If treated as scattering the electron gets scattered to the left or to the right depending on its spin!!

46. Spin Accumulation – Weak SO
The new challenge: understanding spin accumulation
Spin is not conserved; analogy with e-h system
Spin Accumulation – Weak SO
Quasi-equilibrium
Parallel conduction
Spin diffusion length
Burkov et al. PRB 70 (2004)

47. Spin Accumulation – Strong SO?
Mean Free
Path?
Spin Precession
Length

48. SPIN ACCUMULATION IN 2DHG: EXACT DIAGONALIZATION STUDIES
so>>ħ/
Width>>mean free path
Nomura, Wundrelich et al PRB 06
Key length: spin precession length!!
Independent of  !!

49. SHE experiment in GaAs/AlGaAs 2DHG
1.5 m
channel n p y x z
LED 1 2
Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05
10m channel
- shows the basic SHE symmetries
- edge polarizations can be separated
over large distances with no significant
effect on the magnitude
- 1-2% polarization over detection
length of ~100nm consistent with
theory prediction (8% over 10nm
accumulation length)
Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '05

50. 3. Charge based measurements of SHE
Non-equilibrium Green’s function formalism (Keldysh-LB)
Advantages:
No worries about spin-current definition. Defined in leads where SO=0
Well established formalism valid in linear and nonlinear regime
Easy to see what is going on locally
Fermi surface transport

51. PRL 05

52. H-bar for detection of Spin-Hall-Effect
(electrical detection through inverse SHE)
E.M. Hankiewicz et al ., PRB 70, R (2004)

53. New (smaller) sample
sample layout
200 nm
1 mm

54. SHE-Measurement no signal in the n-conducting regime
insulating
p-conducting
n-conducting
strong increase of the signal in the p-conducting regime, with pronounced features
no signal in the n-conducting regime

55. Mesoscopic electron SHE
calculated voltage signal for electrons (Hankiewicz and Sinova) L
L/6
L/2

56. more than 10 time larger! Mesoscopic hole SHE
calculated voltage signal
(Hankiweicz, Sinova, & Molenkamp) L L
L/6
L/2
more than
10 time
larger!