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

OUTLINE

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)

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

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)

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)

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)

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

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

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.

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

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

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.

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.

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

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).

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, 116801 (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

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

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

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

“AHE” in graphene EF Armchair edge Zigzag edge

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

Comparing Botlzmann to Kubo in the chiral basis

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)

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

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

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?

OTHER ANOMALOUS HALL EFFECTS

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

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

INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/0308167) Sinova et al PRL 2004 (cont-mat/0307663) 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)

‘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

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}

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)

First experimental observations at the end of 2004 Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/0410295 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)

OTHER RECENT EXPERIMENTS Transport observation of the SHE by spin injection!! Valenzuela and Tinkham cond-mat/0605423, 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”

QUANTUM SPIN HALL EFFECT (Physics Today, Feb 2008)

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:

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.

Spin-orbit coupling and AHE in atoms x 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)

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

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

EXTRAS

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!!

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)

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

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  !!

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

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

PRL 05

H-bar for detection of Spin-Hall-Effect (electrical detection through inverse SHE) E.M. Hankiewicz et al ., PRB 70, R241301 (2004)

New (smaller) sample sample layout 200 nm 1 mm

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

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

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!