Anomalous Hall transport in metallic spin-orbit coupled systems November 11 th 2008 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research.

Anomalous Hall transport in metallic spin-orbit coupled systems November 11 th 2008 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research fueled by: 1 Hitachi Cambridge Jorg Wunderlich, A. Irvine, et al Institute of Physics ASCR T. Jungwirth, K. Výborný, et al University of Texas A. H. MaDonald, N. Sinitsyn (LANL), et al. University of Nottingham B. Gallagher, K. Edmonds, et al. Texas A&M A. Kovalev, M. Borunda, et al Wuerzburg Univ. L. Molenkamp, E. Hankiweicz et al

2 Anomalous Hall transport: lots to think about Taguchi et al Fang et al Wunderlich et al Kato et al Valenzuela et al SHE Inverse SHE SHE Intrinsic AHE (magnetic monopoles?) AHE AHE in complex spin textures

The family of spintronics Hall effects SHE -1 B=0 spin current gives charge current Electrical detection AHE B=0 polarized charge current gives charge-spin current Electrical detection SHE B=0 charge current gives spin current Optical detection 3

4 1)The basic phenomena of AHE 2)When do we have AHE: necessary and sufficient conditions (P. Bruno) 3)History of the anomalous Hall effect and major contributions 4)Why is anomalous Hall transport difficult theoretically? 5)Cartoon of mechanisms contributing to the AHE 6)Semiclassical theory of AHE: a)Boltzmann Eq. approach for diagonal transport: a warm-up b)Wave-packets of Bloch electrons: birth of Berry’s connection c)Dynamics of Bloch electron wave-packets: birth of Berry’s curvature d)What does it all mean; does it make sense from Kubo formula? e)Boltmann Eq. for Bloch wave-packets: slowly does it. f)Back to the three main mechanisms: clarifying/correcting popular believes 7)Microscopic theory of AHE (Kubo approach) a)Kubo formula microscopic approach to transport b)Does it match the semiclassical approach? c)Other microscopic approaches 8)Spin-injection Hall effect: a new tool to explore spintronic Hall effects 9)Spin Hall Effect 1)The basic idea 2)Short history and developments 3)Spin accumulation 4)Inverse SHE in HgTe OUTLINE

Anomalous Hall effect Simple electrical measurement of magnetization Spin dependent “force” deflects like-spin particles I _ F SO _ _ _ majority minority V InMnAs 5

I F SO _ majority minority V I _ F SO _ _ _ V=0 non-magnetic I spin F SO _ V non-magnetic I F SO V MzMz M z =0 non-magnetic I=0 M z =0 magnetic optical detection AHE SHE SHE -1 SIHE F SO (a)(b) (c)(d)

7 Anomalous Hall effect (scaling with ρ) Dyck et al PRB 2005 Kotzler and Gil PRB 2005 Co films Edmonds et al APL 2003 GaMnAs

Anomalous Hall effect: what is necessary to see the AHE? 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) I _ F SO _ _ _ majority minority V Onsager relations (time reversal, general) 8

Local time reversal symmetry being broken does not always mean AHE present Staggered flux with zero average flux in an electron gas: -- --  -- Is  xy zero or non-zero? Does zero average flux necessary mean zero  xy ? -- -- -- 33 -- -- -- 33 B to –B does not leave the Hamiltonian invariant so  xy ≠0 (Haldane, PRL 88) (P. Bruno– CESAM 2005) 9 Similar argument follows for antiferromagnetic ordering  --  --  --  xy =0 B to –B plus a translation does NOT change the Hamiltonian so  xy (B)=  xy (-B) for this system BUT Onsager says  xy (B)= -  xy (-B)

Is non-zero collinear magnetization sufficient? (P. Bruno– CESAM 2005) In the absence of spin-orbit coupling a spin rotation of  leaves the Hamiltonian the same so  xy (B=0,M)=  xy (B=0,-M) for this system and using Onsager relation one gets  xy =0 (i.e. spins don’t know about left or right) If spin-orbit coupling is present there is no invariance under spin rotation and  xy ≠0 10

(P. Bruno– CESAM July 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 M x =M y =M z =0  xy ≠0 Bruno et al PRL 04 11

COLLINEAR MAGNETIZATION AND SPIN-ORBIT COUPLING vs. CHIRAL MAGNETIC STRUCTURES AHE is present when SO coupling and/or non-trivial spatially varying magnetization (even if zero in average) SO coupled Bloch states: disorder and electric fields lead to AHE/SHE through both intrinsic and extrinsic contributions Spatial dependent magnetization: also can lead to AHE. A local transformation to the magnetization direction leads to an effective SO coupling (chiral magnets), which mimics the collinear+SO effective Hamiltonian in the adiabatic approximation So far one or the other have been considered but not both together 12 (recent exception: J. Shibata and H. Kohno arXiv:0810.0610)

1880-81: Hall discovers the Hall and the anomalous Hall effect The tumultuous history of AHE 1970: Berger reintroduces (and renames) the side-jump: claims that it does not vanish and that it is the dominant contribution, ignores intrinsic contribution. (problem: his side-jump is gauge dependent) Berger 14 Luttinger 1954: Karplus and Luttinger attempt first microscopic theory: they develop (and later Kohn and Luttinger) a microscopic theory of linear response transport based on the equation of motion of the density matrix for non-interacting electrons, ; run into problems interpreting results since some terms are gauge dependent. Lack of easy physical connection. Hall 1970’s: Berger, Smit, and others argue about the existence of side-jump: the field is left in a confused state. Who is right? How can we tell? Three contributions to AHE are floating in the literature of the AHE: anomalous velocity (intrinsic), side-jump, and skew contributions. 1955-58: Smit attempts to create a semi-classical theory using wave- packets formed from Bloch band states: identifies the skew scattering and notices a side-step of the wave-packet upon scattering and accelerating..Speculates, wrongly, that the side-step cancels to zero. The physical interpretation of the cancellation is based on a gauge dependent object!!

The tumultuous history of AHE: last three decades 15 2004’s: Spin-Hall effect is revived by the proposal of intrinsic SHE (from two works working on intrinsic AHE): AHE comes to the masses, many debates are inherited in the discussions of SHE. 1980’s: Ideas of geometric phases introduced by Berry; QHE discoveries 2000’s: Materials with strong spin-orbit coupling show agreement with the anomalous velocity contribution: intrinsic contribution linked to Berry’s curvature of Bloch states. Ignores disorder contributions. 2004-8’s: Linear theories in simple models treating SO coupling and disorder finally merge: full semi-classical theory developed and microscopic approaches are in agreement among each other in simple models.

Intrinsic deflection 16 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. ~τ 0 or independent of impurity density 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. They however come out in a different band so this gives rise to an anomalous velocity through scattering rates times side jump. independent of impurity density STRONG SPIN-ORBIT COUPLED REGIME (Δ so >ħ/τ) Side jump scattering V imp (r) 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. ~1/n i V imp (r) 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) E SO coupled quasiparticles

Why is AHE difficult theoretically AHE conductivity much smaller than σ xx : many usual approximations fail Microscopic approaches: systematic but cumbersome; what do they mean; use non-gauge invariant quantities (final result gauge invariant) Multiband nature of band-structure (SO coupling) is VERY important; hard to see these effects in semi-classical description (where other bands are usually ignored). Simple semi-classical derivations give anomalous terms that are gauge dependent but are given physical meaning (dangerous and wrong) Usual “believes” on semi-classically defined terms do not match the full semi-classical theory (in agreement with microscopic theory) What happens near the scattering center does not stay near the scattering centers (not like Las Vegas) T-matrix approximation (Kinetic energy conserved); no longer the case, adjustments have to be made to the collision integral term Be VERY careful counting orders of contributions, easy mistakes can be made. 17

18 What do we mean by gauge dependent? Electrons in a solid (periodic potential) have a wave-function of the form Gauge dependent car BUT is also a solution for any a(k) Any physical object/observable must be independent of any a(k) we choose to put Gauge wand (puts an exp(ia(k)) on the Bloch electrons) Gauge invariant car

Semi-classical theory of transport for free electrons: warm up I Idea I: to interpret things in classical particles transform description from plane waves to wave-packets (which is a good basis too) ~100 λ F ≈100 Å Valid for fields and disorder that vary in scales larger than the wavepackets Eqs of motion ~1% of BZ 20 Idea II: the distribution of the wave-packets obey Boltzmann classical dynamics. Connection to quantum mechanics is through the treatment of the collisions. Warning: for non-interacting electrons there is NO (1-f)f terms (Born approximation usual: KE conserved; ignores what happens at the scattering center)

Letand look for a solution of the form Step 2: Once we have the non-equilibrium distribution function we can calculate the current and from that deduce the conductivity Same results as in the microscopic calculations Semi-classical theory of transport for free electrons: warm up continued 21 Step one: find the non-equilibrium part of the distribution function

Semi-classical theory of transport for Bloch electrons: the dynamics of wave-packets KNOW about the band structure Idea I: to interpret things in classical particles transform description from plane waves to wave-packets (which is a good basis too) ~100 λ F ≈100 Å Valid for fields and disorder that vary in scales larger than the wavepackets ~1% of BZ Once the information of the periodic potential is imbedded into the wave-packet the choice of the phase factors are important to have it center at r c 22 Are the Eqs of motion Yes No ? ?

Building a wave-packet from Bloch electrons: the birth of the Berry’s connection We want to have w kc (k) such that 23 Berry’s phase connection Influences dynamics of wave-packets (velocities: 2 nd step of semiclassical approach) Influences scattering of wave-packets (collision integral term of Boltzmann eqn. in several parts)

Dynamics of wave-packets of Bloch electrons: the birth of the Berry’s curvature and the anomalous velocity Task: build a Lagrangian from the new dynamic variables r c and k c 24 Applying Lagrange’s equations on the above Lagrangian: Berry’s connection (gauge dependent) where motion perpendicular to E, Hall type motion! already LINEAR EThe whole Fermi sea participates on this current contribution !! Berry’s curvature (gauge invariant) “Anomalous velocity”

Does the intrinsic contribution to the current make sense microscopically? From microscopic linear response to semiclassical linear response 25 Intrinsic contribution matches Kubo linear response theory (clean limit) !!! Kubo formula (1 st order perturbation theory)

Dynamics of wave-packets of Bloch electrons: How do the Berry’s curvature dynamics affect scattering? 26 Tried to interpret it physically BUT it is gauge dependent (i.e. only gauge invariant quantities have measurable physical meaning) ! Early theories (Berge,Smit) noticed that Bloch electron wave-packets seem to experience a side-step upon scattering: (a dangerous way of doing dynamics)

Dynamics of wave-packets of Bloch electrons: How do the Berry’s curvature dynamics affect scattering? 27 Tried to interpret it physically BUT it is gauge dependent (i.e. only gauge invariant quantities have measurable physical meaning) ! Early theories (Berge,Smit) noticed that Bloch electron wave-packets seem to experience a side-step upon scattering: The gauge invariant expressions can be derived using the gauge invariant Lagrangian dynamics shown earlier (Sinitsyn et al 2006) l=(n,k) Side jump scattering

How does side-jump affect transport? 28 Side jump scattering The side-jump comes into play through an additional current and influencing the Boltzmann equation and through it the non-equilibrium distribution function VERY STRANGE THING: for spin-independent scatterers side-jump is independent of scatterers!! 1 st -It creates a side-jump current: 2 nd -An extra term has to be added to the collision term of the Boltzmann eq. to account because upon elastic scattering some kinetic energy is transferred to potential energy. full ω ll’ does not assume KE conserved, T-matrix approximation of ω ll’ (ω T ll’ ) does.

29 Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current As before we do this in two steps: first calculate steady state non- equilibrium distribution function and then use it to compute the current. Set to 0 for steady state solution Only the normal velocity term, since we are looking for linear in E equation order of the disorder potential strength and symmetric and anti-symmetric components 1 st Born approximation 2 nd Born approximation (usual skew scattering contribution) To solve this equation we write the non-equilibrium component in various components that correspond to solving parts of the equation the corresponding order of disorder

30 Semiclassical transport of spin-orbit coupled Bloch electrons: Boltzmann Eq. and Hall current ~V 0 ~V ~V 2 2 nd step: (after solving them) we put them into the equation for the current and identify from there the different contributions to the AHE using the full expression for the velocity

Intrinsic deflection 31 Popular believe: ~τ 1 or ~1/n i WRONG E ~n i 0 or independent of impurity density Skew scattering (2 contributions) term missed by many people using semiclassical approach Side jump scattering (2 contributions) Popular believe: ~n i 0 or independent of impurity density Origin is on its effect on the distribution function

Boltzmann semiclassical approach: easy physical interpretation of different contributions (used to define them) but very easy to miss terms and make mistakes. MUST BE CONFIRMED MICROSCOPICALLY! How one understands but not necessarily computes the effect. Kubo approach: systematic formalism but not very transparent. Keldysh approach: also a systematic kinetic equation approach (equivalent to Kubo in the linear regime). In the quasi-particle limit it must yield Boltzmann semiclassical treatment. Microscopic vs. Semiclassical 33

Kubo microscopic approach to transport: diagrammatic perturbation theory Averaging procedures: = 1/  0 = 0  = + Bloch Electron Real Eigenstates Need to perform disorder average (effects of scattering) n, q Drude Conductivity σ = ne 2  /m * ~1/n i Vertex Corrections  1-cos(θ) Perturbation Theory: conductivity n, q 34

35 intrinsic AHE approach in comparing to experiment: phenomenological “proof” Berry’s phase based AHE effect is reasonably successful in many instances BUT still not a theory that treats systematically intrinsic and ext rinsic contribution in an equal footing n, q n’  n, q 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 et al (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003) colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999). CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004) Experiment  AH  1000 (  cm) -1 Theroy  AH  750 (  cm) -1 AHE in Fe AHE in GaMnAs

36 “Skew scattering” “Side-jump scattering” Intrinsic AHE: accelerating between scatterings n, q m, p n’, k n, q n’  n, q Early identifications of the contributions Vertex Corrections  σ Intrinsic ~  0 or n 0 i Intrinsic  σ 0 /ε F  ~  0 or n 0 i Kubo microscopic approach to AHE n, q m, p n’, k matrix in band index m’, k’

Armchair edge Zigzag edge EFEF “AHE” in graphene: linking microscopic and semiclassical theories 37

Single K-band with spin up In metallic regime: Sinitsyn et al PRB 07 38 Kubo-Streda calculation of AHE in graphene Don’t be afraid of the equations, formalism can be tedious but is systematic (slowly but steady does it) Kubo-Streda formula: A. Crépieux and P. Bruno (2001)

Comparing Boltzmann to Kubo (chiral basis) 39 Kubo identifies, without a lot of effort, the order in n i of the diagrams BUT not so much their physical interpretation according to semiclassical theory Sinitsyn et al 2007

40 Next simplest example: AHE in Rashba 2D system Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984)

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 41

AHE in Rashba 2D system 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 Keldysh and Kubo match analytically in the metallic limit Numerical Keldysh approach (Onoda et al PRL 07, PRB 08) 42

43 Recent progress: full understanding of simple models in each approach Semi-classical approach: Gauge invariant formulation; shown to match microscopic approach in 2DEG+Rashba, Graphene Sinitsyn et al PRB 05, PRL 06, PRB 07 Borunda et al PRL 07, Nunner et al PRB 08 Sinitsyn JP:C-M 08 Kubo microscopic approach: Results in agreement with semiclassical calculations 2DEG+Rashba, Graphene Sinitsyn et al PRL 06, PRB 07, Nunner PRB 08, Inoue PRL 06, Dugaev PRB 05 NEGF/Keldysh microscopic approach: Numerical/analytical results in agreement in the metallic regime with semiclassical calculations 2DEG+Rashba, Graphene Kovalev et al PRB 08, Onoda PRL 06, PRB 08

AHE in Rashba 2D system: “dirty” metal limit? Is it real? Is it justified? what does it mean in the regions where sing of σ AHE changes? Can the kinetic metal theory be justified when disorder is larger than any other scale? 44 No localization physics included (but σ is below the minimum conductivity limit !) Onoda et al 2007,2008

45 AHE in Rashba 2D system: “dirty” metal limit?

47 CONCLUSIONS (AHE) When creating a semiclassical theory be aware of the Gague Wand Usual identifications of contributions to AHE from semiclassical to microscopic theories need to be done carefully (σ xx dependence not the one usually thought off) Now that the metallic theory is clear (and nothing missing), time to study more complex materials in detail (microscopic nano-engineering of AHE) Can we test the simple models? Strongly disorder limit??

Spin-injection Hall effect Spin-injection Hall effect: A new member of the spintronic Hall family JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research fueled by: 49 Hitachi Cambridge Jorg Wunderlich, A. Irvine, et al Institute of Physics ASCR Tomas Jungwirth,, Vít Novák, et al Texas A&M A. Kovalev, M. Borunda, et al

50 Towards a spin-based non-magnetic FET device: can we electrically measure the spin-polarization? ISHE is now routinely used to detect other effects related to the generated spin-currents (Sitho et al Nature 2008) Can we achieve direct spin polarization detection through an electrical measurement in an all paramagnetic semiconductor system? Long standing paradigm: Datta-Das FET Unfortunately it has not worked: no reliable detection of spin-polarization in a diagonal transport configuration No long spin-coherence in a Rashba SO coupled system

Alternative: Alternative: utilize technology developed to detect SHE in 2DHG and measure polarization via Hall probes J. Wunderlich, B. Kaestner, J. Sinova and T. Jungwirth, Phys. Rev. Lett. 94 047204 (2005) Spin-Hall Effect 51 B. Kaestner, et al, JPL 02; B. Kaestner, et al Microelec. J. 03; Xiulai Xu, et al APL 04, Wunderlich et al PRL 05 Proposed experiment/device: Coplanar photocell in reverse bias with Hall probes along the 2DEG channel Borunda, Wunderlich, Jungwirth, Sinova et al PRL 07

material Device schematic - material 52 i p n 2DHG

- i p n 53 trench Device schematic - trench

i p n 2DHG 2DEG 54 n-etch Device schematic – n-etch

VdVd VHVH 2DHG 2DEG VsVs 55 Hall measurement Device schematic – Hall measurement

2DHG 2DEG e h e e ee e h h h h h VsVs VdVd VHVH 56 SIHE measurement Device schematic – SIHE measurement

Photovoltaic Cell Reverse- or zero-biased: Photovoltaic Cell trans. signal Red-shift of confined 2D hole  free electron trans. due to built in field and reverse bias light excitation with = 850nm ( well below bulk band-gap energy) σoσoσoσo σ+σ+σ+σ+ σ-σ-σ-σ- σoσoσoσo VLVL 57 -1/2 +1/2 +3/2 -3/2 bulk Transitions allowed for ħω>E g Transitions allowed for ħω<E g -1/2 +1/2 +3/2-3/2 Band bending: stark effect Transitions allowed for ħω<E g

n2 + + - - Spin injection Hall effect: Spin injection Hall effect: experimental observation n1 (  4) n2 n1 (  4) n2 n3 (  4) Local Hall voltage changes sign and magnitude along the stripe 58

Spin injection Hall effect  Anomalous Hall effect 59

and high temperature operation Zero bias- + + -- + + -- Persistent Spin injection Hall effect 60

THEORY CONSIDERATIONS Spin transport in a 2DEG with Rashba+Dresselhaus SO 61 For our 2DEG system: The 2DEG is well described by the effective Hamiltonian: Hence

62 spin along the [110] direction is conserved long lived precessing spin wave for spin perpendicular to [110] What is special about ? Ignoring the term for now The nesting property of the Fermi surface:

The long lived spin-excitation: “spin-helix” An exact SU(2) symmetry Only Sz, zero wavevector U(1) symmetry previously known: J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003). K. C. Hall et. al., Appl. Phys. Lett 83, 2937 (2003). Finite wave-vector spin components Shifting property essential 63

Spin configurations do not depend on the particle initial momenta. For the same x + distance traveled, the spin precesses by exactly the same angle. After a length x P =h/4mα all the spins return exactly to the original configuration. Physical Picture: Persistent Spin Helix Thanks to SC Zhang, Stanford University 64

65 Persistent state spin helix verified by pump-probe experiments Similar wafer parameters to ours

The Spin-Charge Drift-Diffusion Transport Equations For arbitrary α,β spin-charge transport equation is obtained for diffusive regime For propagation on [1-10], the equations decouple in two blocks. Focus on the one coupling S x+ and S z : For Dresselhauss = 0, the equations reduce to Burkov, Nunez and MacDonald, PRB 70, 155308 (2004); Mishchenko, Shytov, Halperin, PRL 93, 226602 (2004) 66

Steady state solution for the spin-polarization component if propagating along the [1-10] orientation 67 Steady state spin transport in diffusive regime Spatial variation scale consistent with the one observed in SIHE

68 Anomalous Hall effect (scaling with ρ) Dyck et al PRB 2005 Kotzler and Gil PRB 2005 Co films Edmonds et al APL 2003 GaMnAs Strong SO coupled regime Weak SO coupled regime

69 AHE contribution Type (i) contribution much smaller in the weak SO coupled regime where the SO-coupled bands are not resolved, dominant contribution from type (ii) Crepieux et al PRB 01 Nozier et al J. Phys. 79 Two types of contributions: i)S.O. from band structure interacting with the field (external and internal) ii)Bloch electrons interacting with S.O. part of the disorder Lower bound estimate of skew scatt. contribution

Spin injection Hall effect: Theoretical consideration Local spin polarization  calculation of the Hall signal Weak SO coupling regime  extrinsic skew-scattering term is dominant 70 Lower bound estimate

The family of spintronics Hall effects SHE -1 B=0 spin current gives charge current Electrical detection AHE B=0 polarized charge current gives charge-spin current Electrical detection SHE B=0 charge current gives spin current Optical detection 71 SIHE B=0 Optical injected polarized current gives charge current Electrical detection

72 SIHE: a new tool to explore spintronics nondestructive electric probing tool of spin propagation without magnetic elements all electrical spin-polarimeter in the optical range Gating (tunes α/β ratio) allows for FET type devices (high T operation) New tool to explore the AHE in the strong SO coupled regime

73 CONCLUSIONS (SIHE) Spin-injection Hall effect observed in a conventional 2DEG - nondestructive electrical probing tool of spin propagation - indication of precession of spin-polarization - observations in qualitative agreement with theoretical expectations - optical spin-injection in a reverse biased coplanar pn- junction: large and persistent Hall signal (applications !!!)

74 1)The basic phenomena of AHE 2)When do we have AHE: necessary and sufficient conditions (P. Bruno) 3)History of the anomalous Hall effect and major contributions 4)Why is anomalous Hall transport difficult theoretically? 5)Cartoon of mechanisms contributing to the AHE 6)Semiclassical theory of AHE: a)Boltzmann Eq. approach for diagonal transport: a warm-up b)Wave-packets of Bloch electrons: birth of Berry’s connection c)Dynamics of Bloch electron wave-packets: birth of Berry’s curvature d)What does it all mean; does it make sense from Kubo formula? e)Boltmann Eq. for Bloch wave-packets: slowly does it. f)Back to the three main mechanisms: clarifying/correcting popular believes 7)Microscopic theory of AHE (Kubo approach) a)Kubo formula microscopic approach to transport b)Does it match the semiclassical approach? c)Other microscopic approaches 8)Spin Hall Effect 1)The basic idea 2)Short history and developments 3)Spin accumulation 4)Inverse SHE in HgTe OUTLINE

Spin Hall effect like-spin Take now a PARAMAGNET instead of a FERROMAGNET: Spin-orbit coupling “force” deflects like-spin particles I _ F SO _ _ _ V=0 non-magnetic Spin-current generation in non-magnetic systems without applying external magnetic fields. Spin accumulation without charge accumulation excludes simple electrical detection Carriers with same charge but opposite spin are deflected due to SO to opposite sides. 75 Spin Hall Effect (Dyaknov and Perel) Interband Coherent Response  (E F  ) 0 Occupation # Response `Skew Scattering‘ [Hirsch, S.F. Zhang] Intrinsic `Berry Phase’  (e 2 /h) k F [Murakami et al, Sinova et al] Influence of Disorder `Side Jump’’ [Inoue et al, Misckenko et al, Chalaev et al…]

INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/0308167) Sinova et al PRL 2004 (cond-mat/0307663) as there is an intrinsic AHE (e.g. Diluted magnetic semiconductors), there should be an intrinsic spin-Hall effect!!! Inversion symmetry  no R-SO Broken inversion symmetry  R-SO Bychkov and Rashba (1984) (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) n, q n’  n, q 76

‘Universal’ spin-Hall conductivity Color plot of spin-Hall conductivity: yellow=e/8π and red=0 n, q n’  n, q 77

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? the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05) Ladder partial sum vertex correction: 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 n, q n’  n, q + +…=0 For the Rashba example the side jump contribution cancels the intrinsic contribution!! 78

First experimental observations at the end of 2004 CP [%] Light frequency (eV) 1.5051.52 Kato, et al Science Nov 04 Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder) 79 Wunderlich et al PRL Jan 05 SHE in two dimensional hole gas Co-planar spin LED in GaAs 2D hole gas: ~1% polarization

Other experiments followed Also reports of room temperature SHE Sih et al, Nature 05, PRL 05 Valenzuela and Tinkham cond- mat/0605423, Nature 06 Transport observation of the SHE by spin injection!! Saitoh et al APL 06 80

81 1)The basic phenomena of AHE 2)When do we have AHE: necessary and sufficient conditions (P. Bruno) 3)History of the anomalous Hall effect and major contributions 4)Why is anomalous Hall transport difficult theoretically? 5)Cartoon of mechanisms contributing to the AHE 6)Semiclassical theory of AHE: a)Boltzmann Eq. approach for diagonal transport: a warm-up b)Wave-packets of Bloch electrons: birth of Berry’s connection c)Dynamics of Bloch electron wave-packets: birth of Berry’s curvature d)What does it all mean; does it make sense from Kubo formula? e)Boltmann Eq. for Bloch wave-packets: slowly does it. f)Back to the three main mechanisms: clarifying/correcting popular believes 7)Microscopic theory of AHE (Kubo approach) a)Kubo formula microscopic approach to transport b)Does it match the semiclassical approach? c)Other microscopic approaches 8)Spin Hall Effect 1)The basic idea 2)Short history and developments 3)Spin accumulation 4)Inverse SHE in HgTe OUTLINE

Trying to understand spin accumulation Spin is not conserved; analogy with e-h system Burkov et al. PRB 70 (2004) Spin diffusion length Quasi-equilibrium Parallel conduction Spin Accumulation – Weak SO 82

Spin Accumulation – Strong SO Mean Free Path? Spin Precession Length ? 83

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  !! 84

1.5  m channel n n p y x z LED1 2 10  m channel SHE experiment in GaAs/AlGaAs 2DHG - 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) Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05 Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '05 85

First experimental observations at the end of 2004 CP [%] Light frequency (eV) 1.5051.52 Kato, et al Science Nov 04 Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder) 86 Wunderlich et al PRL Jan 05 SHE in two dimensional hole gas Co-planar spin LED in GaAs 2D hole gas: ~1% polarization

Other experiments followed Also reports of room temperature SHE Sih et al, Nature 05, PRL 05 Valenzuela and Tinkham cond- mat/0605423, Nature 06 Transport observation of the SHE by spin injection!! Saitoh et al APL 06 87

1.5  m channel n n p y x z LED1 2 10  m channel SHE experiment in GaAs/AlGaAs 2DHG - 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) Wunderlich, Kaestner, Sinova, Jungwirth, Phys. Rev. Lett. '05 Nomura, Wunderlich, Sinova, Kaestner, MacDonald, Jungwirth, Phys. Rev. B '05 88

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 Charge based measurements of SHE: SHE -1 Non-equilibrium Green’s function formalism (Keldysh-LB) 89

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

91 sample layout Molenkamp et al (unpublished) insulating p-conducting n-conducting Strong SHE -1 in HgTe theory

92 References for AHE (focus on theory; not complete, chosen for pedagogical purposes) Early theories: R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954): on-line;J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955): on-line; J. M. Luttinger, Phys. Rev. 112, 739 (1958): on-lineon-line J Smit, Physica 21, 877 (1955): on-line, Physica 24, 39 (1958): on-line; L. Berger, Phys. Rev. B 2, 4559 (1970): on-line ; P. Leroux-Hugon and A. Ghazali, J. Phys. C 5, 1072 (1972): on-lineon-line AHE in conduction electrons: P. Nozieres, C. Lewiner, J. Phys. France 34, 901 (1973): on-lineon-line Dirac+Pauli approach: A. Crépieux and P. Bruno, Phys. Rev. B 64, 014416 (2001): on-lineon-line “intrinsic” or Berry’s phase AHE: Y. Taguchi, et al Science 291, 5513 (2001): on-line; Jinwu Ye, et al Phys. Rev. Lett. 83, 3737 (1999): on-line;T. Jungwirth, et al PRL. 88, 207208 (2002): on-line, ibid, Appl. Phys. Lett. 83, 320 (2003): on-lineon-line M(r) induced AHW: P. Bruno, al Phys. Rev. Lett. 93, 096806 (2004): on-lineon-line AHE Fermi liquid properties: F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004): on-lineon-line Kubo+Boltzmann: N.A. Sinitsyn et al, Phys. Rev. Lett. 97, 106804 (2006): on-line, Phys. Rev. B 72, 045346 (2005): on-lineon-line Wave-packet dynamics: Ganesh Sundaram and Qian Niu, Phys. Rev. B 59, 14915 (1999): on-line; M. P. Marder, "Condensed Matter Physics", Wiley, New York, (2000)on-line AHE in 2DEG+Rashba:V. K. Dugaev, et al Phys. Rev. B 71, 224423 (2005): on-line; Jun-ichiro InouePhys. Rev. Lett. 97, 046604 (2006): on-line; N.A. Sinitsyn, Phys. Rev. B 75, 045315 (2007): on- line; Shigeki Onoda, et al, Phys. Rev. Lett. 97, 126602 (2006): on-line, Phys. Rev. B 77, 165103 (2008): on-line; N.A. Sinitsyn, et al Phys. Rev. B 75, 045315 (2007): on-line; Mario Borunda et al, Phys. Rev. Lett. 99, 066604 (2007): on-line; Tamara S. et al Phys. Rev. B 76, 235312 (2007): on-line; A. Kovalev, et al Phys. Rev. B 78, 041305 (2008): on-lineon-line on- lineon-line Reviews: L. Chien and C.R. Westgate, "The Hall Effect and Its Applications", Plenum, New York (1980); Jairo Sinova, et al Int. J. Mod. Phys. B 18, 1083 (2004): on-line; N. A. Sinitsyn, J. Phys.: Condens. Matter 20, 023201 (2008): on-lineon-line

DETAILED DERIVATIONS 93 The following slides include detailed derivations that we skipped in the previous slides where the results were stated. Some of these contain important tricks worth learning that appear continuously throughout the literature on anomalous transport.

Building a wave-packet from Bloch electrons: the birth of the Berry’s connection We want to have w kc (k) such that Berry’s phase connection 94 Integration by parts Using periodicity and sum of Integration over a unit cell Detailed derivation

Anomalous Hall transport in metallic spin-orbit coupled systems November 11 th 2008 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research.

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Anomalous Hall transport in metallic spin-orbit coupled systems November 11 th 2008 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research.

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Presentation on theme: "Anomalous Hall transport in metallic spin-orbit coupled systems November 11 th 2008 JAIRO SINOVA Texas A&M University Institute of Physics ASCR Research."— Presentation transcript:

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