Research fueled by: Spintronics Tutorial Session March 20 th, 2011 APS March Meeting JAIRO SINOVA Texas A&M University Institute of Physics ASCR Spin Hall

Research fueled by:

Spintronics Tutorial SessionMarch 20th, 2011

APS March Meeting

JAIRO SINOVATexas A&M University

Institute of Physics ASCR

Spin Hall effect and devices: anomalous and spin Hall effect, spin-helix transistors, and beyond

Hitachi CambridgeJoerg Wunderlich, A. Irvine, et al

Institute of Physics ASCRTomas Jungwirth, Vít Novák, et al

University of Würzburg Laurens Molenkamp, E. Hankiewiecz, et al

University of Nottingham Bryan Gallagher, Richard Campion, et al.

R. Duine (Utrecht); G. Bauer (Delft/Sendai); Y. Tserkovnyak (UCLA), A. Kovalev (Riverside); MacDonald (UT)

2Nanoelectronics, spintronics, and materials control by spin-orbit coupling

I. Introduction: using the dual personality of the electron•Internal coupling of charge and spin: origin and present use•Control of material and transport properties through spin-orbit coupling•Overview of program

II. Anomalous Hall effect: from the metallic to the insulating regime•Anomalous Hall effect basics, history, progress in the metallic regime

III.Spin injection Hall effect: a new paradigm in exploiting SO coupling•Spin based FET: old and new paradigm in charge-spin transport•Theory expectations and modeling•Experimental results: spin FET

Spin Hall effect and devices: anomalous and spin Hall effect, spin-helix

transistors, and beyond

Spin Hall effect and devices: anomalous and spin Hall effect, spin-helix

transistors, and beyond


The electron: the key character with dual personalities

CHARGECHARGEEasy to manipulate: Coulomb interaction

SPIN 1/2SPIN 1/2Makes the electron antisocial: a fermion

quantum mechanics

E=p2/2mE→ iħ d/dtp→ -iħ d/dr

““Classical” external manipulation of charge & spinClassical” external manipulation of charge & spin

special relativity

E2/c2=p2+m2c2

(E=mc2 for p=0)

+ particles/antiparticles & spin

Dirac equation=

4

ee--

Nanoelectronics, spintronics, and materials control by spin-orbit coupling

Internal communication between spin and charge:spin-orbit coupling interaction

(one of the few echoes of relativistic physics in the solid state)

This gives an effective interaction with the electron’s magnetic moment

Classical explanation (in reality it arises from a second order expansion of Dirac equation around the non-relativistic limit)

• “Impurity” potential V(r)Produces

an electric field

In the rest frame of an electronthe electric field generates an effective magnetic field

• Motion of an electron

5

ee--






an electric field

∇V

BBeffeff

pss



Consequence #1:Spin or the band-structure Bloch states are linked to the momentum.


6

ee--






an electric field

∇V

BBeffeff

pss



Consequence #2Mott scattering



Simple electrical measurement of out of plane magnetization (or

spin polarization ~ n↑-n↓)InMnAs

Spin dependent “force” deflects like-spin particles

ρH=R0B ┴ +4π RsM┴

Anomalous Hall Effect: the basics

I

_ FSO

FSO

_ __

majority

minority

V

M⊥

AHE is does NOT originate from any internal magnetic field created by M⊥; the field would have to be of the order of 100T!!!


Anomalous Hall effect (scaling with ρ for metals)

Material with dominant skew scattering mechanismMaterial with dominant scattering-independent mechanism

σxx>106 (Ωcm)-1σxx ~104-106 (Ωcm) -1


Anomalous Hall effect (scaling for insulators)

Diagonal hopping conductivity for most systems showing approximate scaling

σxy~σxx1.4-1.7 over a few decades for σxx <104 (Ωcm) -1

10


•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

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!!

11


The tumultuous history of AHE: last three decades

•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.

•2005-10’s: Linear theories treating SO coupling and disorder finally merge: full semi-classical theory developed and microscopic approaches are in agreement among each other in simple models.

•2010-11: More detailed ab-inito studies of AHE and an ab-initio full formalism developed for AHE (all non-scattering dependent contributions) plus CPA of alloys

12


Cartoon of the mechanisms contributing to AHE

independent of impurity density

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.

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

Intrinsic deflection B

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

Side jump scatteringVimp(r) (Δso>ħ/τ) ∝ λ*∇Vimp(r) (Δso<ħ/τ)

B

Skew scattering

Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. Known as Mott scattering.

~σ~1/niVimp(r) (Δso>ħ/τ) ∝ λ*∇Vimp(r) (Δso<ħ/τ) A

13


Contributions understood in simple metallic 2D models

Semi-classical approach:Gauge invariant formulation

Sinitsyn, Sinvoa, et al PRB 05, PRL 06, PRB 07

Kubo microscopic approach:in agreement with semiclassical

Borunda, Sinova, et al PRL 07, Nunner, Sinova, et al PRB 08

Non-Equilibrium Green’s Function (NEGF) microscopic approach

Kovalev, Sinova et al PRB 08, Onoda PRL 06, PRB 08

Restriction to homogeneous magnetization. Magnetic textures lead to the so called topological AHE.

14


Generalization to 3D: scattering in dependent AHE

Step 1. Use linearized version of Keldysh formalism to obtain

where

Kovalev, Sinova, Tserkovnyak PRL 2010

15


Scattering-independent contribution to the AHE

Well known intrinsic contributionSide jump contribution related to Berry curvature

Remaining side jump contributionKovalev, Sinova, Tserkovnyak PRL 2010

expressed through band structure only !!

16


Results for Luttinger model: simplest model of GaMnAs

Kovalev, Sinova, Tserkovnyak PRL 2010

It gives a formalism that allows for a systematic study of AHE through ab-initio calculations for metals (no alloys)

17


N. Nagaosa, Jairo Sinova, S. Onoda, A. H. MacDonald, and P. Ong, "Anomalous Hall Effect" , Rev. of Mod. Phy. 82, 1539 (2010).

Tentative phase diagram of AHE

Terra incognita

18


General AHE references(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-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-line

AHE in conduction electrons: P. Nozieres, C. Lewiner, J. Phys. France 34, 901 (1973): on-lineDirac+Pauli approach: A. Crépieux and P. Bruno, Phys. Rev. B 64, 014416 (2001): on-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-line

M(r) induced AHE: P. Bruno, al Phys. Rev. Lett. 93, 096806 (2004): on-lineAHE Fermi liquid properties: F. D. M. Haldane, Phys. Rev. Lett. 93, 206602 (2004): on-lineKubo+Boltzmann: N.A. Sinitsyn et al, Phys. Rev. Lett. 97, 106804 (2006): on-line, Phys. Rev. B 72,

045346 (2005): on-lineWave-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)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-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): Nagaosa, Sinova, Onoda, Ong, MacDonald RMP 82, 1539 (2010).

19


Valenzuela et al Nature 06

Inverse SHE

Anomalous Hall effect: more than meets the eye

Wunderlich, Kaestner, Sinova, Jungwirth PRL 04

Kato et al Science 03

IntrinsicExtrinsic

V

Mesoscopic Spin Hall Effect

Intrinsic

Brune,Roth, Hankiewicz, Sinova, Molenkamp, et al

Nature Physics 2010

Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09,

Science 10

Spin-injection Hall Effect

Anomalous Hall Effect

I

_ FS

OFS

O

_ _majority

minority

V

Spin Hall Effect

I

_ FS

OFS

O

_ _

V

Topological Insulators

Kane and Mele PRL 05

20


Towards a realistic spin-based non-magnetic FET device

[001][100]

[010]

Can we achieve direct spin polarization injection, detection, and manipulation by electrical means in an all paramagnetic semiconductor system?

Long standing paradigm: Datta-Das FET (1990)Exploiting the large Rashba spin-orbit coupling in InAs

Electrons are confined in the z-direction in the first quantum state of the asymmetric trap and free to move in the x-y plane.

gate

ky [010]

kx [100]

Rashba effective magnetic field

⊗⊗⊗

21


Can we achieve direct spin polarization injection, detection, and manipulation by electrical means in an all paramagnetic semiconductor system?

Long standing paradigm: Datta-Das FET (1990)Exploiting the large Rashba spin-orbit coupling in InAs

Towards a realistic spin-based non-magnetic FET device

High resistance “0”Low resistance “1”

BUT lMF << LS-D at room temperature

22


Dephasing of the spin through the Dyakonov-Perel mechanism

LSD ~ μm

lMF ~ 10 nm

23


Problem: Rashba SO coupling in the Datta-Das SFET is used for manipulation of spin (precession) BUT it dephases the spin too quickly (DP mechanism).

New paradigm using SO coupling: SO not so bad for dephasing

1) Can we use SO coupling to manipulate spin AND increase spin-coherence?

• Can we detect the spin in a non-destructive way electrically?

24


Spin-dynamics in 2D electron gas with Rashba and Dresselhauss spin-orbit coupling

a 2DEG is well described by the effective Hamiltonian:

α > 0, β = 0[110]

[110]_

ky [010]

kx [100]

Rashba: from the asymmetry of the confinement in the z-direction

α = 0, β < 0 [110]

[110]_

ky [010]

kx [100]

Dresselhauss: from the broken inversion symmetry of the material, a bulk property


25


Spin-dynamics in 2D electron gas with Rashba and Dresselhauss spin-orbit coupling

Something interesting occurs when

• spin along the [110] direction is conserved• long lived precessing spin wave for spin perpendicular to [110]

The nesting property of the Fermi surface:

Bernevig et al PRL 06, Weber et al. PRL 07

Schliemann et al PRL 04

26


Effects of Rashba and Dresselhaus SO coupling

α= -β

[110]

[110]_

ky [010]

kx [100]

α > 0, β = 0[110]

[110]_

ky [010]

kx [100]

α = 0, β < 0[110]

[110]_

ky [010]

kx [100]

27


Spin-dynamics in 2D systems with Rashba and Dresselhauss SO coupling

For the same distance traveled along [1-10], the spin precesses by exactly the same angle.

[110]

[110]_

[110]_

28


Persistent state spin helix verified by pump-probe experiments

Similar wafer parameters to ours

29


Spin-helix state when α ≠ β

Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09

For Rashba or Dresselhaus by themselves NO oscillations are present; only and over damped solution exists; i.e. the spin-orbit

coupling destroys the phase coherence.

There must be TWO competing spin-orbit interactions for the spin

to survive!!!

30


Problem: Rashba SO coupling in the Datta-Das SFET is used for manipulation of spin (precession) BUT it dephases the spin too quickly (DP mechanism).

New paradigm using SO coupling: SO not so bad for dephasing



Use the persistent spin-Helix state and control of SO coupling strength(Bernevig et al 06, Weber et al 07, Wünderlich et al 09)

✓

31


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 01Nozier 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

AHE contribution to Spin-injection Hall effect in a 2D gas


32


Local spin-polarization → calculation of AHE signal

Weak SO coupling regime → extrinsic skew-scattering term is dominant

Lower bound estimate

Spin-injection Hall effect: theoretical expectations



Use the persistent spin-Helix state and control of SO coupling strength

Use AHE to measure injected current polarization electrically

✓

✓

33


Spin-injection Hall device measurements

VL

SIHE ↔ Anomalous Hall


34


T = 250K

Further experimental tests of the observed SIHE

35


VH2

I

VbVH1

x

VH2

VbVH1

x

(a)

(b)

SiHE: new results

SiHE

inverse SHE

Spin Hall effect transitor:Wunderlich, Irvine, Sinova, Jungwirth,

Science 330, 1801 (2010)

36


VH

Vg

I

Vb

VH

VgVb

x

Δx=1μm

σ+

SiHE transistor

Spin Hall effect transitor:Wunderlich, Irvine, Sinova, Jungwirth,

Science 330, 1801 (2010)

37


σ-

σ-σ-

σ-

H1 H2

0+0.1

-0.1

Vg2 [V]

RH1

[Ω]

0 12 RH2

[Ω]0 66 3

SHE transistor AND gate

38


Summary of spin-injection Hall effect device

AHE: metallic theory fairly well understood, insulating regime remains a challenge

New challenge: topological AHE + AHE

spin-FET: spin-injection Hall effect

Basic studies of spin-charge dynamics and Hall effect in non-magnetic systems with SO coupling Spin-photovoltaic cell: solid state polarimeter on a semiconductor chip requiring no magnetic elements, external magnetic field, or bias

SIHE can be tuned electrically by external gate to create a spin-FET that operates in the diffusive regime

39


Spin in Cold Atoms and CM systems

3 day Winter School and 2 day Workshop

40


41


n, q

n’≠n, q

Defining intrinsic anomalous Hall effect in materials where AHE is dominated by scattering-independent mechanisms


The family of spintronics Hall effects

SHE-1

B=0spin current gives

charge current

Electrical detection

AHEB=0

polarized charge current gives

charge-spin currentElectrical detection

SHEB=0charge current gives

spin current

Optical detection

SIHEB=0

Optical injected polarized current

gives charge current

Electrical detection


Nagaosa, Sinova, Onoda, Ong, MacDonald, RMP 10

Phase diagram of AHE

2

I

FSO

FSO

_

majority

minority

V

I

_ FSO

FSO

_ __

V=0non-magnetic

Ispin

FSO

FSO _

V

non-magnetic

IFSO

V

MzMz=0

Mz=0

non-magnetic

I=0

Mz=0

magnetic

optical detection

AHE

SHE SHE-1

SIHE

FSO

45



Phonon-assisted hopping between localized states

The Hamiltonian describing localized states and e-ph interaction (Holstein 1961)

with

Localization:phonon-assisted

hopping

Electric current between two sites:

Ri

RjRk

: direct conductance due to two-site hopping. Responsible for longitudinal conductance.

: off-diagonal conductance due to three-site hopping.

i j k

phonon


Transverse charge transport: Three-site hopping (Holstein, 1961)

Interference term

Hall transition rate

Typical two real phonon process:

Typical one real (two virtual) phonon process:

Direct hoppingIndirect hopping

Geometric phase: break chirality

m: the number of real phonons included in the whole transition.


Macroscopic anomalous Hall conductivity/resistivity via percolation theory

In the thermodynamic limit, we get the AHC:

Unfortunately, too complicated to get an analytical calculation!!!

Critical path/cluster appears when:


The lower and upper limits are given by (note the differences in the constrain)

For the Mott hopping, we find

Physically, the upper limit (γ=1.38) corresponds to the situation that most triads in the system are equilateral triangles (this is actually a regular distribution), while for the lower limit (γ=1.76) the geometry of triads is randomly distributed. Since the impurity sites are randomly distributed, we expect the realistic result is usually close to the lower limit in most systems.

Xiong-Jun Liu, Sinova, unpub. 2010

Anomalous Hall conductivity via percolation theory

Documents

Research fueled by: Spintronics Tutorial Session March 20 th, 2011 APS March Meeting JAIRO SINOVA Texas A&M University Institute of Physics ASCR Spin Hall