Spin Hall EffectMichel Dyakonov

Université Montpellier II, CNRS, France

OUTLINE

Introduction. History. Difference between SHE and HE

Spin current

Coupling between charge and spin currents

Physical consequences

Experimental results

Spin-dependent effects in scattering

Swapping of spin currents

Conclusions

History

• 1879 Hall Effect

• 1881 Anomalous Hall EffectIn ferromagnets:

Explanation: 1951, J. Smit; 1954, R. Karplus, J.M. Luttinger

BRHxy =ρ

MRBR AHHxy πρ 4⋅+=

HAH RR >

Edwin Hall

Spin-orbit interaction

If an observer is moving with a velocity v in an electric field E,

he sees a magnetic field B = (1/c) (v x E), where c is the speed of light.

Ev

BElectron in an atom

So, the electron spin is subject to an effective magnetic field B and has an energy +μB or -μB, depending on the direction of the electron spin.

Consequences: * fine structure of atomic spectra, * values of g-factors different from 2, * spin asymmetry in scattering (Mott effect)

Spin-orbit interaction is strongly enhanced for atoms with large Z !!

Spin-orbit interaction

In solids:

Band spin splitting, leading to spin relaxation (DP mechanism)

Effective g-factors for electrons and holes (Zeeman splitting)

Effective spin-orbit interaction

This interaction makes scattering spin-dependent

Sp )( ⋅∇×= VAH so

Magnus effect and Mott scattering

Magnus effect

A spinning tennis ball deviates from a straight path to the right or to the left, depending on the sense of rotation

Schematic illustration of spin-dependent asymmetry in scattering

Skew scattering or the Mott effect (1929)Sir Neville Mott

Consequences of spin asymmetry: anomalous Hall effect

j = e β E × P

R. Karplus and J.M. Luttinger (1954)

- intrinsic mechanism

Electron spin polarization vector P = 2 S plays the role of magnetic field

Joaquin Mazdak Luttinger

Consequences of spin asymmetry: generation of spin current

Unpolarized beam

Spin current is generated by electron drift

Dyakonov, Perel (1971)

The notion of spin current was introduced for the first time

Mikhail Dyakonov and Vladimir Perel

Leningrad, 1976

Current-induced spin accumulation (Spin Hall Effect)

Electric current leads to accumulation of oppositely directed spins at the boundaries

Hall Effect and Spin Hall Effect

HE SHE

Spin accumulation in a cylindrical wire

The spins wind around the wire

Current

Spin and charge currents

xz qxz - z component of spin is

flowing in the x direction

Generally: qij

Above, the spin current is accompanied by a charge current qx (electric current j = – q/e )

x

z

Now there is a pure spin current qxz

The charge current is zero: q=0

Spin current, like charge current, change sign under space inversion

Unlike charge current, the spin current does not change sign under time inversion

Charge flow density: q = - j/e ( j – electric current density)

Spin polarization flow density tensor: qij (flow of the j-component of spin in direction i )

Spin polarization density: P = 2S, where S is the spin density vector

Without taking account of the spin-orbit interaction:

– normal expression with drift and diffusion

– similar expression (spins carried by drift and diffusion)

i

jjiij

iii

xP

DPEq

xnDnEq

∂

∂−−=

∂∂

−−=

μ

μ

)0(

)0(

Spin and charge currents

Spin-orbit interaction couples the two currents

Coupling of spin and charge currents(phenomenology in isotropic material with inversion symmetry)

γ changes sign under time inversion! => Spin current is dissipationless!

Here γ is a dimensionless coupling parameter proportional to the spin-orbit interaction

εijk is the unit antisymmetric tensor

εxyz= εzxy= εyzx= –εyxz= –εyxz –εyxz=1

qi = qi(0) + γ εijk qjk

(0)

qij = qij(0) – γ εijk qk

(0)

M.I. Dyakonov, PRL, 99, 126601 (2007)

Phenomenological equations(Dyakonov- Perel, 1971)

Inverse Spin Hall Effect

Spin Hall Effect

Diffusive counterpart of SHE

PPEEj curl / δβμ +×+∇+= nDne

)(k

kijkj

jjiij x

nnExP

DPEq∂∂

++∂

∂−−= δβεμ

Anomalous Hall Effect

Dγδγμβ == ,

First observation of the Inverse Spin Hall Effect ( j ~ curl P )

Circularly polarized light creats spin polarization P,however curl P = 0

By applying a magnetic field parallel to the surface, one creates the y component of P. This makes a non-zero curl P and hence an electric current in the x direction.

Proposal: N.S. Averkiev and M.I. Dyakonov, Sov. Phys. Semicond. 17, 393 (1983)

Experiment: A.A. Bakun, B.P. Zakharchenya, A.A. Rogachev, M.N. Tkachuk, and V.G. Fleisher, Sov. Phys. JETP Lett. 40, 1293 (1984)

Spin accumulation near the boundaries (edges)(Dyakonov - Perel, 1971)

0 =+∂∂

+∂∂

s

j

i

ijj Pxq

tP

τContinuity equation for spin density P

Boundary conditions

at y = 0: zyxjqyj ,, 0, ==

Solution:

Pz(y) = Pz(0) exp (– y/Ls), Pz(0) = – βnE Ls /D, Px = Py = 0,

Most optimistic estimate (Si):

Degree of polarization in the spin layer:

213 )0( Degree

/

p

s

F

dz

ττ

vv

nP

⎟⎟⎠

⎞⎜⎜⎝

⎛γ==

Fd /vv ~ Degree

x

yj j

x

y

First observations of the Spin Hall effect

Y.K. Kato, R.C. Myers, A.C. Gossard, and D.D. Awschalom, Science 306, 1910 (2004)

Experiment

Two-dimensional gas of holes in AlGaAs/GaAs heterostructure(optical registration)

J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, PRL, 94, 047204 (2005)

Polarization reversal when the current

direction is changed

Polarization at opposite edges of the sample

First observations of the Spin Hall effect

Science (2004)

Physics Today (2005)

Search and DiscoverySearch and Discovery

Schematics of electron scattering off a charged center

The electron spin sees an effective magnetic field EvB

1. The electron spin rotates → Elliott-Yafet spin relaxation

2. Scattering angle depends on spin → anomalous Hall effect, spin Hall effect

3. Spin rotation is correlated with scattering → swapping of spin currents

rrr×∝

Swapping of spin currents

Spin rotation during scattering is correlated with the direction of scattering!

Consequence

before scattering: spin current qyx

after scattering: a spin current qxy appears!

Scattering results in swapping qyx → qxy

x

y

Swapping of spin currents

Additional term in the second basic equation:

qi = qi(0) + γεijk qjk

(0)

qij = qij(0) – γεijk qk

(0)

j

Spins perpendicular to current

j

Spins parallel to current

In both cases there should be a current-induced rotation of spins near the surface!

Noticed in Dyakonov-Perel, 1971, but not understood at that time

+ κ[qji(0) – δij qkk

(0)]

Swapping of spin currents, suggested experiment

M.B. Lifshits, M.I. Dyakonov PRL, October 30 (2009)

« After all, this is a problem, which falls into the same category of problems related to the orientation of spin. » (from the report of our Referee E)

Three spin-related scattering cross-sections(introduced by Mott and Massey, 1965)

σnnI )( 21×+= BAFThe scattering amplitude with spin-orbit interaction:where n1 and n2 are unit vectors along pi and pf

– scattering cross-section (momentum relaxation)

– cross-section for spin asymmetry (spin-charge current coupling)

– cross-section for swapping of spin currents

– cross-section for spin rotation (EY spin relaxation)

221 ~ BA +σ

)Re(~ *2 ABσ

)Im(~ *3 ABσ

24 ~ Bσ

In the Born approximation the phase difference between A and B is π/2. So, skew scattering does not exist (σ2 = 0). Swapping is more robust (it exists already in the Born approximation)

Swapping constant in the Born approximation

qij = qij(0) – γεijk qk

(0) + κ[qji(0) – δij qkk

(0)]

σk ⋅∇×λ= )( UH SO

In the Born approximation: 22 kκ λ=

Strong spin-orbit splitting: Δ >> Eg , then

Swapping constant:

gmE4

2h=λ

gEE

=κ

Maria Lifshits and Michel Dyakonov, PRL October 30 (2009)

Kinetic equation for the spin density matrix ρ(p,t)

{ } )]Tr([1 ][ , 2

1 2 ρ−−⋅+ρ⋅+−=ρ ρ

τ,ρbaρ

τdtd

spσLσLL

This equation was derived by Dyakonov and Khaetskii (1984)

For the special case of small-angle scattering:

σ1 σ3σ2 σ4

where ppL ∂

∂×−= i is the angular momentum operator in momentum space

Maria Lifshits and Michel Dyakonov (to be published)

Conclusions

The number of theoretical articles exceeds the number of experimental ones by two orders of magnitude!

The direct and inverse spin Hall effects have been observed:

In various semiconductors (3D and 2D)

In metals (Al, Au, Pt, etc)

At cryogenic temperatures, as well as at room temperature

Swapping of spin currents is a related and previously unknown effect worth exploration

It is hard to predict whether SHE will have any practical applications, or it will belong only to fundamental research as a tool for studying spin interactions in solids

THE END

SHE (spin accumulation at the lateral boundaries of a current-carrying sample) is a new transport phenomenon predicted by DP in 1971 and observed for the first time in 2004