# Michel Dyakonov- Spin Hall Effect

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Spin Hall EffectMichel DyakonovUniversit 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

xy = R H B 1881 Anomalous Hall Effect In ferromagnets:

xy = R H B + R AH 4 MR AH > R H

Edwin Hall

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

Spin-orbit interaction

If an observer is moving with a velocity v in an electric field E, he sees a magnetic field Electron in an atom

B = (1/c) (v x E), where c is the speed of light.

B v EConsequences: * fine structure of atomic spectra, * values of g-factors different from 2,

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

* 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

H so = A( p V ) S

This interaction makes scattering spin-dependent

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=eEPR. Karplus and J.M. Luttinger (1954) - intrinsic mechanism

Joaquin Mazdak LuttingerElectron spin polarization vector P = 2 S plays the role of magnetic field

Consequences of spin asymmetry: generation of spin currentSpin current is generated by electron drift

Dyakonov, Perel (1971)

Unpolarized beam

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

Current

The spins wind around the wire

Spin and charge currentsz

x

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 ) z

Now there is a pure spin current qxzThe charge current is zero: q=0 x

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

Spin and charge currents

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:q q(0) i

n = nE i D xi = E i Pj D Pj xi

normal expression with drift and diffusion similar expression (spins carried by drift and diffusion)

(0) ij

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

Spin-orbit interaction couples the two currents

qi = qi(0) + ijk qjk (0) qij = qij(0) ijk qk(0)M.I. Dyakonov, PRL, 99, 126601 (2007)

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

ijk is the unit antisymmetric tensorxyz= zxy= yzx= yxz= yxz yxz=1

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

Phenomenological equations(Dyakonov- Perel, 1971)

Anomalous Hall Effect

Inverse Spin Hall Effect

j / e = n E + D n + E P + curl Pn + ijk ( nE k + q ij = E i Pj D ) x j x k Pj

Spin Hall Effect

= , = D

Diffusive counterpart of SHE

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

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.

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

Pj qij Pj + + =0 xi s tyy

Continuity equation for spin density

P

j x

j

Boundary conditions at y = 0:

q yj = 0, j = x, y, z

x Solution:

Pz(y) = Pz(0) exp ( y/Ls),

Pz(0) = nE Ls /D,

Px = Py = 0,

Degree of polarization in the spin layer:

Pz (0) vd = Degree = n vFDegree ~ v d /v F

3 s p

1/ 2

Most optimistic estimate (Si):

First observations of the Spin Hall effect

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

First observations of the Spin Hall effect

Experiment Two-dimensional gas of holes in AlGaAs/GaAs heterostructure (optical registration)Polarization reversal when the current direction is changed

Polarization at opposite edges of the sample

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

Science (2004)

Physics Today (2005)

Search and Discovery

Schematics of electron scattering off a charged center

r r r The electron spin sees an effective magnetic field B v E

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

Swapping of spin currents

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

Consequence qyx after scattering: a spin current qxy appears!before scattering: spin current y Scattering results in swapping

qyx qxy

x

Swapping of spin currents

Additional term in the second basic equation:

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

qi = qi(0) + ijk qjk (0) qij = qij(0) ijk qk(0) + [qji(0) ij qkk(0)]

j

j

Spins perpendicular to current

Spins parallel to current

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

Swapping of spin currents, suggested experiment 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)

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

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

The scattering amplitude with spin-orbit interaction: where n1 and n2 are unit vectors along pi and pf

F = A I + B ( n1n2)

1 ~ A + B2

2

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)

2 ~ Re(AB )*

3 ~ Im( AB )*

4 ~ B

2

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

Swapping constant in the Born approximation

qij = qij(0) ijk qk(0) + [qji(0) ij qkk(0)]H SO = ( k U )

In the Born approximation:

= 2k 2h2 = 4mE g

Strong spin-orbit splitting:

>> Eg ,

then

Swapping constant:

E = EgMaria Lifshits and Michel Dyakonov, PRL October 30 (2009)

Kinetic equation for the spin density matrix (p,t)This equation was derived by Dyakonov and Khaetskii (1984) For the special case of small-angle scattering:

d 1 1 2 = L + a {L , } + b [ L , ] [ Tr( )] dt 2 p s1 2 3 4

where

L = i p p

is the angular momentum operator in momentum space

Maria Lifshits and Michel Dyakonov (to be published)

ConclusionsSHE (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 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