Spintronics: How spin can act on charge carriers and vice versa

Tomas Jungwirth

University of Nottingham

Institute of Physics Prague

“Mott“ non-relativistic two-spin-channel model of ferromagnets

“Dirac“ relativistic spin-orbit coupling

I

I I

I

Mott, 1936

Dirac, 1928

Two paradigms for spintronics

SHE & STT switching SOT switching

-We see (anti)damping-like torque

-SOT is field-like so we exclude it

- non-relativistic STT in metals is dominated by the (anti)damping torque

-We also see (anti)damping-like torque

-SOT is field-like but maybe there is some (anti)damping-like SOT as well

Ralph, Buhrman,et al., Science ‘12 Miron et al., Nature ‘11

Ohmic “Dirac“ device: AMR

Magnetization-orientation-dependent scattering

Kelvin, 1857

Spin-orbit coupling

Spin-orbit coupling

Extraordinary magnetoresistance: AMR, AHE, SHE, SOT.....

B

V

I

_+ + + + + + + + + + + + +

_ _ _ _ _ _ _ _ _ _ FL

Ordinary magnetoresistance:response to external magnetic field Acting via classical Lorentz force

Extraordinary magnetoresistance:response to internal quantum-relativistic spin-orbit field

ordinary Hall effect 1879 I

_ FSO__

Vanomalous Hall effect 1881

anisotropic magnetoresistance

M

Lord Kelvin 1857

)(21

jiijsAMR

)(21

jiijAAHE

)()( MM jiij

)()2( ,,0 jkn

iknd

d

njiji EgvkdeEj

Linear response: g linear in Ej

nknknknknd

d

kn

knkn

knkn ffWkdE

EfvEe

t

f

dt

df)(

)2(

)(,,,,,

,

,0

,0,,

)( ,0,, knknkn Effg

Classical Boltzmann equation

Non-equilibrium distribution function

Steady-state current in linear response to applied electric field

k

E kn

,

tk

Steady-state solution for elastic (impurity) scattering

Constant quasi-particle relaxation time solution

Steady-state solution for elastic (impurity) scattering

g(i,k)=

if

Transport relaxation time solution: back-scattering dominates

Steady-state solution for elastic (impurity) scattering

g(i,k)=

is isotropic: depends on | - ’| if

No relaxation time solution

Steady-state solution for elastic (impurity) scattering

is anisotropic: depends on k, k’ if

AMR in Rashba 2D system

Rashba Hamiltonian Eigenspinors

)(21

jiijsAMR

anisotropic

AMR in Rashba 2D system

isotropic

.)( constererd rkirki

QM: 1st order Born approximation

)(1 rV

)(rMV xx

M

Heuristic picture from back-scattering matrix elements

Rashba SOI

current

Back-scattering high resistivity

AMR in Rashba 2D system

M

M

)(rMV xx )(rMV yy

Rashba SOI

No back-scattering low resistivity

Mott, N. F. Proc. R. Soc. Lond. A 1929 Dyakonov and Perel 1971

Spin Hall effect in PMs

Electron spin-dependent scattering off Coulomb field of heavy atoms due to spin-orbit coupling

Polarimetry of high-energy electron beams in accelerators

Electron spin-dependent scattering off Coulomb field of dopands in a semiconductor due to spin-orbit coupling

jc

Anomalous Hall effect in FMs 1881

Polarimetry of electrons in FMs

Kato, Awschalom, et al., Science‘04Wunderlich, Kaestner, Sinova, TJ, PRL‘05

jc js

Hirsch PRL‘99

Proposal for electrical spin injection by the spin Hall effect and electrical detection by the inverse spin Hall effect

jc js

Proposal for electrical spin injection by the spin Hall effect and electrical detection by the inverse spin Hall effect

Hirsch PNAS‘05

- index

Theoretical proposal of intrinsic spin Hall effect

FM (Ga,Mn)As Non-magnetic GaAs

TJ, Niu, MacDonald, PRL’02 Murakami, Nagaosa, & S.-C. Zhang, Science’03Proposed detection by polarized electroluminescence

Sinova, TJ, MacDonald, et al. PRL’04Proposed detection by magneto-optical Kerr effect

Intrinsic anoumalous Hall effect in (Ga,Mn)As

Magneto-optical Kerr microscopy Edge polarized electro-luminescence

Extrinsic SHE Kato, Awschalom, et al., Science‘04

Intrinsic SHE Wunderlich, Kaestner, Sinova, TJ, PRL‘05

Optically generated spin current Optically detected charge accummulation due to iSHE

Zhao et al., PRL‘06

fs pump-and-probe: iSHE generated before first scattering in the intrinsic GaAs intrinsic iSHE

Werake et al., PRL‘11

AHE and SHE

)(21

jiijAAHE

AHE and SHE

Skew scattering SHE

Mott (skew) scattering SHE

jiijll )2('

jiija

ll )3(' SHE

AMR

Skew scattering AHE (SHE)

)3('a

ll : not constant, not isotropic, not even symmetric no relaxation time solution

Approximation:

Skew scattering AHE (SHE)

Spin orbit torque

M

Ie

Field-like SOT

Compare with AMR or skew-scattering SHE

)()2( ,,0 jkn

iknd

d

ni Egvkdej

)()2( ,,0 jkn

iknd

d

ni Egkds

s

E=Ex x

Field-like SOT

s

E=Ex x

isotropic

(r)

.)( constererd rkirki

)()2( ,,0 jkn

iknd

d

ni Egkds

Field-like SOT

isotropic

(r)

.)( constererd rkirki

g(i,k)=

)()2( ,,0 jkn

iknd

d

ni Egkds

Field-like SOT

s

E=Ex x

sMJdtMd ex

yEmes xtr ˆ21

3

MJH exex

Rex HH

Intrinsic spin Hall effect in PMs

FM (Ga,Mn)As Non-magnetic GaAs

TJ, Niu, MacDonald, PRL’02 Murakami, Nagaosa, & S.-C. Zhang, Science’03Sinova, TJ, MacDonald, et al. PRL’04

Intrinsic anoumalous Hall effect in FMs

Werake et al., PRL‘11Wunderlich, Kaestner, Sinova, TJ, PRL‘05

Boltzmann theory : non-equilibrium distribution function and equilibrium states

Linear response I.

pAmce

mpA

cep

mˆ

2ˆ

)ˆ(21 2

2

ccevEietU ti .ˆ)(ˆ

ti

ll

ti

ll

til

ll eelvElli

eelt

''

|ˆ|''||)(|

mp

mpr

iHr

iv

ˆ]

2ˆ

,ˆ[1],ˆ[1ˆ2

)ˆ(2ˆ 2

rVm

pH tA

ceE ti

1 tie

iEcA

Perturbation theory: equilibrium distribution function and non-equilibrium states

Linear response II.

tA

ceE ti

1

)]ˆ(ˆ

[2ˆ

)]ˆ(ˆ[)ˆ(21 2

2 zmpA

ce

mpA

cepzA

cep

m

ti

ll

ti

ll

til

ll eelvElli

eelt

''

|ˆ|''||)(|

)ˆ(]ˆˆ[2ˆ 2

rVpzm

pH SO

)ˆ(ˆ

]]ˆˆ[2ˆ

(,ˆ[1],ˆ[1ˆ2

zmppz

mpr

iHr

iv

ccevEietU ti .ˆ)(ˆ

tieiEcA

Perturbation theory: equilibrium distribution function and non-equilibrium states

Linear response II.

)()(|ˆ|)( 0 llzyll

zy ftjtJ

ti

ll

ti

ll

til

ll eelvElli

eelt

''

|ˆ|''||)(|

Perturbation theory: equilibrium distribution function and non-equilibrium states

Intrinsic SHE (AHE)

xE

zyj

Linear response II.

0 0

pz

pxpy

pz

pxpy

E=Ex x

0, yeffB tEpB xxyeff ~~, xz Es ~

xz Es ~

00

)(1

2

2

dtds

dtsd

Bsdt

ds

zy

eqeffz

y

Heuristic picture: Bloch equations

Field-like SOT

Compare with AMR or skew-scattering SHE

)()2( ,,0 jkn

iknd

d

ni Egvkdej

)()2( ,,0 jkn

iknd

d

ni Egkds

s

E=Ex x

Intrinsic antidamping SOT from linear response II.

Compare with intrinsic SHE

0 0

0 0

pz

pxpy

pz

pxpy

pz

pxpy

pz

pxpy

Intrinsic SHE: transverse spin current

Intrinsic SOT: spin polarization

Hex=0

Hex >> HR tEpB xxyeff ~~,

tEpB xxyeff ~~, xz Es ~

xz Es ~

xz Es ~

xz Es ~

pz

pxpy

pz

Intrinsic SHE: transverse spin current

Intrinsic SOT: spin polarization

tEpB xxyeff ~~, xz Es ~

xz Es ~

pxpy

tEpB xxyeff ~~,

xz Es ~

xz Es ~

Mxsd

z eEMJ

ss cos2 22

pxpz eEp

ss

sin

2 22

2

,

dtdB yeff /,

2)( equileffB

dtdB yeff /,

2)( equileffB

pz

pxpypxpy

tEpB xxyeff ~~, xM ˆ||

pz

pxpypxpy

tEpB xxyeff ~~,

)]ˆ([~ MzEMsMJdtMd ex

Intrinsic SOT is antidamping-like

yM ˆ||

0zs

0zs

SHE & STT switching SOT switching

-We see (anti)damping-like torque

-SOT is field-like so we exclude it

- non-relativistic STT in metals is dominated by the (anti)damping torque

-We also see (anti)damping-like torque

-SOT is field-like but maybe there is some (anti)damping-like SOT as well and maybe we found it intrinsic SOT analogous to intrinsic SHE

Ralph, Buhrman,et al., Science ‘12 Miron et al., Nature ‘11