manipulation of spin current & spin hall effects in ... · 0 2.5 5-5-2.5 0 2.5 5 0 2.5 5-5-2.5...

MANIPULATION OF SPIN CURRENT & MANIPULATION OF SPIN CURRENT & SPIN HALL EFFECTS SPIN HALL EFFECTS

IN METALLIC SYSTEMSIN METALLIC SYSTEMS

Institute for Solid State Physics, University of Tokyo

Takashi Kimura

Y. Otani

1. Nonlocal spin injectionPlanar F/N hybrid structureElectrical detection of spin accumulationSpin absorption effect

2. Spin Hall effectElectrical detection of SHE by spin absorptionPossible origins for SHESHEs for various transition metals

ContentsContents

Mainly two terminal structure

Difficult to makemulti-terminal devices

Planar structures

Easily expand to multi-terminal and functional devices

Advantage of planar Advantage of planar spintronic spintronic devicesdevices

Detailed study of spin current diffusion in F/N hybrid structures

Conventional

Development of novel spintronic devices.

MTJLEAD

MRAM

Optimizing structures for efficient operation

Spin accumulation

ε ε

εF

N↑ N↑N↓ N↓

Accumulated spins diffuse with spin-flip scattering.

Spin diffusion length

ε

xλ

2

2 S

V V VDt x τ

∂Δ ∂ Δ Δ= −

∂ ∂2

20 S

V VDx τ

∂ Δ Δ= −

∂

exp expx xV V Vλ λ+ −

⎛ ⎞ ⎛ ⎞Δ = − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

SDλ τ=

Electric currents are injected from F into N.

e Vμ μ μ↑ ↓Δ = −

= Δ

D: Diffusion constantτS : Spin life time

Spin injection from F to NSpin injection from F to N

0e VΔe VΔ

Diffusion equation General solution

F NIe

F NR R R= +

F N BR R R R= + + B F N, R R R

NF II ↑↑ = NF II ↓↓ =

NF ↓↓ = μμ NF ↑↑ = μμ

1. Conservation of spin current at the interface

2. Continuity of the electrochemical potential

van Son, Phys. Rev. Lett. (1987)

x

x

μ

BμΔ

Boundary resistance due to spin accumulationBoundary resistance due to spin accumulation

Resistance change is too small.

2

2(1 )SR

S Pλ

σ≡

−

2(1 )F N

BN F F N

P IP S

λ λμλ σ λ σ

Δ =− +

Spin resistance

SF SNBB

SF SN

PR RRI R RμΔ

≡ =+

V

λ : Spin diffusion lengthS : Cross sectionP : Spin polarization

Detection of spin accumulationDetection of spin accumulation

ε εε

εF εF

D↑ D↑D↑D↓ D↓D↓

ε εε

εF

D↑ D↑D↑D↓ D↓D↓

Two Fs are in parallel

Small spin accumulation in NSmooth current flow Low resistance

I

V

HAg

100 nmNiFe (Py)

NiFe

Large spin accumulation in NHigh resistance

Two Fs are in anti-parallel

-1000 -500 0 500 10000.38

0.39

0.4

Magnetic field (Oe)

R(Ω

) 18 mΩ

Magnetoresistance due to spin accumulationMagnetoresistance due to spin accumulationT. Kimura et al. Appl. Phys. Lett 85, 3501 (2004)

BAP BP2 18 meI

μ μΔ − Δ= Ω

Spin-polarized current

I

Ferromagnet

Nonmagnet

Nonlocal spin injection and pure spin currentNonlocal spin injection and pure spin current

x

μ

μ↑

μ↓

Jx

μ↑↑

∂∝

∂

Jx

μ↓↓

∂∝

∂

Driving force for spins is the diffusion into the equilibrium state.

0J J↑ ↓+ =

Without charge flow

0J J↑ ↓− ≠

M. Johnson, B. J. van Wees

Only spin current

NiFe1

I

H

V+

V-

NiFe2

Ag

100 nm

Nonlocal spin valve measurement Nonlocal spin valve measurement

V

NiFe1

Ag

NiFe2

I

-1000 -500 0 500 1000-10

-5

0

5

10

I

Nonlocal Nonlocal spin valve measurement spin valve measurement

Magnetic field (Oe)

V/I

(mΩ

)

ΔRS

Voltage does not include any back ground signal.

Sensitive detection of spin information

0Pμ μΔ = Δ

Ag

NiFe2

Δμ

NiFe1

I

H

V+

V-

NiFe2

Ag

100 nm

Δμ0

700 nmλ =

3 μmλ =

Spin diffusion length of Ag wireSpin diffusion length of Ag wire

PI = 0.17 @ RT 0.26 @ 77K

Py1 Py2

Ag

d

T. Kimura & Y. Otani. Phys. Rev. Lett. 99, 196604 (2007)

Spin current absorption into Spin current absorption into Py Py wire wire

V

I

H600 nm

V

I

H 550 nm

Cu

PyPyPy

Without middle wire

With middle wire

Py Py

Cu

0.2 mΩ

Magnetic field (Oe)

0.05 mΩ

V/I

(mΩ

)

Magnetic field (Oe)

-800 -400 0 400 800-0.4

-0.2

0

0.2

-600 -300 0 300 600-0.4

-0.2

0

0.2

Spin current absorption into Spin current absorption into Py Py wire wire

V

I

H600 nm

V

I

H 550 nm

Cu

PyPyPy

Without middle wire

With middle wireV

/I(m

Ω)

RTPy Py

Cu

RT

Drastic reduction of the spin signal due to the middle Py insertion

T. Kimura et al. APL (2004)

Ie

Is Is

2

2 2

1x

μμλ

∂ ΔΔ =

∂

Ie

Is Is

Spin accumulation simply expressed by the exponential decay.

exp( / ) exp( / )x xμ μ λ μ λ− +Δ = − +General solution

exp( / )xμ μ λ−Δ = −

Taking into account the spin diffusion into additional contact

exp( / ) exp( / )x xμ μ λ μ λ− +Δ = − −

x x

(x > 0)

In single F/N junction With additional contactμ

Influence of additional contact Influence of additional contact

μ

-800 -400 0 400 800

-1.0

-0.5

0

Magnitude of the spin signal strongly depends on the spin resistance of the middle insertion, but is not related to whether ferromagnet or nonmagnet.

Magnetic field (Oe)

ΔR

(mΩ

)

VI

Middle insertion

600 nm

@RT

2

2(1 )SR

S Pλ

σ≡

−

Spin signals with insertion for various materialsSpin signals with insertion for various materials

Without

Cu

Au

Ptλ : Spin diffusion lengthS : Cross sectionP : Spin polarization

Spin resistance

ΔRS

Spin resistances for several metalsSpin resistances for several metals

Material ρ (μΩcm) λ (nm) P RS (Ω)

Cu 2.1 500 0 1.25

Py 15.4 3 0.2 0.15

Au 5.24 60 0 0.31

Co 24 20 0.2 0.46

Pt 15.6 10 0 0.15 S=(100 nm)2

Spin resistance

2

2(1 )SR

S Pλ

σ≡

−

: Spin diffusion lengthS : Effective cross section for spin current

: Spin polarization

λ

P

T. Kimura et al. PRB (2005)

Single interface

Spin injector

Spin currents diffuse isotropically.

Spin sink effectSpin sink effect

Additional contact

Additional contact

When the spin resistance for the additional contact is small, spin currents are preferably absorbed into the contact.

Spin injector N CS SR R>>

Summary 01Summary 01

1. Nonlocal spin injection can generate pure spin current.

2. Spin accumulation in N can be detected by using a F voltage probe.

3. An additional ohmic contact with a small spin resistance strongly modify the distributions of the spin current and spin accumulation. (Spin sink effect)

1. Nonlocal spin injectionPlanar F/N hybrid structureElectrical detection of spin accumulationSpin absorption effect

2. Spin Hall effectElectrical detection of SHE by spin absorptionPossible origins for SHESHEs for various transition metals

ContentsContents

Spin Hall effect Spin Hall effect

Spin-orbit interaction

Ie

Is

↑ ↓∝ × ,F s k

Trajectories of electrons are affected by spin-orbit interaction.Scattering direction depends on the spin.

Transverse spin current is induced.

AHE in Ni film

1.5 nm

15 nm

60 nmBoth of the spin and charge are accumulated at the side edge.

Voltage generation due to charge accumulation

Anomalous Hall effect in ferromagnetAnomalous Hall effect in ferromagnet

MVH

Ieθ sinHVI

θ∝

I

H

V

Spin-orbit interaction

↑ ↓∝ × ,F s k

I

H

V

Spin orbit interaction induces spin-dependent scattering. No charge accumulation

Electrical detection is impossible ?

Anomalous (Spin) Hall effect in nonmagnetAnomalous (Spin) Hall effect in nonmagnet

Is

I

H

VS

Spin accumulation can be detected electrically by using ferromagnetic voltage probe.

Anomalous (Spin) Hall effect in nonmagnetAnomalous (Spin) Hall effect in nonmagnet

Is

Spin Hall effect & inverse spin Hall effect Spin Hall effect & inverse spin Hall effect

Novel way for generation & manipulation of spin current

Sc JSJ ×∝

Unpolarized charge current induces transverse spin current.

cJ

SJ

Spintronics without ferromagnets

- - - - - -

∝ ×S cJ S J

Spin current induces transverse charge current.

Inverse SHE (Reciprocal SHE)SJ

cJ

Direct SHE : Transformation from charge to spin currents

Transformation from spin to charge currents

Novel technique for manipulating spin current

Experimental study of Spin Hall effectExperimental study of Spin Hall effectOptical detection in semiconductor

Electrical detection using a Hall cross in metallic systems

T. Seki et al (FeMn/Au, Nature mat. 2008)

J. Wunderlich et al. Phys. Rev. Lett. (2005)

Y. K. Kato et al. Science (2004)

Valenzuela & Tinkham, Nature (2006)　CoFe/Al hybrid structure

Limited materials (λ > 100 nm)Weak SO interaction

Quantitative study of large SHE in materials with large SO interaction(Transition metals : Pt, Pd, ….)

E. Saitoh et al. APL (2006)　Using spin pumping

Qualitative analysis

qjSj

x

y

S q xyy x

xx

j jσσ

=

Spin current & spin accumulation with SO interaction Spin current & spin accumulation with SO interaction

Transverse spin current due to SHE.

Spin accumulation in x-z plane

w

Electron with z spin axis

z

0 100 200 300

-1

0

1

0 100 200 3000

0.5

1

1.5

y (nm)

qjSj

x

y

( )S q xy yyy x

xx

j je y

σ σδμ δμ

σ ↑ ↓

∂= + −

∂

, δμδμ

↑

↓

Syj

S qxy

y xyy

j jσσ

=


/ /

S q / / / /

1 11 e ey yw w

xyy w w w w

xx

e ej je e e e

λ λλ λ

λ λ λ λ

σσ

−−

− −

⎛ ⎞⎛ ⎞− −= − +⎜ ⎟⎜ ⎟⎜ ⎟− −⎝ ⎠⎝ ⎠

Using boundary conditions in steady state :

JSy = 0 @ y=0 & w

Pt wire with λ=10 nm, w = 300 nm

Transverse spin current due to SHE.

Spin accumulation in x-z plane

w

spin current due to spin accumulation

Electron with z spin axis

z

λ

Structure fabrication is very difficult.

Junction size is less than the spin diffusion length (< 10 nm).

Structure for electrical detection of SHE Structure for electrical detection of SHE

Spin accumulation appears only in the vicinity of the edges.

, δμδμ

↑

↓

Je V

FF

FFλ

( )S qxz zz

z xxx

j je z

σ σ δμ δμσ ↑ ↓

∂= + −

∂

0 2.5 5-0.2

-0.1

0

0.1

0.2

0 2.5 50

0.025

0.05

z (nm)

, δμ δμ↑ ↓

Syj

λ=10 nm, t = 5 nm

JSz = 0 @ z=0 & t

Surface spin accumulation in x-y plane can be detected by putting a ferromagnetic probe on the surface.

z

jqx


Spin accumulation in x-y plane

t

x

y

Electron with y spin axis

Using boundary conditions in steady state :

Influence of additional contactInfluence of additional contact

zjqx

y

( )S qxz zz

z xxx

j je z


∂= + −

∂

0 2.5 5-5

-2.5

0

2.5

5 0 2.5 5-5

-2.5

0

2.5

5 0 2.5 5-5

-2.5

0

2.5

5

Spin accumulation is suppressed by the Py contact.

Taking into account the absorption of the spin current into additional contact.

Without contact

With Py contact

With Cu contact

z (nm)

, δμ δμ↑ ↓

Influence of additional contactInfluence of additional contact

zjqx

Py

Cu

x

y

Cu PtS SR R>>Spin accumulation is not modified by the Cu contact so much.

Py PtS SR R≈

( )S qxz zz

z xxx

j je z


∂= + −

∂

Spin accumulation in Pt

Detection of spin accumulation in Detection of spin accumulation in xx--yy plane plane

Spin accumulation due to SHE is extracted by Cu wire without disturbing the spin distribution.

Spin accumulation in the Cu wire is measured by ferromagnetic probe.

, δμ δμ↑ ↓

μΔ

jqx

Js

V+Pt

Py

Cu

Cu 1 μm

y

V-

λ ≈

x

y

z

∝ ×S cJ S J

Js

Jc

S

jq

Jq’

jS jS

Detection of charge accumulation along Detection of charge accumulation along xx axis axis

Induced spin current is preferably absorbed into the Pt wire.

Injected spin current is transferred to the charge current along the Pt wire.

PtPy

Cu

Direction of the spin current in the Pt wire is perpendicular tothe junction (along z axis).

x

y

z

, δμ δμ↑ ↓

y

JcJs

S

Sc JSJ ×∝

s μ∝ ∇J

VC

IeM

H

VS

IeM

H

Direct SHE

Inverse SHE

Sample dimension

Pt : 80 nm wide, 4 nm thickPy : 1 μm wide, 30 nm thick Cu : 100 nm wide, 80 nm thick

Sample structure for electrical detection of SHESample structure for electrical detection of SHET. Kimura, Y. Otani et al. PRL (2007)

x

y

-2000 -1000 0 1000 2000

-0.05

0

0.05

VC

IeM

H

Magnetic field (Oe)Δ

Vc/

I(m

Ω)

Sweep direction

RT

B

A

Observation of inverse SHE

IeIs

Is

sIe

Is

IeIs

Is

sIe

Is

Direction of the spin axis of generating spin current is reversed by the magnetic field.

Sc JSJ

A B

×∝x

y

z

Jc

S

Js

-2000 -1000 0 1000 2000-0.1

-0.05

0

0.05

0.1


Vc/

I(m

Ω)

Sweep direction

10 K

B

A

Observation of inverse SHE

IeIs

Is

sIe

Is

IeIs

Is

sIe

Is

Direction of the spin axis of generating spin current is reversed by the magnetic field.

BA

VC

IeM

H

Sc JSJ ×∝x

y

z

JsJc

S

-2000 -1000 0 1000 2000-0.1

-0.05

0

0.05

0.1

VS

IeM

H


Vs/

I(m

Ω)

Sweep direction

10 K

Observation of direct SHEObservation of direct SHE

Py PtCu Py PtCu

A

B

A B

Positive voltage appears. Negative voltage appears.

∝ ×S cJ S J

Ie

Is

Is

s

IeIs

x

y

zJs

Jc

S

( ) ( )Py PyS

CPy Pt Cu Py Pt

Cu Cu

cosh sinh

P RII d dR R R R R

λ λ

≈⎛ ⎞ ⎛ ⎞

+ + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Estimation of spin Hall conductivity Estimation of spin Hall conductivity

2 CSHE SHE

S

Iw RI

σ σ⎛ ⎞

= Δ⎜ ⎟⎝ ⎠JSy = 0 @ y=0 & t

4SHE 2.4 10σ ≈ × (Ωm)-1 for Pt at RT

10 ∼104 times larger than previously reported values(Al at 4.1 K, GaAs at 20 K, ZnSe at RT)

Spin Hall angleSHEσα σ≡ 3

Pt 3.7 10α −= ×

Larger than previously reported values

Spin Hall conductivity

, SHE /c y y ZJ E eσ σ δμ= + ∇

Possible mechanism of SHEPossible mechanism of SHE

SHEρ ρ∝ 2SHEρ ρ∝

Intrinsic SHE in Pt

Band structure & Berry phase

The present Pt wire is polycrystalline. Extrinsic SHE

G. Y. Guo et al. PRL (2008)

Extrinsic SHE in Pt

J. Smit (1956) L. Berger (1970)

10 cmxxρ μ> ΩQuadratic term is dominant.

Side jump (?)

Relation between Relation between ρρSHESHE & & ρρ

2xy xx xxa bρ ρ ρ= +

31.0 10a −= ×42.4 10b −= × (μΩcm)-1

0 10 200

0.02

0.04

0.06

0.08

0.1

ρ t ( cm)μΩP

ρSH

E (cm

)μ

Ω

2 CSHE SHE

S

Iw RI

σ σ⎛ ⎞

= Δ⎜ ⎟⎝ ⎠

2SHE SHEρ ρ σ≈

Intrinsic contribution is also proportional to ρ2

-4000 -2000 0 2000 4000-0.1

-0.05

0

0.05

0.1

-4000 -2000 0 2000 4000-0.1

-0.05

0

0.05

0.1

Magnetic field (Oe)

Nb

ΔRSHE = 60 μΩ

ΔV

c/I(

mΩ

)

ΔRSHE = 80 μΩ

Pd

Magnetic field (Oe)

ΔV

c/I(

mΩ

)

-4000 -2000 0 2000 4000-0.1

-0.05

0

0.05

0.1

Mo

Magnetic field (Oe)

ΔV

c/I(

mΩ

)

Inverse Inverse SHEsSHEs for various materialsfor various materials

ΔRSHE = 120 μΩ

H

Py

Cu

I+V+ V-I-

Positive

Negative Negative

Changing the material of the middle insertion

SH conductivity for various materials SH conductivity for various materials

Material

Pt (10K)

Pd (10 K)

Au (10 K)

Cu (10 K)

σ (Ωm)-1

8.0 x 106

2.2 x 106

2.0 x 107

5.0 x 107

σSHE (Ωm)-1

3.3 x 104

1.1 x 103

2.0 x 104

2.0 x 103

σSHE /σ

4.1 x 10-3

0.5 x 10-3

1.0 x 10-3

0.4 x 10-3

Nb (10 K) 2.7 x 106 -2.0 x 103 -0.7 x 10-3

Mo (10 K) 2.8 x 106 -1.4 x 103 -0.5 x 10-3

5 6 7 8 9 10 11-2

0

2

4

Numerical calculation intrinsic SHE

Number of electron

Experimentally obtained Hall angle

σ SH

E/σ

10-3

The experimental results seem to reproduce the numerical results.

Origin of SHEOrigin of SHE

H. Tanaka et al. PRB (2008)

4d5d Pt

AuPd

MoNb

Summary 02Summary 02

1. Spin Hall effects in transition metals are efficiently detected by means of spin current absorption.

2. Pt was found to have a large SH conductivity and Hall angle.

3. Temperature dependence of SHE in Pt suggests the relationship “ρ ρSHE

2”.

4. SHEs for other metals shows suggest the intrinsic origin.

∝

d

ΔV0

In single F/N junction With additional contact

ΔV0’

'0 0V VΔ Δ

An additional contact with a small spin resistance is strongly modify the spin accumulation and spin current.

d

Influence of additional contact Influence of additional contact

Mainly two terminal structure

Difficult to makemulti-terminal devices

Planar structures

Advantage of planar spintronic devicesAdvantage of planar spintronic devices

Conventional

Development of novel functional spintronic devices.

MTJLEAD

MRAM

Easily expand to multi-terminal devices

Dual spin injection (T. Kimura, Y. Otani & P. Levy, PRL 2007)

Difficult to detect spin informations. Optimization of the structure

Detailed study of spin current diffusion in F/N hybrid structures

Taniyama et al. PRL 99

The magnetizations at the corner easily align with the transverse (y) direction.

Spin Hall voltage will be induced even at the remanent state. Bi-stable spin Hall signal will appear.

+-

Ie

+ -

Ie

Py L-shaped wirePd wire

Cu wire

H

-8000 -4000 0 4000 8000-0.02

-0.01

0

0.01

0.02

Magnetic field (Oe)

ΔR(m

Ω)

SHE with zigzag spin injectorSHE with zigzag spin injector

Sc JSJ ×∝

SJ : Perpendicular to the junction

S : should be parallel to the Cu wire.

Is

V

SpinSpin--currentcurrent--induced Hall effect in nonmagnetinduced Hall effect in nonmagnet

Spin current induces charge accumulation, which can be detected by the conventional voltmeter.

Spin current can be injected by spin current absorption.

IC

0 100 200 3000

0.1

0.2

T (K)

-2000 -1000 0 1000 2000

Magnetic field (Oe)

V/I

(mΩ

)

0.1 mΩΔR

SHE

(mΩ

)

Temperature dependence of inverse SHE for Pt wireTemperature dependence of inverse SHE for Pt wire

10 K

100 K

200 K

290 K

ΔRSHE

ΔRSHE increases as the temperature declines. This is because the spin diffusion length of the Cu wire increases with decreasing the temperature, leading to the increment of the injected spin current.

-1000 -500 0 500 10000.38

0.39

0.4

Magnetic field (Oe)

R(Ω

)

18 mΩ

Magnetoresistance due to spin accumulationMagnetoresistance due to spin accumulationT. Kimura et al. Appl. Phys. Lett 85, 3501 (2004)

RAg

RB(P)RB(P)

Py1 Py2Ag

RAg

RB(AP)RB(AP)

Py1 Py2AgB(AP) B(P) 18 mΩR R− =

μμ

-1000 -500 0 500 1000-10

-5

0

5

10

μ

I

V

Magnetic field (Oe)V

/I(m

Ω)

Py

1 exp x dμσ λ↑

↑

⎛ ⎞−∝ −⎜ ⎟⎜ ⎟

⎝ ⎠ Py

1 exp x dμσ λ↓

↓

⎛ ⎞−∝ − −⎜ ⎟⎜ ⎟

⎝ ⎠

σ σ↑ ↓>

Positive voltage appears


Charge neutral point shifts upward.

0I I↑ ↓+ =

Ag

Py2

eVP

Δμ

μ

I

V

Py

1 exp x dμσ λ↑

↑

⎛ ⎞−∝ −⎜ ⎟⎜ ⎟

⎝ ⎠ Py

1 exp x dμσ λ↓

↓

⎛ ⎞−∝ − −⎜ ⎟⎜ ⎟

⎝ ⎠

σ σ↑ ↓<

-1000 -500 0 500 1000-10

-5

0

5

10


/I(m

Ω)


Negative voltage appears

0I I↑ ↓+ =

Charge neutral point shifts downward.

Ag

Py2

eVAP

μ

I

V-1000 -500 0 500 1000-10

-5

0

5

10


/I(m

Ω)

ΔRS

Δμ

Cu

Py2


Voltage does not include any back ground signal.

Sensitive detection of spin information

Δμ0 0Pμ μΔ = Δ

x

y

μ

μ

Cu

Pt

IS

In Cu wire,

Spin currents flow along the Cu wire. SI x

In Pt wire, Spin currents are sucked into the Pt wire because of the short spin diffusion length. SI y

When S x ∝ ×C SI S I

Flow of spin currents in Cu & Pt wires

SHE will be induced.

Single interface

Spin injector

Spin currents diffuse isotropically.

Spin sink effectSpin sink effect

Additional contact

Additional contact

When the spin resistance for the additional contact is small, spin currents are preferably absorbed into the contact.

Spin injector N CS SR R>>

, SHE

, SHE N /c y y

s z z

J EJ e

σ σσ σ δμ

−⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟′ −∇⎝ ⎠⎝ ⎠⎝ ⎠

: Spin-Hall conductivity for the charge current

SHEσ

SHE'σ: Charge-Hall conductivity for the spin current

Basic transport formulation with SHE

Reciprocal relation between spin & charge currentsReciprocal relation between spin & charge currents

-2000 -1000 0 1000 2000-0.1

-0.05

0

0.05

0.1

Magnetic field (Oe)-2000 -1000 0 1000 2000-0.1

-0.05

0

0.05

0.1

Magnetic field (Oe)

ΔRISHEΔRDSHE

Py Py SHE 1DSHE 2

Cu CusinhS

S

R P dR R wσ

λσ− ⎛ ⎞Δ ≈ ⎜ ⎟

⎝ ⎠Py Py SHE 1

ISHE 2Cu Cu

sinhS

S

R P dR R wσ

λσ−

′ ⎛ ⎞Δ ≈ ⎜ ⎟⎝ ⎠

DSHE ISHER RΔ = ΔSHE SHE'σ σ=means

Experimental demonstration of Onsager reciprocal relation

: Spin-Hall conductivity for the charge current

SHEσ SHE'σ : Charge-Hall conductivity for the spin current

DSHE ISHE

manipulation of spin current & spin hall effects in ... · 0 2.5 5-5-2.5 0 2.5 5 0 2.5 5-5-2.5...

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