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 0...
Transcript of 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 0...
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
=
Spin current & spin accumulation with SO interaction Spin current & spin accumulation with SO interaction
/ /
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 current & spin accumulation with SO interaction Spin current & spin accumulation with SO interaction
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
σ σ δμ δμσ ↑ ↓
∂= + −
∂
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
σ σ δμ δμσ ↑ ↓
∂= + −
∂
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
Magnetic field (Oe)Δ
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
Magnetic field (Oe)Δ
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
Nonlocal spin valve measurement Nonlocal spin valve measurement
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
Magnetic field (Oe)V
/I(m
Ω)
Nonlocal spin valve measurement Nonlocal spin valve measurement
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
Magnetic field (Oe)V
/I(m
Ω)
ΔRS
Δμ
Cu
Py2
Nonlocal spin valve measurement Nonlocal spin valve measurement
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