Spintronics: How spin can act on charge carriers and vice versa
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Spintronics: How spin can act on charge carriers and vice versa
Tomas Jungwirth
University of Nottingham
Institute of Physics Prague
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“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
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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
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Ohmic “Dirac“ device: AMR
Magnetization-orientation-dependent scattering
Kelvin, 1857
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Spin-orbit coupling
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Spin-orbit coupling
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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
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)()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
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Steady-state solution for elastic (impurity) scattering
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Constant quasi-particle relaxation time solution
Steady-state solution for elastic (impurity) scattering
g(i,k)=
if
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Transport relaxation time solution: back-scattering dominates
Steady-state solution for elastic (impurity) scattering
g(i,k)=
is isotropic: depends on | - ’| if
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No relaxation time solution
Steady-state solution for elastic (impurity) scattering
is anisotropic: depends on k, k’ if
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AMR in Rashba 2D system
Rashba Hamiltonian Eigenspinors
)(21
jiijsAMR
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anisotropic
AMR in Rashba 2D system
isotropic
.)( constererd rkirki
QM: 1st order Born approximation
)(1 rV
)(rMV xx
M
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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
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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
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jc js
Hirsch PRL‘99
Proposal for electrical spin injection by the spin Hall effect and electrical detection by the inverse spin Hall effect
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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
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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
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Magneto-optical Kerr microscopy Edge polarized electro-luminescence
Extrinsic SHE Kato, Awschalom, et al., Science‘04
Intrinsic SHE Wunderlich, Kaestner, Sinova, TJ, PRL‘05
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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
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AHE and SHE
)(21
jiijAAHE
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AHE and SHE
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Skew scattering SHE
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Mott (skew) scattering SHE
jiijll )2('
jiija
ll )3(' SHE
AMR
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Skew scattering AHE (SHE)
)3('a
ll : not constant, not isotropic, not even symmetric no relaxation time solution
Approximation:
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Skew scattering AHE (SHE)
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Spin orbit torque
M
Ie
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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
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Field-like SOT
s
E=Ex x
isotropic
(r)
.)( constererd rkirki
)()2( ,,0 jkn
iknd
d
ni Egkds
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Field-like SOT
isotropic
(r)
.)( constererd rkirki
g(i,k)=
)()2( ,,0 jkn
iknd
d
ni Egkds
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Field-like SOT
s
E=Ex x
sMJdtMd ex
yEmes xtr ˆ21
3
MJH exex
Rex HH
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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
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Boltzmann theory : non-equilibrium distribution function and equilibrium states
Linear response I.
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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.
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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.
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)()(|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
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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
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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
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Intrinsic antidamping SOT from linear response II.
Compare with intrinsic SHE
0 0
0 0
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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 ~
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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
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pz
pxpypxpy
tEpB xxyeff ~~, xM ˆ||
pz
pxpypxpy
tEpB xxyeff ~~,
)]ˆ([~ MzEMsMJdtMd ex
Intrinsic SOT is antidamping-like
yM ˆ||
0zs
0zs
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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