非慣性系におけるスピン輸送現象
松尾 衛 (中国科学院大学カブリ理論科学研究所)
in collaboration with：(Theory)
Y. Ohnuma, J. Ieda & S. Maekawa(Experiment)
H. Chudo, R. Takahashi, M. Ono, K. Harii,
Y. Ogata, M. Imai, S. Okayasu, & E. Saitoh (JAEA)
R. Iguchi (NIMS)
D. Kobayashi, Y. Nozaki (Keio Univ.)Ref.

松尾・齊藤・前川「非慣性系のスピントロニクス」物理学会誌（2017年９月）
MM et al., “Spin-mechatronics”, JPSJ 86, 011011 (2017)

MM et al., “Spin-mechatronics” Chap. 25 in Spin current 2nd ed.(Oxford)

What is electron?
Electronics
Charge[electricity]

Charge current = flow of charged spheres

Spin[magnetism]

Spin current = flow of spinning gears

Spintronics

fragile ⇒ controlled by nanotechnology
to utilize magnetism and rotation

robust ⇒ easy to control

Mechanical generation of spin current
Dirac eq. in non-inertial frame [spin connection]

H = βmc 2 + (cα − v) ⋅ (p + eA) + eA0 − S ⋅ ω(p + eA)2eλ
H=+ eϕ − S ⋅ γB − S ⋅ ω − S ⋅ p × (E + (ω × r) × B)
2mℏSurface acoustic wave

Rigid rotation

Fluid vorticity

NCuHg,GaInSn

PRL 106, 076601 (2011)
PRB(R)2017

P H Y S I PRB(R)2013
CAL REVIEW LETTERS
PRL2017Nat.Phys. 2016Pt

PHYSICAL REVIEW LETweek ending
PRL201118 FEBRUARY 2011

Effects of Mechanical Rotation on S
Sci.Rept. 2018

Mamoru Matsuo,1,2 Jun’ichi Ieda,2,3 Eiji Saitoh,2,3,4 aects of Mechanical Rotation on Spin Currents

“Spin-mechat=onics”
ChargeSpin

e-Charge current & mechanical motion

Spin current & mechanical motion

Observation of spin-current generation byPH
Y Nat.
SICA
L R2016)EVIEWPRL
106, 076601• Liquid metal
motionin Hg (2011)(R.Takahashi etal.,Phys.
• Surface acoustic wave in Cu (D.Kobayashi et al., PRL 2017
• Rigid rotation in Pt (A.Hirohata et al., Sci.Rept.2018)

LETT)

Effects of Mechanical Rotation on Sp

How to detect? Rotation at 10kHz
Rotation as gravity
0.4 million G !!
(@ 1 mm from rotation axis )rΩ = 1mm × ( 2π × 10 s2gravity on white dwarf star
0.1 million G66=4×10m/s~0.4×10G)

4 −1 2Rotation as magnetic field
Gyromagnetic ratio of electron:
Geomagnetism in Tokyo
1T~30GHz
46 ± 0.05 μT
10kHz→0.3μTe
B = Ω / γ e , γ e = = 1.76 × 1011 rad ⋅s −1 ⋅ T −1
mChallenge: How to use mechanical rotation to manipulate spins?

ObserYation of spin-rotation coupling・Ferromagnets: Barnettʼs original exp. (1915)

H Spin-rotation = −S ⋅ Ω

Theoretical predictions:・MM et al., PRL(2011), …
Spin-rotation coupling arise universally in rotating materials
・Paramagnetic states (Gd, Tb, Dy):
Ono et el., PRB(2015),
Ogata et al., APL(2017); JMMM(2017)

・Ferrimagnetic states

Imai et al., APL(2018)
・Nuclear spin:

Chudo et al., APEX(2014), JPSJ(2015)

Spin-current generation by SRC
・Liquid metal ﬂow: Takahashi et al, Nat.Phys.(2016)

・Surface acoustic wave: Kobayashi et al., PRL(2017)

Contents

Gyromagnetic effectSpin current generation by rigid, fluid, elastic motion

Mechanical generation of spin current
Dirac eq. in non-inertial frame [spin connection]

H = βmc 2 + (cα − v) ⋅ (p + eA) + eA0 − S ⋅ ω(p + eA)2eλH=+ eϕ − S ⋅ γB − S ⋅ ω − S ⋅ p × (E + (ω × r) × B)
2mℏ
Surface acoustic wave

Rigid rotation

Fluid vorticity

NCu

Hg,GaInSn

PRL 106, 076601 (2011)
PRB(R)2017

P H Y S I PRB(R)2013
CAL REVIEW LETTERS
PRL2017Nat.Phys. 2016Pt

PHYSICAL REVIEW LET
week ending
PRL201118 FEBRUARY 2011

Effects of Mechanical Rotation on S
Sci.Rept. 2018

Mamoru Matsuo,1,2 Jun’ichi Ieda,2,3 Eiji Saitoh,2,3,4 aects of Mechanical Rotation on Spin Currents

Spin cur=entSpin current↑↓zJs,x= Jc,x− Jc,xℏ
σ<Ji,s = Tr[ {σ, vk,i}Gkω,σ]∫ω,k2<G1,2,σ:= − iTrρψ1†ψ2<R (2)Gkω,σ= 2iImGkω
fkω,σSpin connection: Spin couples to space-time rotation
CarHan (1922)

CurYed spacetime w/ spin & torsion
0 ( x ')e µ0
e µ ( x)
3 ( x ')e µ
e µ ( x)3e1µ ( x )e2 µ ( x )e µ(1x ')x ')(µe2gµν ( x ) = ηab ea µ ( x ) ebν ( x )⎡ eµ ( x ) ~ gµν ( x ) ⎤⎣⎦()

L = ψ ⎡⎣ieµ aγ a pµ − ω µ ab Σ ab − m ⎤⎦ψ

spin connection:µω µ dx = e ⋅ deabab⎛ σc O ⎞!
Σ ab = ε abc ⎜⎟⎜⎝ O σ c ⎟⎠2

CarHan theor, (1922)
Gravit, w/ spin & torsion

Spin connection gives rise to “spin gauge field”
mechanical control of spin & spin current

Pauli-Schrödinger eq. in rotating Iame
Low energ, limit of Dirac eq. in rotating Iame12
H=( p + eA ) + eφ −L ⋅ Ω2mCoulomb
Coriolis& Lorentz
elect=ic/magKetic ﬁeld + rotation → charge− µ Bσ ⋅ ( B + ( m / e ) Ω )Spin-rotation
ZeemanmagKetic ﬁeld+ rotation → spinλ+ σ ⋅ ⎡⎣( p + eA ) × (−e) ( E + ( Ω × r ) × B ) ⎤⎦!

Spin Orbit Int.

NEW!

elect=ic ﬁeld / magKetic ﬁeld x rotation→spin cur=ent

B + 2 ( m / e) Ωe-e-E

Coulomb & Lorentz
Se-Zeeman

Coriolis

B + ( m / e) !

BarKeeEinstein-de Haas?

Mechanical Spin Hall Eﬀect due to rotation
Mechanical Spin-Orbit Int.
λσ ⋅ ⎡⎣( p + eA ) × (−e) ( E + ( Ω × r ) × B ) ⎤⎦!

deλ
r=v−σ × ( E + (Ω × r ) × B)
dt!
±Azimuthal±z

Spin cur=ent generated
in azimuthal direction

Erot RadialΩBe-erErot = ( Ω ⋅ B ) rNPt; B = 1T, Omega = 10kHz, r=0.1m→ Spin cur=ent 108A/m2
MM et al., PRL 106, 076601(2011)
Hirohata et al., Sci.Rept. 2018

Mechanical generation of spin current
Dirac eq. in non-inertial frame [spin connection]

H = βmc 2 + (cα − v) ⋅ (p + eA) + eA0 − S ⋅ ω(p + eA)2eλ
H=+ eϕ − S ⋅ γB − S ⋅ ω − S ⋅ p × (E + (ω × r) × B)
2mℏSurface acoustic wave

Rigid rotation

Fluid vorticity

NCu

Hg,GaInSn

PRL 106, 076601 (2011)
PRB(R)2017

P H Y S I PRB(R)2013
CAL REVIEW LETTERS
PRL2017Nat.Phys. 2016Pt

PHYSICAL REVIEW LET
week ending
PRL201118 FEBRUARY 2011

Effects of Mechanical Rotation on S
Sci.Rept. 2018

Mamoru Matsuo,1,2 Jun’ichi Ieda,2,3 Eiji Saitoh,2,3,4 aects of Mechanical Rotation on Spin Currents

Mechanical analog*e 0f SterK-Gerlach eﬀect
H Zeeman = −S ⋅ γ B⇒ F = −∇H Zeeman = S ⋅∇ (γ B )Spin current is generated

along gradient of mag. ﬁeld.

H Spin−rotation = −S ⋅ Ω⇒ F = −∇H Spin−rotation = S ⋅∇Ω
Spin current is generated

along rotation-gradient.

How to create rotation-gradient?
→ 1. Surface acoustic wave,

2. Fluid motion of liquid metal !!

Spin cur=ent by vorHicit, g=adient
Fluid motion

Elastic motion(surface acoustic wave)

zx

R. Takahashi, MM. et al.,Nature Physics 2016
MM et al., PRB(R)2017

yScience, Editorʼs choice

MM et al., PRB(R)2013

Nature Materials, N&V

PRL2017 (Editorsʼ Suggestion)

Nature Physics, N&V

Kobayashi, MM et al.,rtvo

Vorticity:
local rotation of ﬂuid

ic

ity-gradient

Rotation (vorHicit,) -g=adient in a pipe ﬂow of liquid metal

∇×v
v(r )
v(r + δ r )Spin current generation
along vorticity-gradientExQerimental set*p for spin hydrodyKamic generation

PulsedPressureNano-voltmeter

LO

HgHI

Quartz Pipe
φ0.4mmGroundLiquid Mercury
Turbulent Flow
Platinum Electrode Wire(chemical reactivity : weak,relative Seebeck coef. : small )
R. Takahashi, MM et al., Nat. Phys. 12, 52-56 (2016)

Result - Spin-hydrodyKamic sigKal measurement

R. Takahashi, MM et al., Nat. Phys. 12, 52-56 (2016)

Mechanism of Spin-hydrodyKamic voltage generation“Spin-hydrodynamic generation”
1. Spin current generation
along vorticity gradient

2. Spin current is converted+into charge current by ISHE

1. Spin current generation
along vorticity gradient

ﬂow of liquid metal: Hg
2. Spin current is converted
into charge current due toInverse Spin Hall Eﬀect

SHD bridges spint=onics and hydrodyKamics
Navier-Stokes-like eq.

L :pipe length
r0 : pipe radius

Spin diffusion with spin-vorticity
Solution of “Navier-Stoke” equation:
Turbulent ﬂow in a pipe

MM et al.,PRB(R) (2017)

SHD signal
22θλr04ev
VSHD =× SHE sf × ξ × *L!
σ0κsize

ElectronFluidelectronLχ 0 zz :susceptibility
γ : gyromagnetic ratiofluid⎡⎣ λ sf ≪r0 ⎤⎦r0

η : viscosityρ :mass densityκ : Karman constant =0.41
σ 0 : conductivity
θ SH :Spin Hall Angle
λ sf :Spin diffusion length

λ sf = 10nm
θ SHE = 0.01
v* :average velocity of turbulernt flow

R. Takahashi, MM et al., Nat. Phys. 12, 52-56 (2016)

Observation of spin-rotation coupling in quark-gluon plasma (Nature 2017)

LETTER

doi:10.1038/nature23004

Global Λ hyperon polarization in nuclear collisions
The STAR Collaboration*

The extreme energy densities generated by ultra-relativisticcollisions between heavy atomic nuclei produce a state of matter thatbehaves surprisingly like a fluid, with exceptionally hightemperature and low viscosity1. Non-central collisions have angularmomenta of the order of 1,000ћ, and the resulting fluid may have astrong vortical structure2–4 that must be understood to describe thefluid properly. The vortical structure is also of particular interest
because the restoration of fundamental symmetries of quantum
chromodynamics is expected to produce novel physical effects inthe presence of strong vorticity5. However, no experimental
indications of fluid vorticity in heavy ion collisions have yet beenfound. Since vorticity represents a local rotational structure of thefluid, spin–orbit coupling can lead to preferential orientation ofparticle spins along the direction of rotation. Here we present
measurements of an alignment between the global angularmomentum of a non-central collision and the spin of emitted
particles (in this case the collision occurs between gold nuclei andproduces Λ baryons), revealing that the fluid produced in heavy
ion collisions is the most vortical system so far observed. (At highenergies, this fluid is a quark–gluon plasma.) We find that Λ and Λhyperons show a positive polarization of the order of a few per cent,consistent with some hydrodynamic predictions6. (A hyperon is aparticle composed of three quarks, at least one of which is a strange
quark; the remainder are up and down quarks, found in protons
and neutrons.) A previous measurement7 that reported a null result,that is, zero polarization, at higher collision energies is seen to beconsistent with the trend of our observations, though with larger
statistical uncertainties. These data provide experimental access tothe vortical structure of the nearly ideal liquid8 created in a heavy
ion collision and should prove valuable in the development ofhydrodynamic models that quantitatively connect observations to
the theory of the strong force.
The primary objective of the Relativistic Heavy Ion Collider(RHIC) at Brookhaven National Laboratory is to produce a large(relative to the size of a proton) system of matter at temperatures ofRHIC is essential to understand quark confinement and the originof hadron mass.
A collaboration of physicists from 13 countries operates the STARdetector system12, which has recorded billions of collisions at RHIC.
A rendering of the STAR experiment is shown in Fig. 1. Opposing
beams of gold nuclei collide in the centre of the time projection
chamber (TPC), generating a spray of charged particles. The TPC
signal from a single event is shown in Fig. 2. Forward- and backwardtravelling particles and fragments that experience only a small deflection are measured in the beam–beam counters.
Most collisions at RHIC are not head-on, and so involve substantialangular momentum, of the order of 1,000 ћ (where ћ is the reducedPlanck constant) for a typical collision. A slight sideward deflection of
the forward- and backward-travelling fragments13 from a given collision allows experimental determination of the direction of the overall
angular momentum Jˆsys. as shown schematically in Fig. 3.
Recently, Takahashi et al.14 reported the first observation of a coupling between the vorticity of a fluid and the internal quantum spinof the electron, opening the door to a new field of fluid
21spintronics.
In their study, the vorticity ω—a measure of the ‘swirl’ of the velocity flow field around any point (non-relativistically, ω = ½∇ × v)—isgenerated through shear viscous effects as liquid mercury flows next
to a rigid wall.
In a heavy ion collision, shear forces generated by the interpenetrating nuclei may present an analogous situation, introducing vorticity tothe fluid. Indeed, hydrodynamic calculations15 with initial conditions
tuned to reproduce measured momentum anisotropies predict tremendous vorticity in the fluid at RHIC. So far, no experimental evidenceof vorticity at RHIC has been reported, and its role in the evolutionof the fluid has not been explored extensively at the theoretical level.

Au+Au, non-central collision in Relativistic Heavy Ion Collider (RHIC)

Au

Au

vorticity ~ 10 Hz (~ 1017 T)!!

Time of flightTPCttedandeavyhighnd Λcent,n is aange
tons
sult,
o bergerss toeavy
nt ofns toiderarge
es ofng at
thanangular momentum Jsys. as shown
inandFig. 3.colleagues have
RyoschematicallyTakahashi
spin-dependent phenomena
uncoveredintheirarms
are pulled closer to their
14
Recently,
Takahashi
et al.
reported
observationofa couthis field.
Now, writingin Nature
Physics, the firstspinningbodies.
But
theroleof mechanicaldemonstrated
awaytoharness
spins
3
Ryo
Takahashi
andcolleagues
have
angularmomentumisusually
overlookedpling between the vorticity of a fluid and the internal quantum4 spindemonstrated a way to harnessin
spins
in spintronics experiments, despite thefluids,
establishing
the principles
oof the electron,
openingthedoortoanew
field
offluid
spintronics.
in fluids, establishing the principles offact that a conversion between magnetic
fluid
spintronics.
In their fluid
study,
the vorticity ω—ameasureof the‘swirl’angularof thevelo- wasspintronics.
andmechanicalmomentum
One aroundof the centralindemonstratedover
a century
ago by Einsteincity flow field
anychallenges
point (non-relativistically,ω5 =½∇
×6 v)—isOneofthecentral
challenges
inspintronics involves exploiting the fact thatand de Haas , and Barnet .
generated through shear viscous effects as liquid mercury flows nextspintronics involves exploiting the fac
to a rigid wall.
Ref.14: R.Takahashi et al., Nature Physics 12, 52 (2016)ba
In a heavy ion collision, shear forces generated by the interpenetrating nuclei may present an analogous situation, introducing
news &vorticity
views to15
ja
the fluid. Indeed, hydrodynamic calculations
with initialconditions
FLUID SPINTRONICS
tuned to reproduce measured momentum anisotropies predict tremendous vorticity in the fluid at RHIC. So far, no experimental evidence
The rotational motion of liquids can induce a flow of electron spins, and could enable ultra-small spinof vorticity
at RHIC has been reported, and its role in the evolutionhydrodynamicc generators that operate with liquid metals.
of thefluid has not been explored extensively at the theoretical level.
Igor Žutić and Alex Matos-Abiagues

Cause a stirS

pintronics research1,2 has delivered
remarkable improvements inmagnetic information storage andpromised us spin-enhanced information
processing. But the focus so far has beenon solid-state systems; liquids have beenconspicuously absent from thejswealth ofspin-dependent phenomena uncovered inthis field. Now, writing in Nature Physics,
Ryo Takahashi and colleagues3 havedemonstrated a way to harness spinsin fluids, establishing the principles offluid spintronics.

angular momentum is conserved when it isTakahashi et al.3 focused on theconverted between different subsystems, such conversion of angular momentum in a
as electrons, photons and even elementaryspin current. Rather than describing aexcitations. In perhapsTime
its simplest
flow of charge, spin currents7 have anof flightmechanical manifestation, the conservation
imbalance of electron spins pointing inof angular momentum causes the rotational
directions, or different velocities
TPCdifferent
frequency of ice skaters to increase whenfor opposite spins, and can be used totheir arms are pulled closer to theirreverse the direction of magnetization ofspinning bodies. But the role of mechanicalmacroscopic objects7 — enabling spin8
angular momentum is usually overlooked
based logic, superior lasers9 and highBeamline
4in spintronics experiments , despite theperformance interconnects10. In liquids,
fact that a conversion between magnetic
the role of spin currents has not been
and mechanical angular momentum waspreviously explored, so two questions

cjq

Spin cur=ent Iom Surface Acoustic Wave
Spin cur=ent∝Gradient of rotation

zxy

10-6m @ GHz
MM et al., Phys. Rev. B87, 180402(R) (2013)

Spin Hall Eﬀect

Spin-rotation

St=ong Spin-Orbit

w/o Spin-Orbit

ShorH Spin Lifetime Long Spin Lifetime
PtCu

Cu can be utilized for spin-cur=ent source! → Rare metal Iee spint=onics

Howspincurrent
causedbySRC
Howtotodetectdetectspincurrent
causedbySRC
How to detect AC spin cur=ent by SAW?
How to detectspin current caused by SRCVVHallHall
jcjc

jsjs
jsInverse
Inversespin
spinHallHalleffecteffect

VHall

jc

Pt

σσσ

Cu

Hall voltage caused byCu
Cu

Non-uniform spin current
Hall
Hallvoltage
voltagecaused
causedby
byis Hall
compensated…
voltage caused by

nonuniform
nonuniformspin
spincurrent
current
nonuniform spin current
isiscompensated.
compensated.
is compensated.

SAWSAWSAW
Prof. Nozaki’s beautiful idea!

jsjjssInverse spin Hall effectPtPt Inverse
SHE:

Kobayashi, Nozaki, MM et al., PRL2017

NiFe
NiFe
NiFe
NiFe
Attenuationof SAW
owing
toAttenuation
ofofSAWowing
Attenuation
SAWowingtoto
Spin torque FMR
magnetic
dampingmagnetic
dampingCu Cu
magnetic
damping
CuCu

σσσ
SAW

SAW
SAW

Attenuation of SAW
Due to magnetic damping!

First obserYation of spin-cur=ent generation in Cu by spin-rotation coupling

マイクロ波吸収強度の層構(a)(c)
NiFe(20 nm)

Microwave
absorption

Cu(180 nm)
NiFe(20 nm)(b)
Cu(180 nm)
SiO2(20 nm)
NiFe(20 nm)

依存性•

NiFe/Cu界面にスピン抵抗の大きなSiO2を挿入するとDirectマイクロ波吸収が減少
excitation of FMR due to microwave is small.
•Cu/NiFe
NiFe単層ではマイクロ波吸収が10分の1以下
interface!!
PRL 106, 076601 (2011)

PHYSICAL REVIEW L

マイクロ波による直接FMR励起は小さく、Cu/NiFe界面が重要

Kobayashi, Nozaki, MM et al., PRL201715/20

Effects of Mechanical Rotation o