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Dissipationless quantum spin currentat room temperature

Shoucheng Zhang (Stanford University)

Collaborators:

Shuichi Murakami, Naoto Nagaosa(University of Tokyo)

Tsinghua Spring School 2004/04

Science 301, 1348 (2003)



Can Moore’s law keepgoing?

Power dissipation=greatest obstacle for Moore’s law! Modern processor chips consume ~100W of power of which

about 20% is wasted in leakage through the transistorgates.

The traditional means of coping with increased power pergeneration has been to scale down the operating voltage of the chip but voltages are reaching limits due to thermal

fluctuation effects.

0

100

200

300

400

500

0.5 0.35 0.25 0.18 0.13 0.1 0.07 0.05

Active Power

Passive Power (Device Leakage)

350 250 180 130 100 70 50

500

500

400

300

200

100

0

Technology node (nm)

Power density (W/cm)

2



Spintronics

•The electron has both charge and spin.

•Electronic logic devices today only used the charge propertyof the electron.•Energy scale for the charge interaction is high, of the order

of eV, while the energy scale for the spin interaction is low,of the order of 10-100 meV.

•Spin-based electronic promises a radical alternative, namelythe possibility of logic operations with much lower powerconsumption than equivalent charge based logic operations.

• Spin-based electronics also promises a greater integrationbetween the logic and storage devices



Spintronic devices

Devices• Spin valve• Magnetic tunneling junction• Datta-Das current

modulator etc.

Applications• Read heads in magnetic

recording• Nonvolatile memory• Nonvolatile reprogrammable

logicetc.

Spin valve Magnetic tunneling junction

Conductance changes depending on the magnetization direction.



Spintronic devices with semiconductors

• spin injection into semiconductor • Ohmic injection from ferromagnet Low efficiency

(Difficulty):• Ferromagnetic metal :

conductivity mismatch spin polarization is almost lost at interface.

• Ferromagnetic semiconductor (e.g. Ga1-x MnxAs) :

Curie temperature much lower than room temp.• Ferromagnetic tunnel junction.

• spin detection by ferromagnet

• spin transport in semiconductorspin relaxation time

•Optical pump and probe



Quantum Hall effect in higher D?

k ijk spin

i

j E J ε σ =

• Since the spin is a vector, the spin current is a tensor.

An electric field along the z direction can induce a spincurrent flowing along the x direction, where the spinsare polarized along the y direction.

•Murakami, Nagaosa and Zhang, Science, valence band•Sinova et al, cond-mat, conducting band

Spin current generated by the electric fieldthrough the spin-orbit interaction

k jk H j E J ε σ =



Dissipationless spin current induced by the electricfield



Time reversal symmetry and dissipativetransport

•Microscopic laws physics are T invariant.•Almost all transport processes in solids break T

invariance due to dissipative coupling to theenvironment.

•Damped harmonic oscillator:

)(,2

l k k heE J F F jj ∝= σ σ

• Only states close to the fermi energy contribute to thedissipative transport processes.

•Electric field=even under T, charge current=odd under T.

•Ohmic conductivity is dissipative!

0=+− kxxxm

η



Only two known examples of dissipationlesstransport in solids!

• Supercurrent in a superconductor is dissipationless,since London equation related J to A, not to E!

•Vector potential=odd under T, charge current=oddunder T.

• In the QHE, the Hall conductivity is proportional to themagnetic field B, which is odd under T.

•Laughlin argument: all states below the fermi energycontribute to the Hall conductance.

•Streda formula, TKNN formula relates the Hallconductance to the 1st Chern number.

t

A

cE AJ

j

jjS j ∂

∂−==1

,ρ

BE J H H ∝= σ ε σ ν µ ν µ ,



Dissipationless transport at roomtemperature?• Room temperature superconductivity?• QHE at room temperature would require a very high

magnetic field!•The achieve dissipationless quantum transport at room

temperature is the main objective of condensed matterphysics!

• Spin current=even under T.

• spin transport can be non-dissipative!

F spink ijk spin

i

j ek E J ∝= σ ε σ ,

• It works because of spin-orbit coupling, which can belarge even at room temperature.

• In fact, the spin conductivity is entirely topological, canbe expressed as the integral of a gauge curvature inmomentum space.

•Similar to Streda, or TKNN formula in QHE.



Valence band of Si



Valence band of Ge



Valence band of GaAs



p-orbit (x,y,z)× spin ↑,↓= 6 states

split-off band (SO)heavy-hole band (HH) doubly degeneratelight-hole band (LH) (Kramers)

Valence band of GaAs

Luttinger Hamiltonian

( : spin-3/2 matrix, describing the P3/2 band)S

+ spin-orbit coupling

( )

⋅−

+=

2

2

2

21 22

5

2

1S k k

mH

γ γ γ

−−

=

=

−−

−=

2/3000

02/100

002/10

0002/3

02/300

2/3010

0102/3

002/30

02/300

2/300

002/3

002/30

zyx S S

i

ii

ii

i

S



( ) )(22

5

2

1 2

2

2

21 xV S k k m

H

+

⋅−

+= γ γ γ

Unitary transformation

)()()(22

5

2

1)()(

22

2

2

21 k U xV k U S k k m

k HU k U H z

++ +

−

+==′ γ γ γ

Diagonalize the first term with a local unitary transformation

HH

LH

LH

HH

m

k

:

:

:

:

2

2

2

2

2

23

21

21

23

21

21

21

21

2

−=−===

−+

+−

λ

λ

λ

λ

γ γ

γ γ

γ γ

γ γ )()()()( DV k U iV k U k

=∂ +

i

i

i Ak

iD −∂∂

=

)()( k U k

k iU Ai

i

+

∂∂

−= : gauge field in k!

z y S iS i

z eek U kS k U S k k U ϕ θ

==⋅+ )(,)()(

Helicity basis S k

⋅= ˆλ

)ˆ(k U

)'ˆ(k U



Local gauge field in k space

HH

LH

LH

HH

d id d

id d d id d

id d d id d

id d d

dk A ii

:

:

:

:

cos)(sin

)(sincossin

sincos)(sin

)(sincos

23

21

21

23

23

2

3

2

3

21

21

2

3

2

3

23

−=−===

−+−

+−−+−

=

λ

λ

λ

λ

ϕ θ θ ϕ θ

θ ϕ θ ϕ θ θ ϕ θ

θ ϕ θ ϕ θ θ ϕ θ

θ ϕ θ ϕ θ

Adiabatic transport = potential V does not cause inter-band transitions only retain the intra-band matrix elements

Abelian approximation = retain only the intra-helicity matrix elements

HH

LH LH

HH

d id d

id d d id d id d d id d

id d d

dk A ii

:

::

:

cos)(sin

)(sincossinsincos)(sin

)(sincos

2

3

21

2

1

2

3

2

3

2

3

2

3

2

1

2

1

2

3

2

3

2

3

−=−==

=

−+− +−−

+−

=

λ

λ λ

λ

ϕ θ θ ϕ θ

θ ϕ θ ϕ θ θ ϕ θ θ ϕ θ ϕ θ θ ϕ θ

θ ϕ θ ϕ θ



)(2

2eff xV

m

k H

+=λ

)(~

k A

k

iDx i

i

ii

−

∂

∂=≡

Effective Hamiltonian for adiabatic transport

k jijk

i

iii k E k m

k

xE k ε

λ

λ

3,−=−=

ijjiijjiji iF xxik xk k −=== ],[,],[,0],[ δ

Eq. of motion

3k

k F k

ijk ij λ ε =

(Dirac monopole)

ik

E

∂∂

=Drift velocity Topological term ij

jF

eE

−=

Nontrivial spin dynamics comes from theDirac monopole at the center of k space, witheg=λ :



Non-commutativegeometry

Heisenberg uncertainty principle:Non-commutativity in phase space x

H p

p

H xipx ijji ∂

∂−=

∂∂

=⇒= ,],[ δ

2D QHE:x

V y

y

V xil yxyxV H

∂∂

−=∂∂

=⇒== ,],[,),( 2

3D spin current:Non-commutativity inmomentum space

33],[

k

k k x

k

k ixx k

jijk ik

ijk ji λ ε λ ε −=⇒−=

Real space monpole:33

],[x

xxp

x

xipp k

jijk ik

ijk ji λ ε λ ε =⇒−=



Eq. of motion:

It can be integrated:

Real-space trajectory within Abelianapproximation

k

//

x

y

z E //

k jijk i

iii k E k m

k xE k ε

λ

λ 3

, −=−=

( )

( )

( ) 202

0

2

0

0

2

0

2

0

00

0

20

20

20

0

2

0

2

0

000

200

000

)(

,)(

,2

)(

,,,)(

z z yx

z z

yx

xy

z z yx

z z

yx

yx

z z

z z yx

k t E k k

k t E

k k

k t

m

k xt y

k t E k k

k t E

k k

k t

m

k xt x

t m

E t

m

k z t z

t E k k k t k

−++

−+

−+=

−++

−

+++=

−+=

−=

λ

λ

λ

λ

λ λ

Hole spin

Side jump ( )( )S k

//⊥

0>λ

0<λ

,36)(

31

,4)(

3

1

2,

2,

21

23

π

π

λ

λ

λ

λ

L

F z y

k

Lxy

H

F z y

k

H

xy

k E k nS xj

k E k nS xj

==

==

∑

∑

±=

±=

Spin motion can be known from orbital motion since .Spin current (spin//y, velocity//x)

k S ˆλ =



3D motion projection onto xy plane: side-jump perpendicular to and

0: >λ

0: <λ

E z //

E z

//

Spin direction

Real-Space trajectory for the HH band

S k

( and : antiparallel)

( and : parallel)S k

S

E



Conservation of total angular momentum

In the presence of the E field, Jz is conserved.

Total angular momentum:

0],[,0],[, 0 ==+×= z J H J H S k xJ

0ˆ)()( =+×+×=

••

••

k k xk xJ z z z λ

,

36

)(

3

1

,4

)(31

2

,

2,

21

23

π

π

λ

λ

λ

λ

L

F z y

k

L

xy

H

F z y

k

H

xy

k E k nS xj

k E k nS xj

==

==

∑

∑

±=

±=

The total angular momentum conservation directly leads to the spincurrent.



Luttinger model

Expressed in terms of the Dirac Gamma matrices.



Full quantum calculation of the spin current based onKubo formula

Definition of the conserved spin current in the presence of the spin orbit

coupling:

( ) ijk

L

F

H

F

k

k ijH Lijk

k ijk

j

i

k k e

k Gk nk nV

E j

ε π

σ

σ

−=

−=

=

∑26

)()]()([4

Final result for the spin conductivity: (Similar to the TKNN formula for theQHE. Note also that it vanishes in the limit of vanishing spin-orbitcoupling).



Non-abelian gauge field in k space

Gauge field in the 3D k space is induced from the SU(2) gauge field in

the 5D d space.



Dissipationless spin current induced by the electricfield



Spin current induced by an electric field

x: current direction

y: spin directionz: electric field

SU(2) analog of the QHE• topological origin• dissipationless• All occupied states in the

valence band contribute.

GaAs

E

( ) z sLF

H F

z xy E k k eE j σ π ≡−= 26

z

y

x

External electric field does not break time-reversalsymmetry.Spin current is allowed in this system with time-reversal

symmetry

Direct Kubo formula calculation yieldsessentially the same result.



Application in spintronics : Effective source of spin currents

At present, efficiency of spin injection is still very low.Electric-field-induced spin currents can overcome this difficulty!

p-GaAsFerro.

V

Example:

Depending on the direction of magnetization of theferromagnet, the voltage dropwill change.



carrierdensity

mobility Chargeconductivity

Spin (Hall)conductivity

1019 50 80 73

1018 150 24 34

1017 350 5.6 16

1016 400 0.64 7.3

)cm( 3−n )cm( -11−Ωσ /Vs)cm( 2µ )cm( -11−Ωsσ

3/1nk

en

F S ∝∝

=

σ

µ σ

As the hole density decreases, both and decrease.decreases faster than .

σ σ S σ

S σ

Order of magnitude estimate (at room temperatur



Rapid relaxation of hole spins

Spin relaxation time at RT:

hole : momentum relaelectron:

≈≈ secf 100sτ

• Because of strong spin-orbit coupling in thevalence band, deviation of spin/momentum

distribution away from equilibrium relaxes rapidlyfor holes.

Our spin current is free from this rapid relaxation, becausethe spin/momentum distribution is in equilibrium.(The spin current originates from anomalous velocity.)

secp100≈sτ



x

y

z

ferro.

p-GaAs

I

x

y

z

n-GaAs

GaAs

(In,Ga)As

GaAs

p-GaAs

+σ

Detection of spincurrent

(a) Measuring theconductance difference

by attachingferromagnetic electrode

(b) Measuring the circularpolarization of emitted lightby

attaching n-GaAs

J

J

sJ

sJ



Spin injection by ferromagneticsemiconductor Ga1-x MnxAs

Ohno et al., Nature 402,790 (1999)



Spin accumulation at the boundary

x0

s

yxy

yy t xs

x

t xj

x

t xs

Dt

t xs

τ

),(),(),(),(2

2

−∂

∂

−=∂∂

−∂∂

p-GaAs :Spin current :

0≤x)()( xjxj xyxy −= θ

Diffusion eq.

p-GaAs

xyj

Steady-state solution: s

Lxsxy

y DLeD

jxs τ τ

≡= ,)( /

x0

ys

sDL τ ≡

sxyjs τ =totalTotalaccumulatedspins:



Charge current :

At room temperature:

(c) Accumulation of holespins

n-GaAs

Detection of spin current by measuring accumulatedspins

p-GaAs

+σ +

σ

p-GaAs

(d) Convert holespins

into electron spins

29total cm/103 Bsxyjs µ τ ×==

secf 100=sτ

nm4=L

24A/cm10=J

secp30=sτ

212

total cm/10 Bsxyjs µ τ ==

At room temperature :

secp100=sτ

21312 cm/10 Bsxyj µ τ −=m11.0 µ −=L

At 30K :



Conclusion & Discussion

• A new type of dissipationless quantum spin transport,

realizable at room temperature.•Similar to the edge transport of the QHE. Can be

viewed as the 3D edge transport of the 4D QHE.•Topological origin, spin conductivity is an integral

over the monopole field strength, over all states

below the fermi energy.• Instrinsic spin injection in spintronics devices.•Spin injection without magnetic field or ferromagnet.•Spins created inside the semiconductor, no issues

with the interface.•Room temperature injection.•Source of polarized LED.

•Reversible quantum computation.

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