The spin Hall effect

Shoucheng Zhang (Stanford University)

Collaborators:Shuichi Murakami, Naoto Nagaosa (University of Tokyo)Andrei Bernevig, Congjun Wu, Taylor Hughes (Stanford University)Xiaoliang Qi (Tsinghua), Yongshi Wu (Utah)

APCTP 2005/08

Science 301, 1348 (2003)PRB 69, 235206 (2004), PRL93, 156804 (2004)cond-mat/0504147, cond-mat/0505308,…

My view on the status of the SHE

Fspinkijkspini

j ekEJ ∝= σεσ

he

qpEJ HjijHi

2

== σεσ

• Quantum Hall effect exists in D=2, due to Lorentz force.

• Natural generalization to D=3, due to spin-orbit force:

• 3D hole systems (Murakami, Nagaosa and Zhang, Science 2003)• 2D electron systems (Sinova et al, PRL 2004)

My view on the status of the SHE• The intrinsic spin Hall conductivity vanishes in the n-type Rashba

model due to vertex corrections. But an uniform magnetization isinduced by the electric field.

• The intrinsic spin Hall conductivity is finite in the p-type Luttingermodel in 3D and the p-type Rashba model in 2D. In fact, the impurity vertex correction vanishes identically for delta-function scatters. Interestingly, no uniform magnetization is induced by the electric field. =>An interesting complimentarity principle?

• The UCSB effect could either be intrinsic or extrinsic. Maybe they can be distinguished by the sign of the effect.

• The Hitachi-Cambridge effect could be intrinsic. The best opportunity to compare theory with experiment.

• The non-conservation of spin current, and relationship between the spin current and spin accumulation are still outstanding theoretical issues, but maybe the naïve expectations are basically correct.

• The new frontier in SHE theory is the quantum SHE.

Rashba model:

Intrinsic spin Hall conductivity (Sinova et al.(2004))

+ Vertex correction in the clean limit (Inoue et al (2003), Mishchenko et al,Sheng et al (2005))

Effect due to disorder

0=Sσ

πσ

8e

S =

+ spinless impurities ( -function pot.)

πσ

8vertex e

S −=

( )xyyx kkm

kH σσλ −+=2

2

Green’s function method

xJzyJ

+ ⋅⋅⋅+xJ

zyJ

δ

Luttinger model:

Intrinsic spin Hall conductivity (Murakami et al.(2003)) )(6 2

LF

HFS kke

−=π

σ

+ spinless impurities ( -function pot.)

0vertex =Sσ

( ) ( )yxxy SkSkSkm

kH ⋅−⋅+⋅+= 22

1

2

2λλ

xJzyJ

+ ⋅⋅⋅+xJ

zyJ

δ

Vertex correction vanishes identically!(Murakami (2004), Bernevig+Zhang (2004)

Quantum Spin Hall

• Can one have a quantum spin Hall effect without any external magnetic field and T breaking?

• Landau level problem:

raE =

raE =chargeρ

raE =chargeρ

GaAs

E

• 2D momenta and E field, σz only:

raE =chargeρ

• Hamiltonian for spin-orbit coupling:

( ) σµ⋅×++= Ep

mcgear

mpH B2

2

2

BrAm

AepHrrr

rr

×=−

=21

2)( 2

• Example of such a field: inside a uniformly charged cylinder

Quantum Spin Hall • In semiconductors without inversion symmetry, shear strain is like an

electric field in terms of the SO coupling term

dh TO ⎯⎯ →⎯breakingsymmetry inversion

cubic gp symm gp: Ixyz ≡ (rotation part only, inversion not a symmetry)

⎪⎩

⎪⎨

⎧

⇒⎪⎩

⎪⎨

⎧

+++

⇒≡

y

x

z

EEE

yzxxzxyzyzyxxy

Ixyz~~~

~~z~

xz

yz

xy

εεε

ayaxraE

xz

yz

xy

===

↔=εεε 0

zyyzxxz ppCDrm

pH σεε )(22

322

−++=h

(shear strain gradient creates the same SO coupling situation as a radialyincreasing electric field)

zxyyx ypxpRyxppH σ)(2222 −++++= (up to a coordinate re--scaling)

aDmCR 2

23

h=

Quantum Spin Hall

GaAs

E• Hamiltonian for electrons:

zxyyx ypxpRyxppH σ)(2222 −++++=

• Tune to R=2

( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

+

−= 2

2

00

ApApH

)0,,( xyA −=

• Spin up

( )21 ApH −=

effectiveB

• Spin down

( )21 ApH +=

effectiveB

Quantum Spin Hall

• P,T-invariant system

0arg =xyechσ

πσ

42

22

2 eeh

espin ==

h

• Spin up

effectiveB

∗−=

zzn

n en

z 21

!πφ

• Spin down

effectiveB

( ) ∗−∗

=zz

m

m em

z 21

!πφ

( ) ( ) ( )njiji

jim

jijiji

jiji

mjii zzzzzzz ∏∏∏

=↓=↑<

∗

=↓<

∗∗

=↑<

−−−=↓↑Ψ,;,;,;

),,(

• Halperin-like wavefunction

Quantum Spin Hall

• Purely electrical detection measurement, measure xx

echargρ

ν

xyσxxσ

xxρ

• More effort to directly measure , open question. spinσ

• Landau Gap and Strain Gradient

aCELandau 3=∆ m/s108 53 ×=h

Cstrain gradient=a

m10over %1for µ=a mKELandau 10=∆

Spin-Hall insulator: dissipationless spin transport without charge transport (PRL 93, 156804, 2004)

• In zero-gap semiconductors, such as HgTe, PbTe and α-Sn, the HH band is fully occupied while the LH band is completely empty.

• A charge gap can be induced by pressure. In this case, charge conductivity vanishes, but the spin Hall conductivity is maximal.

ae

s 1.0−≈σ

Topological Quantization of the AHEMagnetic semiconductor with SO coupling (no Landau levels):

charge Hall conductance topological quantized to be n/2π

Charge Hall effect of a filled band:

Topological Quantization of SHE

LH

HH

SHE is topological quantized to be n/2π

Paramagnetic semiconductors such as HgTe and α-Sn:

In the presence of mirror symmetry z->-z, d1=d2=0! In this case, the H becomes block-diagonal:

Topological Quantization of Spin Hall • Physical Understanding: Edge states

In a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states.

Energy spectrum on stripe geometry.

Laughlin’s Gauge Argument:

When turning on a flux threading a cylinder system, the edge states will transfer from one edge to another

Quantum spin Hall effect in graphene (Haldane, Kane&Mele)

• SO coupling opens up a gap at the Dirac point.• One pair of TR edge state on each edge.• Numerical calculation indicate stability (Sheng et al)

Topological Quantization of Spin Hall • Physical Understanding: Edge states

When an electric field is applied, n edge states with Γ12=+1(−1) transfer from left (right) to right (left).

Γ12 accumulation Spin accumulation

Conserved Non-conserved

+=

Stability at the edge• The edge states of the QSHE is the

1D helical liquid. Opposite spins have the opposite chirality at the same edge.

• It is different from the 1D chiralliquid (T breaking), and the 1D spinless fermions. T2=1 for spinlessfermions and T2=-1 for helical liquids.

• One particle backscattering is forbidden by the T symmetry. (Kane&Mele), however, two particle backscattering is allowed.

• An new kind of stability!

Conclusion & Discussion

• A new type of dissipationless quantum spin transport.• Natural generalization of the quantum Hall effect.• Lorentz force and spin-orbit forces are both velocity

dependent.• U(1) to SU(2), 2D to 3D.

• Quantum SHE.• A new type of 1D metal: the helical liquid.• Standard semiconductors with a strain gradient, narrow gap

semiconductors and monolayers of graphene.• More experiments!