Topological Aspects of the Spin Hall Effect

Yong-Shi Wu

Dept. of Physics, University of Utah

Collaborators:

Xiao-Liang Qi and Shou-Cheng Zhang

(XXIII International Conference on

Differential Geometric Methods in Theoretical Physics

Nankai Institute of Mathematics; August 21, 2005)

Motivations

• Electrons carry both charge and spin• Charge transport has been exploited in

Electric and Electronic Engineering:

Numerous applications in modern technology • Spin Transport of Electrons

Theory: Spin-orbit coupling and spin transport

Experiment: Induce and manipulate spin currents

Spintronics and Quantum Information processing• Intrinsic Spin Hall Effect:

Impurity-Independent Dissipation-less Current

kijksH

ij EJ

• Key advantages:• Electric field manipulation, rather than magnetic field• Dissipation-less response, since both spin current and electric field are even under time reversal• Intrinsic SHE of topological origin, due to Berry’s phase

in momentum space, similar to the QHE• Very different from Ohmic current:

lkh

ewhereEJ Fcjcj

22

Electric field induces transverse spin current due to spin-orbit coupling

The Spin Hall Effect

p-GaAs

E

x

y

z

Family of Hall Effects

• Classical Hall Effect Lorentz force deflecting like-charge carriers

• Quantum Hall Effect Lorentz force deflecting like-charge carriers (Quantum regime: Landau levels)

• Anomalous (Charge) Hall Effect Spin-orbit coupling deflecting like-spin carriers (measuring magnetization in ferromagnetic materials)

• Spin Hall Effect Spin-orbit coupling deflecting like-spin carriers (inducing and manipulating dissipation-less spin currents without magnetic fields or ferromagnetic elements)

Time Reversal Symmetry and Dissipative Transport

• Microscopic laws in solid state physics are T invariant• Most known transport processes break T invariance due to dissipative coupling to the environment• Damped harmonic oscillator

lkh

ewhereEJ Fjj

22

(only states close to the Fermi energy contribute!)

kxxxm • Ohmic conductivity is dissipative: under T, electric field is even charge current is odd

• Charge supercurrent and Hall current are non-dissipative:

t

A

cEAJ j

jjSj

1

, BEJ HH /1,

under T vector potential is odd, while magnetic field is odd

Spin-Orbit Coupling

• Origin: ``Relativistic’’ effect in atomic, crystal, impurity

or gate electric field = Momentum-dependent magnetic field Strength tunable in certain situations

• Theoretical Issues: Consequences of SOC in various situations? Interplay between SOC and other interactions?

• Practical challenge: Exploit SOC to generate,manipulate and transport spins

The Extrinsic Spin Hall effect(due to impurity scattering with spin-orbit coupling)

D’yakonov and Perel’ (1971) Hirsch (1999), Zhang (2000)

• The Intrinsic Spin Hall Effect Berry phase in momentum space Independent of impurities

impurity scattering = spin dependent (skew) Mott scattering plus side-jump effect

Spin-orbit couping

up-spin down-spinimpurity

Cf. Mott scattering

Berry Phase (Vector Potential) in Momentum Space from Band Structure

xdk

uuikn

kknikA d

i

knkn

ini

cellunit

*)(

( : periodic part of the Bloch wf. )

knu

xki

knknexux

)()(

)()( kAkB nkn

: Magnetic field

in momentum space : Band index n

Wave-Packet Trajectory in Real Space

)(, )( kBEe

m

kxEek

Anomalous velocity (perpendicular to and )

k

//

zE //

Hole spin

S

0

0

E

Spin current (spin//x,velocity//y)

Chang and Niu (1995); P. Horvarth et al. (2000)

Intrinsic Hall conductivity (Kubo Formula) Thouless, Kohmoto, Nightingale, den Nijs (1982) Kohmoto (1985)

kn

nznFxy kBkEnh

e

,

2

)( )(

)()( kAkB nkn

: field strength; : band index n

(Degeneracy point Magnetic monopole)

Field Theory Approach

• Electron propagator in momentum space

• Ishikawa’s formula (1986):

• Hall Conductance in terms of momentum space topology

),p( with )( 0pppSF

(p)]S(p)S(p)Str[p εdh

e

π

iσ FλFνFμ

μνλxy

11132

224

p-GaAs

E

x

y

z

Cf. Ohm’s law: Ej

: odd under time reversal = dissipative

response

: even under time reversal = reactive response

(dissipationless)

i: spin directionj: current directionk: electric field

kijksij Ej

s

• Nonzero in nonmagnetic materials.

Intrinsic spin Hall effect in p-type semiconductors

In p-type semiconductors (Si, Ge, GaAs,…), spin current is induced by the external electric field.

(Murakami, Nagaosa, Zhang, Science (2003))

Valence band of GaAs

Luttinger Hamiltonian

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

2

22

21 22

5

2

1Skk

mH

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 SS

i

ii

ii

i

S

S

P

S

P3/2

P1/2

Luttinger model

Expressed in terms of the Dirac Gamma matrices:

Spin Hall Current (Generalizing TKNN)

ijkLF

HF

k

kijHLijk

kijkj

i

kke

kGknknV

Ej

26

)()]()([4

xJzyJ

• Of topological origin (Berry phase in momentum space)• Dissipation-less 　　　　　　　　

　　　　　　　　• All occupied state contribute

Spin Analog of the Quantum Hall EffectAt Room Temperature

(Sinova, Culcer, Niu, Sinitsyn,Jungwirth, MacDonald, PRL(2003))

8

es

Rashba Hamiltonian

m

kikk

ikkm

k

km

kH

xy

xy

z

2)(

)(2

2 2

2

2

Intrinsic spin Hall effect for 2D n-type semiconductors in heterostructure

Kubo formula : zSyx JJ

independent of zySy SJJ z ,

2

1

2D heterostructure

x

y

z

Effective magnetic field

SHE: Spin precession by “k-dependent Zeeman field” Note: is not small even when the spin splitting is small. due to an interband effect

S

)ˆ(int kzB

Spin Hall insulator

• Motivation: Truly dissipationless transport Gapful band insulator (to get rid of Ohmic currents)

• Nonzero spin Hall effect in band insulators: - Murakami, Nagaosa, Zhang, PRL (2004)

• Topological quantization of spin Hall conductance: - Qi, YSW, Zhang, cond-mat/0505308 (PRL)

• Spin current and accumulation: - Onoda, Nagaosa, cond-mat/0505436 (PRL)

Theoretical Approaches

• Kubo Formula (Berry phase in Brillouin Zone)

Thouless, Kohmoto, Nightingale, den Nijs (1982)

Kohmoto (1985)

• Kubo Formula (Twisted Phases at Boundaries) Niu, Thouless, Wu (1985)

(No analog in SHE yet!)

• Cylindrical Geometry and Edge States

Laughlin (1981)

Hatsugai (1993) (convenient for numerical study)

Cylindrical Geometry and Edge States

Laughlin Gauge Argument (1981):

•Adiabatically changing flux

•Transport through edge states

Bulk-Edge Relation:

(Spectral Flow of Edge States)

(Hatsugai,1993)

Topological Quantization of the AHE (I)

)coscos2(

sin,sin

syxz

xyyx

ekkcd

kdkd

Magnetic semiconductor with SO coupling in 2d

(no Landau levels)

Model Hamilatonian:

Topological Quantization of the AHE (II)

Two bands:

)()()( kVdkkE

Charge Hall conductance is quantized to be n/2

Charge Hall effect of a filled band:

Band Insulator: a band gap, if V is large enough, and only the lower band is filled

4 0 0,

4 2 1,

40 ,1

ss

s

s

eoreif

eif

eifn

(c>0)

Topological Quantization of the AHE (III)

)5.0,3/,1( setVcOpen boundary condition in x-direction Two arrows: gapless edge states The inset: density of (chiral) edge states at Fermi surface

Topological Quantization of Spin Hall Effect I

SHE is topologically quantized to be n/2

Paramagnetic semiconductors such as HgTe and -Sn:

are Dirac 4x4 matrices (a=1,..,5)

With symmetry z->-z, d1=d2=0. Then, H becomes block-diagonal:

a

Topological Quantization of Spin Hall Effect II

LH

HH

yxs

yx

yx

kkekd

kkkd

kkkd

coscos2)(

)cos(cos3)(

sinsin3)(

5

4

3

2/],[ 2112

For t/V small: A gap develops between LH and HH bands.

Conserved spin quantum number is

4 0 0,

4 2 1,

40 ,1

ss

s

s

eoreif

eif

eifn

Topological Quantization of Spin Hall Effect III• Physical Understanding: Edge states IIn a finite spin Hall insulator system, mid-gap edge states emerge and the spin transport is carried by edge states

Energy spectrum for cylindrical geometry

Laughlin’s Gauge Argument:

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

Topological Quantization of Spin Hall Effect IV• Physical Understanding: Edge states II

Apply an electric field

n edge states with transfer from left (right) to right (left).

accumulation

Spin accumulation

Conserved

Non-conserved

+=

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

0S

+ spinless impurities ( -function pot.)

8vertex e

S

xyyx kkm

kH

2

2

(Green’s function method)

8

eS xJz

yJ

xJ

zyJ

Luttinger model: Intrinsic spin Hall conductivity (Murakami et al,2003)

+ spinless impurities ( -function pot.)

0vertex S

yxxy SkSkSkm

kH 2

21

2

2

)(6 2

LF

HFS kk

e

xJz

yJ

xJ

zyJ

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

Topological Orders in Insulators

• Simple band insulators: trivial• Superconductors: Helium 3 (vector order-parameter)• Hall Insulators: Non-zero (charge) Hall conductance 2d electrons in magnetic field: TKNN (1982) 3d electrons in magnetic field: Kohmoto, Halperin, Wu (1991)• Spin Hall Insulators: Non-zero spin Hall conductance 2d semiconductors: Qi, Wu, Zhang (2005) 2d graphite film: Kane and Mele (2005) • Discrete Topological Numbers: in 2d systems Z_2: Kane and Mele (2005); Z_n: Hatsigai, Kohmoto , Wu (1990)• 2d Spin Systems and Mott Insulators: Topological Dependent Degeneracy of the ground states Fisher, Sachdev, Sethil, Wen etc (1991-2004)

Conclusion & Discussion

• Spin Hall Effect: A new type of dissipationless quantum spin transport, realizable at room temperature

• Natural generalization of the quantum Hall effect• Lorentz force vs spin-orbit forces: both velocity dependent• U(1) to SU(2), 2D to 3D

• Instrinsic spin injection in spintronics devices• Spin injection without magnetic field or ferromagnet• Spins created inside the semiconductor, no interface problem• Room temperature injection• Source of polarized LED

• Reversible quantum computation? • Many Theoretical Issues: Effects of Impurities Effects of Contacts Random Ensemble with SOC Topological Order of Quantized Spin Hall Systems