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```									Spin-Hall Effect in 2D Electron Systems with Spin-Orbit Couplings

L. Sheng, Q. Wang, and C. S. Ting
Texas Center for Superconductivity, University of Houston Collaborators: D. N. Sheng (CSUNorthridge) F. D. M. Haldane (Princeton)

Contents
l. ll. lll. Introduction Can Spin-Hall Effect Survive Disorder Scattering? (Rashba Spin-Orbit Coupling) Finite size calculations:

a) Dissipative Spin Hall Conductance b) Nondissipative Quantized Spin Hall Conductance lV. Discussion

L. Sheng et al., Phys. Rev. Lett. 94, 016602 (2005). Phys. Rev. Lett. 95, 136602 (2005) Q. Wang et al., cond-mat/ 0505576 (2005)

I. Introduction
• Spin-Hall Effect and Spin-Hall Conductivity (SHC)—due to the spin-orbit coupling

1 1 SHC    siz viy   (J  J ) E i 2eE
Current for Spin-up Electrons

2DEG

Electric Field E
Current for Spin-down Electrons

Iy=J↑+J↓=0

Charge Current

1. Introduction--Spin-Orbit Coupling
• The origin of the spin-orbit coupling comes from the orbital motion of an electron around an ion which gives rise an induced magnetic field B and the Zeeman coupling between B and the spin of the electron. • Spin-orbit interaction can be presented by a term Hso in the one-electron Hamiltonian:

 denotes the periodic crystal potential plus addition electric fields where V (r ) due to lack of inversion symmetry.  The expression of the spin-orbit coupling may be different due to V (r ) in
different crystals.

    V (r )  p  . H so  2 2 4m c

Origin of Inversion Asymmetry

•

•

•

Bulk inversion asymmetry (BIA) – E.g. Zincblende structures – Due to the polar nature of the ionic bonding – Dresselhaus, or Luttinger terms – Important for III-V or II-VI materials Structure inversion asymmetry (SIA) – External field V  )  E z V ((()) Ez zzzz ˆ V rrr  E ˆˆ – Rashba term : Interface asymmetry – Noncommon atom

z

SHC in Rashba Model
J. Sinova et al., Phys.Rev.Lett. 92, 126603 (2002)

p2 H  [σ  (e z  p)] 2m p2    (σ x p y  σ y p x ) 2m

The Rashba interaction originates from the inversion asymmetry of the confining electric potential for the 2DEG

    H so  2 2 V (r )  p   . 4m c

 ˆ V (r )  Ez z

Rashba spin-orbit interaction
ˆ For a two-dimensional electron gas (DEG) , V (r )  Ez z
This is known as Rashba spin-orbit interaction and is usually written as
   ˆ    ( z  p) 

)  Ez z ˆ V r ˆ V ((r)  Ez z 

, where  is the Rashba coupling constant.

Because of the spin-orbit coupling, a momentumdependent effective magnetic field make the spins (red arrows) to align perpendicularly to the momenta (green arrows). Eigen-states of the plane waves are ,

where

, and

p H  σ  (e z  p) 2m
[p, H] = 0 (p is a good quantum number), but [σz, H] ≠ 0 Rashba coupling can be regarded as a momentumdependent Zeeman coupling

2

BR (p)  2 (e z  p)
As a result, the energy band splits into two Rashba subbands. In equilibrium, the electron spin s is confined in the xy plane and perpendicular to p

s

BR(p)

p

I. Introduction
In the presence of electric field, the electrons are accelerated. Therefore, p changes with time p = p(t) and so does BR(p)=BR(t). The semiclassical equation of motion for electron spin reads

dn(t )  n(t )  BR (t ) dt

Bloch Equation

n(t) - a unit vector representing the direction of the electron spin

I. Introduction
Consider a small and very slow change in BR(t) due to electric field (adiabatic approximation). By solving the equation of motion perturbatively, one obtains:

1 dBR (t ) nz (t )  (1) 2 dt [ BR (0)]

( 2)

BR(t) = 2λ[ez x p(t)]→ BR(1)(0) at t = 0

p(t)= p(0) –eEt,

BR(2)(t) = BR(t)·ey

I. Introduction
Substituting dpx/dt = -eE and the expression for the Rashba field into the above equation, one obtains

n z (p) 

 ep y E 2 p
3

In response to the applied electric field, the electron spins are tilted out of the xy plane toward the positive z direction for py < 0 and the negative z direction for py > 0. This effect leads to a nonzero SHC.

I. Introduction
The SHC can be calculated from

d p nz (p) p y Js, y    f ( p )  sH E 2 ( 2 ) 2 m 
Here,  is the index for the two Rashba bands. When both the Rashba bands are occupied, the SHC takes a universal value

2

sH

e  8

[J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)]

II. Can the spin-Hall effect survive disorder scattering?

For the Rashba model, spin-Hall conductance in an Infinite system disappears for arbitrarily small disorder-J. I. Inoue, G. E. W. Bauer, and L. W. Molenkamp, cond mat/0402442 (PRB, 2004) E. G. Mishchenko, A. V. Shytov, and B. I. Halperin; O. V. Dimitrova, (PRL, 2005) Liu and Lei, (PRB, 2004)

IIl Finite Size Calculation
• Examine Whether the SHC of a Rashba Spin-Orbit Coupling System can survive in finite sizes when defect scattering is in presence— Landauer-Buttiker Approach: • Nikolic, Zarbo and Souma, (PRB, 2005) • Hankiewicz, Molenkamp, Jungwirth and Sinova, (PRB, 2004) • Sheng, Sheng and Ting, (PRL 94, 016602 (2005))

Dissipative Spin-Hall Conductance
• Our Numerical Calculation Based Upon The 4Terminal Landauer-Buttiker Formula

lx →

xx ≠ 0

• Rashba Model in The Tight-Binding Representation

Here, ei randomly distributed between [-W/2, W/2] represents nonmagnetic disorder. (For infinite systems, if we take the Fourier transform, we see that the above Hamiltonian reduces to the continuous Rashba model near the band bottom.)

• Multi-Terminal Landauer-Buttiker Formula (M. Buttiker, Phys. Rev. Lett. 57, 1761 (1986))

e2 Il  2

T
l ' l

ll '

(U l '  U l )

Here, Ul is the electrical potential in lead l, and Tll’ is electron transmission coefficient from lead l’ to lead l.
Transmission Coefficient in Terms of Green’s Function (Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1994))

Tll '  Tr( l G l' G )
r a

IIl Finite Size Calculation
For tight-binding model with nearest neighbor hopping, Hlc Couples the device to only the first layer in lead l. Therefore

 l  H g H lc  H S 2 ( H10 S1 ) H lc
(l ) 0

 lc

 lc

1

• Numerical Results

U1  0 U2  U3  U0 / 2 ( I 3  I 3 ) SHC  2U 0

IIl Finite Size Calculation

Fig. 1 SHC GsH for some disorder strengths as a function of electron Fermi energy E. Here, the system size L = 40 and the spin-orbital coupling Vso = 0.5t.

IIl Finite Size Calculation

Fig. 2 SHC as a function of system size L for different disorder strengths at E=-2t and Vso=0.5t. Inset: SHC as a function of disorder strength for L=40 and 60.

IIl Finite Size Calculation
Finite Size Scaling-- Localization length calculated on long bar as a function of disorder strength for different cross-section L.

One Remark on Our Finite Size Calculation

Why Is The SHC in Our LB Calculation Non-Universal?
• In finite-size systems with boundary, the SHC Is sensitive to the boundary condition. For example, hard-wall boundary may enhance and stabilize the spin currents near the boundary. Our results are consistent with all existing finite size calculations.

IV. Recent Experimental Evidence—Spin Accumulation
• “Experimental Discovery of The Spin-Hall Effect in 2Dimensional Spin-Orbit Coupled Semiconductor System” J. Wunderlich, B. Kastner, J. Sinova and J. Jungwirth, condmat/0410295 (PRL, 2005).

• “Observation of the Spin Hall Effect in Semiconductors” Y. K. Kato, R. C. Myers, A. C. Cossard, D. D. Awschalom (Science 306, 1910 (2004) Both of these works did not directly measure the spin-Hall conductance, Instead, they measured the spin accumulation or polarization near the lateral edges of the sample using optical method.

Spin-Hall Effect---Spin Polarization

Jx
The spins near the two edges of the system have opposite polarizations along z-direction and they can be observed by the Kerr rotation microscopy using a circular polarized light.

Quantized Spin Hall Conductance
• The quantized Hall conductance---In the presence of a strong magnetic field
σxy =< jy jx >/E = n[e2/h], with n=1, 2, 3…., and jx=evx • The quantized Spin-Hall conductance—

SHC=< (jy)sjx >/E = n[e/4],
current operator and sz=ħ/2

with (jy)s = szvy as the spin

Quantized Spin-Hall Conductance
• But in order to observed the quantized SHC, one needs a spin-orbit coupling Hso such that [sz, Hso]=0. • One of the systems being proposed to be a graphite film with the honeycomb lattice ----graphene

The quantized Spin Hall Effect
The Hamiltonian for a graphene 2DES:

• Vso is the instrinsic SO coupling allowed by the symmetry of the honeycomb lattice, and it conserves the z-component of the electron spin, i.e. [H, sz] =0. • With Vso=0 the system is a semimetal, and with Vso 0 the system becomes an band insulator

Vso=0

  

=0

Vso 0

=0

lx →

xx  0

Dissipationless Transport
This system in the insulating phase [xx=0] has the following property: ( Haldane (PRL, 1988)); Onoda and Nagaosa (PRL, 2003)

The spin up electrons and the spin down electrons are decoupled, the spin up electrons would see a positive magnetic field B generated by their orbital motion while the spin down electrons see a negative B and thus gives rise to a quantized Hall conductance

σxy(↓)= e2/h, similarly we also have σxy(↑)=- e2/h
σxy = σxy(↓) + σxy(↑)=0, and σxx(↑)= σxx(↓)=0 The resistivity for spin ↑ and ↓ electrons are:

xx(↑)= σxx(↑)/[σxx2(↑) + σxy2(↑)]=0, and xx(↓)= σxx(↓)/[σxx2(↓) + σxy2(↓)]=0
The transport is thus nondissipative

Quantized Spin-Hall Conductance
• In this case, the spin-Hall conductance is also quantized as: SHC= [Jy(↓)-Jy(↑)]/E = (ħ/2e)[σxy(↓) - σxy(↑)]=2(e/4) [Kane and Mele (PRL, (2005)] • When the graphene film is placed on a substrate, asymmetric potential is present along z-direction, we may have the Rashba SO coupling VR and also the disorder effect:

The effect of the the Rashba SO coupling VR is to destablize the quantized spin Hall effect.

Effect Rashba SO Coupling on Quantized SHC

No Disorder

Effect of Disorder on Quantized SHC

Conclusion
A. In this talk I only emphasized on the Rashba spin-orbit coupling in 2-dimensional electron systems, not on the Luttinger spin-orbit coupling for p-doped 3d semiconductors which has been done extensively by S. C. Zhang and coworkers.

B. For finite size calculation, the results in dissipative transport
depend critically on the sample sizes. For nondissipative transport, the results are robust and are practically in dependent sample sizes.

C.

The idea of current driving spin transport has been a hot subject recently in condensed matter physics because it may have great potential in spintronics applications. But so far the spin Hall conductance has not yet been experimentally measured, this could become a negative factor for motivating people to do further investigation in this field.

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