Spin-dependent transport in the presence of spin-orbit interaction

L.Y. Wanga(王律堯 ), C.S. Tangb

and C.S. Chua

aDepartment of Electrophysics, NCTU

bPhysics Division, NCTS

2005.08.05

Outline:

1. Introduction of Rashba-type spin-orbit interaction

2. Spin-pumping via ac-biased finger- gates

3. Spin current generation involving a quantum dot

4. Conclusions

InGaAs

InAlAs2DEG

Asymmetric heterostructure

EV

Beff

I

I

effB H EV

Rashba 0 ˆH p z

0 : Rashba constant

E

Rashba Effect (Spin-orbit interaction)

-0.2 -0.1 0 0.1 0.2-0.01

0

0.01

0.02

0.03

0.04

0.05

Energy dispersion relation Energy dispersion relation

20*

1ˆ

2H p p z V y

m

The Hamiltonian of an electron in a quantum channel

20

1: spin up

;

1: spin-down

x xE k k

E

xk

No SOI

EE

Fig: Dispersion relation for 11

0 0.13 (3 10 eV m)

0: right-going spin- wave vector

: left-going spin- wave vector

RL

LR

k

k

LRk

RLk LRk

LRk

Lx

(APL. Vol.56 p665, Datta and Das.)

045 . . .

1 1 0

1 0 1pol z pol y pol

. . .

1 1 0

1 0 1x pol z pol z pol

Rashba-SOI

VG改變介電常數

VG改變 SOI-coupling constant

AC gate & spin current generation: Schematic distribution of spin currents induced by a time dependent circular gate

Under the gate, electrons with opposite spins move in opposite direction (dashed-line arrows).

Arrows outside the gate area show the accumulated spin polarization during a half period /.

Mal’shukov, C.S. Tang, C.S. Chu and K. A. Chao (PRB 68, 233307)

Nitta. et al. Phys.Rev.B 60, 7736(1999)

Gate voltage

Ras

hba

cons

tant

Tuning Rashba constant by a metal gateTuning Rashba constant by a metal gate0

InGaAs

InAlAs2DEG

gVGate

Experimental data

2

1

0

0

1ˆ ˆ, , ;

2; out of a AC-gate

,; under a AC-gat

( 1) for (-down) sta1 t spin-

co

up

e

e

s

x x x x

t

H p x t p p x t

x t

The Hamiltonian of an ac-biased finger-gate in the Rashba-type quantum channel

2DEG2DEG

ac-biased

FG

The mechanism of spin pumping under the ac-biased finger-gate

The mechanism of spin pumping under the ac-biased finger-gate

The longitudinal Hamiltonian:

2 2, ,

ˆˆ2 4xx t x t

H t i z xx

Spin-dependent vector potential:

,ˆˆ

2

x tA t z x

The spin-dependent vector potential gives rise to a spin-resolved driving electric field.

AE t

t

It turns out that the linear term of vector potential is given by

ˆ xxi x A Akx

, manages to give rise to nontrivial spin-resolved transmission.

LRk LRk

LRk RLk

xk

E

0

The spin-up transition amplitude is stronger than spin-down one.

1

2

2

2

2

1

2

1

,

i

4

s n2

x UUH tt

xt

t

U

U x

t

x

t

After unitary transformation

Spin-indep. Oscillating term

Spin-dep. Oscillating term

There are both of spin-independent and spin-dependent oscillating terms to pump spin.

Spin-up

Spin-down

What is spin current?What is spin current?

Charge current

Spin current (up)

( )S C

RL LR

RL LR LR RL

I I I

dEf E T T T T

Fig: spin-resolved current transmissions (solid) and (dashed) versus the incident energy . Parameters N=1, , , l=20 (80nm), and with (a) 0.03, (b) 0.04, and (c) 0.05. The corresponding dc spin current is plotted in (d).

0 0.13 0.002( 28 )f GHZ 1

RLT RLT

/

The main dip structures are corresponding to the process due to the spin-resolved inelastic mechanism. The brown circle is related to the

process due to the stronger pumping strength.

1

2

Numerical Results:

Time-dependent perturbation approachTime-dependent perturbation approach

1

1

12 2

2

R L

R L

i k E k E L R R

L R

i k E

R

k E

e k E k Er E

k E k E

er

k

E

E

1

2

L R R

L R

R k E E

k E

k k

E

E

k

Here, only is right (left)-going spin-dependent wave vector. ( )R Lk E

(1) For :E R R T T

E (2) For :

R R T T 0.0 0.5 1.0 1.5

0.0

0.5

1.0

/

Tra

nsm

issi

on

T

T

Fig: Current transmission versus for FG number N= (a) 1, and (b) 2. Pumping spin per cycle is plotted in (c) for N=1 (think curve) and N=2 (thick curve) with driving frequency . Other parameters are , , and the edge-to-edge separation .

/

0.001( 14 )f GHz 0 0.13 1 0.065

88l nm

2DEG2DEG

ac-biased

Two ac finger-gate with a phase difference Two ac finger-gate with a phase difference

InGaAs

InAlAs2DEG

L L

1 cos t 1 cos t

The phase difference is maintained between the ac biases of the two finger gates. In such case, the charge current is nonzero due to the breaking the symmetry. This configuration would yield spin-polarized charge current by tuning .

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

( )

Sp

in (

char

ge)

Cu

rren

t (n

A)

Fig: Spin (charge) Current vs. phase difference . , , , , .

0 0.13 1 0.065 20L 20L 0.002 (28GHz)

Negative

charge current

Positive

charge current

Spin current

Charge current

1FG-1QD-1Fg System configuration

Fig : A quantum dot locates between two ac-biased finger-gate in a Rashba-type quantum channel.

0 1V x x 0 1V x x

L

2DEG 2DEG

LFD FDDD

QD

1(a) 0.02

1(b) 0.04

1(c) 0.06

0

0

0.13

0.001 ( 14 )

0.4

35 (140nm)

20 (80nm)

40 (160nm)

f GHz

V

L

FD

DD

p

q

r p=3.93q=4.93r=5.93 RRL LT T

RRL LT T

RRL LT T

Spin-dependent transport with varying 1

Fig :

1st resonance peak 2nd resonance peak

1 4.93E

2 20.13E

1E

2E

Spin-current with varying 1

Fig: The polarized direction of a spin current can be changed in opposite direction when an electron is incident in the front and in the back of resonance energies (switching point) within a quantum dot.

But the net charge current is zero due to the symmetry configuration.

1 4.93E 2 20.13E

S

spin current:

I 2[ ]FD RL RLd f T T

x y

x

|(x

,y)|

2

|(x

,y)|

2

y=5Spin-downSpin-up

For the main-peak (resonance energy) q=4.93 ( )RRL LT T

Fig : The magnitude of the spatial wave function is plotted when the incident energy is approached to the resonance energy.

1 0.06

DotFG FG

|(x

,y)|

2

|(x

,y)|

2

xy

x

y=5

Spin-downSpin-up

For the satellite-peak p=3.93 ( )RRL LT T

Fig : The magnitude of the spatial wave function is plotted when the incident energy is equal to .1E

1 0.06 DotFG FG

|(x

,y)|

2

|(x

,y)|

2

xx

y

y=5

Spin-downSpin-up

For the satellite-peak r=5.93 ( )RRL LT T

Fig : The magnitude of the spatial wave function is plotted when the incident energy is equal to .1E

1 0.06 Dot

FG FG

One-sideband approximation in an ac-FG

2

1 11

2

1 11

1 cos ( )

2 42

1 cos ( )

2 42

RL

RL

LT

LT

k

k

1( ) T

L

1 ( )T 0th-order

1st-order

1st-order

0

1

0.13

0.02

0.001 ( 14 )

35 (140nm)

f GHz

L

One-sideband contribution and approximation:

Fig:

The transmission of spin-up electron is larger than spin-down one in T1 and T-1 processes.

The transition rate of a spin-up electron is larger than spin-down one

1E1E

1E

1E1E

1 2E

1E RRL LT T

RRL LT T

(a)V0=0.4

(b)V0=0.8

(c)V0=1.2

0

1

0.13

0.06

0.001 ( 14 )

35 (140nm)

40 (80nm)

20 (160nm)

f GHz

L

FD

DD

The satellite peaks of the 2nd resonance peak can be resolved by increasing V0

The spin-resolved peaks become more narrow via increasing V0 .

Fig. 8:

0

1

0.13

0.06

0.001 ( 14 )

35 (140nm)

40 (80nm)

20 (160nm)

f GHz

L

FD

DD

Fig. 9:

The switching point of the spin-polarized direction would be shifted toward the higher energy with increasing V0.

Conclusion:1. We have proposed a generation of dc spin current witho

ut charge current via ac-biased FGs in a Rashba-type quantum channel in the absence of magnetic field.

2. The two ac-biased FGs with a fixed phase difference can generate the charge current with spin current.

3. We propose a mechanism to switch the polarized direction of a spin current in the 1FG-1QD-1FG structure.

Bound state:

0x 1x W 2x W x L

0V1V

0

E

0.75 0.25

40

In Al As

nm0.75 0.25

13.5

In Ga As

nm

4

InAs

nm

0.75 0.25

2.5

In Ga As

nm

*0: 0.023 ; 0.356Direct

gInAs m m E eV

0.75 0.25 : 0.556DirectgIn Ga As E eV

0.75 0.25 : 0.9625DirectgIn Al As E

1 0.20325V eV

0 0.30325V

0

1

1 0

the lowest energy 0.20692

2 lowest energy 0.12865

0.078

14 0.058

E eV

nd E eV

E E E eV

GHz E meV

E E