Electrical Means of Manipulating Electron Spins in Semiconductors

NTHU Colloquium 11.22.2006

Electrical Means of ManipulatingElectron Spins in Semiconductors

C. S. ChuDept. of Electrophysics

National Chiao Tung University

NCTU

Collaborators: A.G. Mal’shukov (RAS) L. Y. Wang (NCTU)


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Spintronics, or spin electronics, involves the study of active control and manipulation of spin degrees of freedom in solid-state systems.

in Reviews of Modern Physics, vol. 76, p.323-410, 2004, by I. Žutić, J. Fabian, and S. Das Sarma.

Quoted from the abstract of “Spintronics: Fundamentals and applications”

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How about transport of spins in non-magnetic semiconductor using only electrical control?

Schemes making use of spin Hall effect

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Schemes other than spin Hall effect


NCTUA simplest version of a spin Hall effect:

An electrical current passes through a sample with spin-orbit interaction, and induces a spin polarization near the lateral edges, with opposite polarization at opposing edges (M.I. D’yakonov and V.I. Perel’, JEPT Lett., 13, 467 (1971)).

This effect does not require an external magnetic field or magnetic order in the equilibrium state before the current is applied.

M.I. D’yakonov and V.I. Perel’ (1971) proposed an extrinsic mechanism for the spin Hall effect in the paper: “Possibility of orienting electron spins with current”. cond-mat/0603306 H. Engel, E.I. Rashba, B.I. Halperin


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V.M. Edelstein, Solid State Commun. 73, 233 (1990)“Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems”

S. Murakami, N. Nagaosa, S.C. Zhang, Science 301, 1348 (2003)“Dissipationless quantum spin current at room temperature”

J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth, and A.H. MacDonald, Physical Review Letters 92, 126603 (2004)“Universal Intrinsic Spin Hall Effect”


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A simple picture for the extrinsic spin Hall effect

J.E. Hirsch, PRL 83, 1834 (1999)


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2( ) ( )

2pH V r p Vm

2( ) ( )

2pH V r p Em

( ) ( )mr V E p E

E

Potential energy

For electron incident upon theLHS of the attractive scatterer, again the spin up particle is deflected more tothe left and the spin down particle is deflected more to the right.

More on the simple picture for the extrinsic spin Hall effect


NCTU A “simple picture” for the intrinsic spin Hall effect

PRL 92, 126603 (2004)J. Sinova, D. Culcer, Q. Niu et al.


Asymmetric heterostructure that has Spin-orbit interaction

Schematic layer structure of an inverted In0.

53Ga0.47As / In0.52Al0.48As heterostructure.(Nitta et al. Phys. Rev. Lett.78, 1355(1997))

Calculated conduction band diagram (solid line) and electron distribution (dash line).(Nitta et al. Physica E, 2, 527(1998))


Rashba effect (spin-orbit interaction )

analogE

InGaAs

InAlAs

2DEG

Heterostructure:

Structure inversion asymmetry:

E


EV

In Lab. frame

Beff

In the rest frame of an electron

I

I

Effective magnetic field induced by the effective current I.

Effective magnetic field induced by the effective current I.

An electron moves between

two charged plane

An electron moves between

two charged plane

The SOI hamiltonian is given by

B H V Eeff

Rashba 0 ˆH p z

where is called the Rashba constant.

0


Fig.3.Dispersion relation for a 2D Rashba-type system and the Rashba constant .

ky

kx

E

0 0.13

2 2 2 22 0 D x y x yE k k k k +


Rashba spin-orbit interaction (SOI)

: normal to interface: the Pauli spin operato

Rashba term:

r

so ˆ

ˆ

H

v

p v

• SOI is significant in narrow gap semiconductor heterostructures.

• Large variation (up to 50%) of the SOI coupling constant , tuned by metal gates, has been observed experimentally.

[ Nitta et. al. PRL 78 (1997)

Engels et. al. PRB 55 (1997) Grundler, PRL 84 (2000) ]

• Static gate control of has been the focus of previous proposals on spin polarized transistors. [ Datta et. al. APL 56 (1990), ……]


Tuning of the coupling constant by a metal gateTuning of the coupling constant by a metal gate0

Spin-orbit coupling parameter of the first (circle) and second (square) subband as a function of the gate voltage: including (solid) and not including (open) band nonparabolicity correlation.

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

InGaAs

InAlAs

2DEG2DEG

Gat

e

gV


NCTU A “simple picture” for the intrinsic spin Hall effect

E

J. Sinova, et al PRL 92, 126603 (2004)

This picture is subject to change

since it has not incorporated the scattering picture

as well as the form of the spin-orbit interaction


Experimental observation of extrinsic spin Hall Effect in thin 3D layersY.K. Kato, R.C.Myers, A.C. Gossard, D.D. Awschalom, Science 306, 1910 (2004)


Experimental confirmation of spin Hall Effect in a 2D hole gasJ. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005)


SHE in n-type Rashba spin-orbit systems vanishes in the presence of weak disorder

J.I. Inoue, et al, Phys. Rev. B 70, 041303 (2004)E.I. Rashba, Phys. Rev. B 70, 201309 (2004)O. Chalaev et al, Phys. Rev. B 71, 245318 (2005)E.G. Mishchenko, et al, Phys. Rev. Lett. 93, 226602 (2004)A.A. Burkov, et al, Phys. Rev. B 70, 155308 (2004)O.V. Dimitrova, Phys. Rev. B 71, 245327 (2005)R. Raimondi et al, Phys. Rev. B 71, 033311 (2005)A.G. Mal’shukov et al, Phys. Rev. B 71, 121308(R) (2005)B.A. Bernevig and S.C. Zhang, Phys. Rev. Lett. 95, 016801 (2005)

Vertex correction is important !


SHE is found in Dresselhaus-type spin-orbit systems

A.G. Mal’shukov et al, Phys. Rev. B 71, 121308(R) (2005)

SHE is found in cubic Rashba-type hole systems

B.A. Bernevig and S.C. Zhang, Phys. Rev. Lett. 95, 016801 (2005)

Would the entire intrinsic spin Hall story collapse

due to the presence of impurities?


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Thus far, the research on spin Hall focused onphysical quantities such as:

spin current; spin Hall conductivity.

The system of interest was mostly Rashba-typerather than Dresselhaus-type.

Spin accumulation at the edges was essentially obtained from the bulk spin current plus someplausible arguments.

Explicit calculation of the spin accumulation at the edges in a diffusive sample was in order at the time.


Question: What sort of spin accumulation could instrinsic Rashba SOI or Dresselhaus SOI induce near a diffuse sample boundaries?

Outline: Derivation of a diffusion equation for the

spin and charge densities in a 2D strip

Spin accumulation at the strip edges and its symmetry properties

Connection between the spin flux and the spin densities

Summary


Derivation of a diffusion equation for the spin and charge densities in a 2D strip

2

0 = + 2 p

pH h

m

Rashba SOI:

ˆ = ( )k

h k z

InGaAs

InAlAs

2DEG

Asymmetric heterostructureDresselhaus SOI:

2 2

2 2

= ( );

= ( )

xk x y

yk y x

h k k

h k k

Symmetric quantum well


2

0 = + 2 p

pH h

m

Four vector dens (ity , ) : iD r t

0 ( , ) = ( , )D r t n r t

( , ) = 2 ( , )i iD r t S r t

0 imp = + + 'H H HV

0where 1, and for , , .i i i x y z

i

ii trtrBtrVH ),( ),(),( '

Random distribution of Isotropic scatterers


†ˆ ˆ, , , ,ii l l down upD r t T r t r t S

, ,, ,ii G rD r t i T rr t t

†ˆ ˆ, , ', , , l

loop

i T r t r t i H dG r r t t

2 ˆ ˆ' , , ,jjH d r r r r

Linear response:

< ….. > denotes averaging over impurity configuration


2 '

2

, , , ' , , '

, ,

jj r

dd r G r r G r r

G r r

, , , ,, , r aFn G r r G r rG r r

r

a

G G G

G G G

2 2

2 2

,

,

kr

k k

a

k k

k

k k

G ki h

G

h

i h

i

hik


2 0, ' ,',, ,i j ij

ij rD r d r r rr D

'

, , ' , , ' 2

'

, ,

a rF ji

ij

dndi Tr G r r G r r

d

r r

0

2 [ ', , ' , ', '

',

', ,

,

'' '

' , ', '

2

]

jr rj

ja a

i

iF

j

i

r Tr G r r G r r

G r r G r r

D r

di d r n


1 1 1 1, , ' , , 'r rG p k q G k p q

0Evaluation of ( , )iD r

q q

1,k q

1, p

1, p q

1, k

Q

1k q ''

' '

1p

1k

1p q

Ladder diagrams do not contribute


00, 2 ( ) ( , )i F iD q N E q

1p

1p

1p q

1p q

ji

q q

0 ( ) is the density of states per spin

at the Fermi energyFN E

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2Fourier transform of ' , ', ' :,ij jd r rr r

0 2, , ' , , ',i i ij jj

D r D r d r r r r

Derivation of the diffusion equation of ( , )iD r

'

, ' ,'2

, ' ', , ''

r a

pj

iF j

p

G p p qq G p p qdni

dd


+

p

p q

'p p

p

'p

p q

'p q

+ + +…..

Treating the disorder within the ladder diagram approximation

, ',K q

+

+

+….

'

' '

.

''

, ', ' '

, ' , '

', ' ' , ' , ' '

, '

,

,

r app

r a r

m

a

r a

i

G p G p

G p p G p q

q

G p G p q G p G p

q

K q

p


, ',K q

+

+

+….

12, ', 1 , ',i

S

cK q V q

V

, ', q

2, ' , 'r ai

Sp

cV G p G p q

V

p

p q

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Treating the disorder within the ladder diagram approximation

'

' '

.

''

, ', ' '

, ' , '

', ' ' , ' , ' '

, '

,

,

r app

r a r

m

a

r a

i

G p G p

G p p G p q

q

G p G p q G p G p

q

K q

p

12, ', 1 , ',i

S

cK q V q

V

'

, ' ,'2

, ' ', , ''

r a

pj

iF j

p

G p p qq G p p qdni

dd

0 2, , ' , , ',i i ij jj

D r D r d r r r r


'' ' '

' '

1

2n m

nmO O

'' ' '

1

2i j

ijij

O O

21, ', , ' , '

2is i a s ri

Sp

cq V Tr G p q G p

V

To get some feeling, let’s consider the case h0:

2

0, ', 1is is

hq i Dq

2

0

2

1

2

/ 2

i S F

F

c V N E

D v

F

F

p

E

qv

h

, ,

:limitDirty


First order in

, ', ?is

hq

2linear in h and =0, ',

F

ismis m

F p

iq q v h

Need to evaluate up to first order in . q

kkhh

Angular averagePrecession of the inhomogeneous spin polarization about the effective SOI field.


2 2

22

3

2

3

2 0, =0 term

, ',

' '

1 1

2 ''

1

' '

is

p p qi s

p p q

pi siS

p p qp

p qi s

p p q

qh

q

h hTr

i i

hcV Tr

V ii

hTr

i i

It is diagonal, and is nonzero for i, s limited to x, y, and z.


2

0, =0 term

, ', , , , ,ijqh

q i j x y z

2 2

2 0, =0 term

, ', 4F

ij ij i jp k kq

hq h n n

Charge-spin coupling

kkkhhn

ˆ

D’akonov-Perelspin relaxation

0 3 02

, ', , ',F

F

lpl l

p

nq h iq q

p


Diffusion equation of ( , )iD r

0 2, , ' , , ',i i ij jj

D r D r d r r r r

0

32 2

24 4 F F

F F

ijjj

ip pij ij ijm m m ij i j

p F m p p p

D D D i D

h nD D h v h n n

p

ijmmR ij 0iM

0

0 0 0 00 0 0

(e > 0)

2 ; 0 ; 0x y z

eEx

D N eEx D D D D


Equation for the spin densities for Rashba-typesemiconductor strip

y

SRS

y

SD

yzyyzzz

z

2

2

y

x

E

0 2

2

xxx

x

Sy

SD

x

DM

y

SRS

y

SD

yx

zyzyyyy

y

0

00

2

2

2

Bulk spin density : Sx = Sz = 0

eENx

DMS yyy

xy

0

000

b 2

ˆ = ( )k

h k z V.M. Edelstein

Solid State Comm. 1990J.I. Inoue et al, PRB 2003

Rashba-type strip


What boundary conditions do we have for the solving of the spin densities ?

Answer: Connecting spin flux and spin densities

2, ' ' , ', ' ', 'l li ij j

j

I r t d r dt r r t t r t

, ',

''[ , ', ' , ', ' ]

2 '

lij

F a li

jr

r r

ndi Tr G r r G rJ r

4ill

ili vvJ

*l

l kl

kv h

m k


Connecting spin flux and spin densities

b

1

2

iy ijy j j

i iz SH

SI r D R S S I

y

20

b2 2

1 zjy j y kSH F k

x z

N hI R S eE v h

k


Rashba-type strip

)(2

1byyzyy

zyz SSR

y

SDI

zyzyy

yy SR

y

SDI

2

1

y

SDI

xyx

0 2

:conditionBoundary

dI yi

0SHI

NO Spin Accumulation at edges for Rashba-type strip.


Equation for the spin densities for Rashba-typesemiconductor strip

y

SRS

y

SD

yzyyzzz

z

2

2

y

x

E

0 2

2

xxx

x

Sy

SD

x

DM

y

SRS

y

SD y

x

zyzyyyy

y

0

002

2

Bulk spin density : Sx = Sz = 0

eENx

DMS yyy

xy

0

000

b 2

ˆ = ( )k

h k z V.M. Edelstein

Solid State Comm. 1990J.I. Inoue et al, PRB 2003

Rashba-type strip


Equation for the spin densities for Dresselhaus-typesemiconductor strip

y

SRS

y

SD

xzxyzzz

z

2

2

y

x

E

0 2

2

yyy

y

Sy

SD

x

DM

y

SRS

y

SD x

x

zxzyxxx

x

0

002

2

Bulk spin density : Sy = Sz = 0

)48(

)2(162

2244

2240

000

b

FF

Fxxxx

x

kk

keEN

x

DMS

Dresselhaus-type strip


Spin density diffusion equation in a 2D stripdriven by a homogeneous electric field

y

SRS

y

SD

zxzyxxx

x

2

2

0 2

2

yyy

y

Sy

SD

y

SRS

y

SD

xzxyzzz

z

2

2

Edges of the strip are at y = ± d/2

0 yS

parity opposite of bemust and zx SS 0 2

:conditionBoundary

dI yi



Spin density diffusion equation in a 2D stripdriven by a homogeneous electric field

y

SRS

y

SD

zxzyxxx

x

2

2

y

SRS

y

SD

xzxyzzz

z

2

2

Edges of the strip are at y = ± d/2

parity opposite of bemust and zx SS

0 2

:conditionBoundary

dI yi

x

yx

SI y D

y

1

2

zy zxy xz SH

SI y D R S I

y

odd

even z

x

S

S



Spin densities for i = x, z as a functions of its width d.

The inset shows the dependence of Sz(y) on the transverse coordinate y. Lengths are measured in unit of .

(PRL.95, 146601(2005))

/ 2 iiS d S

2 / 2SO F Fy kyl v v h


xx

ZS ZS

yy

Spin densities of Sz are of odd parity in a 2D strip with /k=1.3 for the strip width d = (a) 1; (b) 10, respectively.

(a) (b)

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Summary

1. A diffusion approximation has been derived for the spin density.

2. Spin accumulation at the two edges of a Dresselhaus 2D strip associated with the spin Hall effect is obtained.

3. The spin accumulation exhibits damped oscillations as a function of the strip width.

4. Our analysis shows that the spin current decreases as τ2 whereas the strip spin density decrease as τ. This explains why we still obtain noticeable spin polarization in our dirty regime examples.

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Electrical Means of Manipulating Electron Spins in Semiconductors

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