Spin-dependent transport phenomena in strongly spin-orbit coupled mesoscopic systems: spin Hall effect and Aharonov-Casher

JAIRO SINOVA

Collaborators: Allan MacDonald, Dimitri Culcer, Ewelina Hankeiwc, Qian Niu, Kentaro Nomura, Nikolai Sinitsyn, Laurens Molenkamp, Hartmut Buhmann, Charlie Becker, Volker Daumer, Yongshen Gui Matthias König, Jian Liu, Markus Schäfer, Joerg Wunderlich, Bernd Kästner, Tomas Jungwirth, Branislav Nikolic, Satofumi Souma, Liviu Zarbo, Mario Borunda Hong Kong, August 17 th 2005 Collaborators supported by:

OUTLINE

  

Spin dependent transport in SO coupled systems: Das-Datta transistor paradigm

Spin-Hall effect   Basic pheonemonlogy Some settled issues: mini-workshop on spin-Hall effect  Spin Hall effect in the mesoscopic regime    Why study the mesoscopic regime Transport indications of SHE in the mesoscopic regime Spin accumulation in ballistic and coherent systems Aharonov-Casher effect in mesoscopic rings

Spin-orbit coupling interaction (one of the few echoes of relativistic physics in the solid state) Ingredients: -“Impurity” potential V(r) - Motion of an electron Produces an electric field 

E

   1

e



V

(

r

) In the rest frame of an electron the electric field generates and effective magnetic field 

B eff

    

k



cm

   

E

This gives an effective interaction with the electron’s magnetic moment

H SO

   

Beff

   

e S



mc

      

k



mc



r

 1

er dV dr

(

r

)     

S

  

L

CONSEQUENCES

•

If part of the full Hamiltonian quantization axis of the spin now depends on the momentum of the electron !!

•

If treated as scattering the electron gets scattered to the left or to the right depending on its spin!!

Spin Hall effect

Take now a PARAMAGNET instead of a FERROMAGNET:

Spin orbit coupling “force” deflects like-spin particles

F SO _ F SO

non-magnetic

I V=0 Carriers with same charge but opposite spin are deflected by the spin-orbit coupling to opposite sides.

Spin-current generation in non-magnetic systems without applying external magnetic fields Spin accumulation without charge accumulation excludes simple electrical detection Refs: Dyakonov and Perel (71), J. E. Hirsch (99)

INTRINSIC SPIN-HALL EFFECT:

Murakami et al Science 2003 (cond-mat/0308167) Sinova et al PRL 2004 (cont-mat/0307663)

as there is an intrinsic AHE (e.g. Diluted magnetic semiconductors), there should be an intrinsic spin-Hall effect!!!

n, q (differences: spin is a non-conserved quantity, define spin current as the gradient term of the continuity equation. Spin-Hall conductivity: linear response of this operator) n’  n, q  sH

xy

Inversion symmetry  no R-SO

H k

  2

k

2

m

2  0   (

k x



y

Broken inversion symmetry  R-SO 

k y



x

) 2

k

2

m

2  0  

k

  

e

   8 

e

for 8 

n

2

D n

2 *

D n

2

D

for 

n

* 2

D n

2

D



m

2   

n

* 2

D

 2 4

Disorder effects: beyond the finite lifetime approximation for Rashba 2DEG Question: Are there any other major effects beyond the finite life time broadening? Does side jump contribute significantly?

Inoue, Bauer, Molenkamp PRB 04 Also: Mishchenko et al, PRL 04 Raimondi et al, PRB 04, Dimitrova PRB05, Loss et al, PRB 05 Ladder partial sum vertex correction:   NOTE: the vertex corrections are zero for 3D hole systems (Murakami 04) and 2DHG (Bernevig and Zhang 05)

Experimental observations Wunderlich, Kästner, Sinova, Jungwirth, cond-mat/0410295 PRL in press: 1 0

Experimental observation of the spin-Hall effect in a two dimensional spin-orbit coupled semiconductor system

-1 Co-planar spin LED in GaAs 2D hole gas: ~1% polarization 1.505

1.52

Light frequency (eV) Kato, Myars, Gossard, Awschalom, Science Nov 04

Observation of the spin Hall effect bulk in semiconductors

Local Kerr effect in n-type GaAs and InGaAs: ~0.03% polarization (weaker SO-coupling, stronger disorder)

SHE controversy

•Does the SHE conductivity vanish due to scattering? Seems to be the case in 2DRG+Rashba (Inoue et al 04), does not for any other system studied •Dissipationless vs. dissipative transport •Is the SHE non-zero in the mesoscopic regime?

•What is the best definition of spin-current to relate spin-conductivity to spin accumulation •……

A COMMUNITY WILLING TO WORK TOGETHER APCTP Workshop on Semiconductor Nano-Spintronics: Spin-Hall Effect and Related Issues August 8-11, 2005 APCTP, Pohang, Korea

http://faculty.physics.tamu.edu/sinova/SHE_workshop_APCTP_05.html

Semantics agreement:

The intrinsic contribution to the spin Hall conductivity is the the spin Hall conductivity in the limit of strong spin orbit coupling and  >>1. This is equivalent to the single bubble contribution to the Hall conductivity in the weakly scattering regime.

General agreement

•The spin Hall conductivity in a 2DEG with Rashba coupling vanishes in the absence of a magnetic field and spin-dependent scattering. The intrinsic contribution to the spin Hall conductivity is identically cancelled by scattering (even weak scattering). This unique feature of this model can be traced back to the specific spin dynamics relating the rate of change of the spin and the spin current directly induced, forcing such a spin current to vanish in a steady non-equilibrium situation.

•The cancellation observed in the 2DEG Rashba model is particular to this model and in general the intrinsic and extrinsic contributions are non-zero in all the other models studied so far. In particular, the vertex corrections to the spin-Hall conductivity vanish for p-doped models.

OUTLINE

   Spin dependent transport in SO coupled systems: Das-Datta transistor paradigm Spin-Hall effect   Basic pheonemonlogy Some settled issues: mini-workshop on spin-Hall effect  Spin Hall effect in the mesoscopic regime    Why study the mesoscopic regime Transport indications of SHE in the mesoscopic regime Spin accumulation in ballistic and coherent systems Aharonov-Casher effect in mesoscopic rings

SHE in the mesoscopic regime

Non-equilibrium Green’s function formalism (Keldysh-LB) Advantages: •No worries about spin-current definition. Defined in leads where SO=0 •Well established formalism valid in linear and nonlinear regime •Easy to see what is going on locally

6

Spin Hall effect in the mesoscopic regime, simplifying the debate

Hankiewicz, Molenkamp, Jungwirth, Sinova, PRB 70, 241301(R) (2004).

Also: Sheng et al, PRL 05 Nikolic et al, PRB 05

HgTe-QW D R = 5-15 meV

Actual gated H-bar sample

5  m Gate Contact ohmic Contacts Unfortunately the device is too large to observe coherent transport

Spin accumulation in mesoscopic systems

Nikolic, Souma, Zarbo, and Sinova, PRL 05

100x100, E

F

=-3.8t, t

so

=0.1t

Rashba Model

S z

  Non-linear regime

eV

Right  

eV

2

eV

2DEG 100 80  0 60 40 20

y x eV

Left 

eV

2

OUTLINE

   Spin dependent transport in SO coupled systems: Das-Datta transistor paradigm Spin-Hall effect   Basic pheonemonlogy Some settled issues: mini-workshop on spin-Hall effect  Spin Hall effect in the mesoscopic regime    Why study the mesoscopic regime Transport indications of SHE in the mesoscopic regime Spin accumulation in ballistic and coherent systems

Aharonov-Casher effect in mesoscopic rings

HgTe Ring-Structures

Three phase factors: Aharonov-Bohm Berry Aharonov-Casher

s

 and  , parallel and anti  parallel to

B tot b

  1 for  ,   

B ext

,

B tot

;

B tot



B ext



B eff

High Electron Mobility



> 3 x 10 5 cm 2 /Vsec

180 160 140 120 100 80 60 40 20 0 0 q1867 n Hall = 1,79*10 12 cm -2 n SdH = 1,74*10 12 cm -2 µ= 301000 cm 2 /Vs 1 2 3 B (T) 4 5 6 2,5 2,0 1,5 1,0 0,5 7 0,0

Rashba Effect in HgTe

8 x 8

k



p

band structure model 160 140 120 = 0.2 V, Sym. Case =  2.0 V, Asym. Case 15 10  2.0 V  100 5 80 60 40 20 0.0

0.05

0.1

0.15

0.2

(nm -1 ) 0.25

0.3

0.35

A. Novik et al., PRB 72, 035321 (2005).

0.4

Y.S. Gui et al., PRB 70, 115328 (2004).

0 0.0

0.05

0.1

0.15

(nm -1 ) 0.2

 0.25

Rashba splitting energy D

R

, max  30 meV

HgTe Ring-Structures

Modeling E. Hankiewicz, J. Sinova, Concentric Tight Binding Model + B-field

CONCLUSION

     Spin Hall effect is robust in the mesoscopic regime Coherent transport can in principle be used as a spin injector.

Need to connect the two regimes (bulk, mesoscopic) Need a consistent spin-accumulation theory (in terms of the chiral states) Aharonov-Casher effect in HgTe ring nanostructure consistent with theory

Rashba Splitting

(Bychkov-Rashba) subband splitting due to macroscopic asymmetric potential spin orbit coupling in an asymetric potential  Rashba hamiltonian

H

   2   2 2

m

*    

i

   

E

    Rashba term  : effective mass parameter  : vector of Pauli spin matrices

E

: confining electric field energy dispersion

E

 

E i

  2

k

2 2

m

*  .....

 

k

3 

k

in case of a hole system E k y k x

Band Structure of HgTe QWs

4 nm QW 0.20

normal 0.15

semiconductor 0.10

15 nm QW 0.20

inverted 0.15

E2 semiconductor 0.10

0.05

0.00

-0.05

k

=(k x ,k y )

k

|| (1,0)

k

|| (1,1) -0.10

-0.15

-0.20

6 5 4 3

k

(0.01

2 -1 ) 1 0.6

0.8

1.0

d HgTe (100 ) 1.2

1.4

H1 H2 0.50

0.00

-0.05

-0.10

L1 -0.15

1 2 3

k

(0.01

4 -1 ) 5 6 -0.20

Asymmetric HgTe-QW D R = 5-15 meV

First Data

Inverted Bandstructure



6 HgTe HgCdTe HgCdTe



8

type-III QW

HgTe-QW D R = 5-15 meV Signal due to depletion...

First Data

Other Wafer

Symmetric HgTe-QW D R = 0-5 meV Signal less than 10 -4 -5.00E-008 -1.00E-007 -1.50E-007 -2.00E-007 -2.50E-007 -3.00E-007 I: 1->4 U:7-10 -2 -1 0 -V_gate14 [V] 1 2 3

HgTe: Semimetal or Semiconductor

bandstructure zero gap: 1000 500 8 0 -500 6 -1000 7 -1500 -1.0

-0.5

k

0.0

(0.01

) 0.5

1.0

fundamental gap

E

 6 

E

 8   300 meV D.J. Chadi et al. PRB, 3058 (1972)

Using SO: Datta-Das spin FET

B eff

v

B eff

1000

HgTe Quantum-Well

HgTe Hg 0.32

Cd 0.68

Te 6 500 8 0 VBO 8 6 -500 1000 500 0 -500 -1000 7 -1500 -1.0

-0.5

k

0.0

(0.01

) 0.5

well 1.0

-1.0

-1000 7 -0.5

k

0.0

(0.01

) 0.5

1.0

-1500 VBO = 570 meV barrier