Spintronics with topological insulator

Takehito Yokoyama, Yukio Tanaka*, and Naoto Nagaosa

Department of Applied Physics, University of Tokyo, Japan*Department of Applied Physics, Nagoya University, Japan

arXiv:0907.2810

Edge states

From http://www.physics.upenn.edu/~kane/

Quantum Spin Hall state

Kane and Mele PRL 2005

Generalization of quantum Spin Hall state to 3DFu, Kane & Mele PRL 07Moore & Balents PRB 07Roy, cond-mat 06

One to one correspondence between spin and momentum

B. A. Bernevig and S.-C. Zhang PRL2006

Spin filtered edge states

3D Topological insulator Bi1-xSbx

D. Hsieh et al. Nature (2008) & Science (2009)

Observation of Dirac dispersion! H k σ

In topological insulator, electrons obey Dirac equation

2.Zeeman field acts like vector potential

( )k σ k H σ

H k σ

1.This corresponds to the infinite mass Rashba model

Therefore, spintronics on topological insulator seems promising!

Motivation

We study magnetotransport on topological insulator.

Current

Setup

Formalism

1 2

( ) ( )

( )( )

( )

( , , ) (sin cos ,sin sin ,cos ), (0,0, )

( ) ( )1( 0)2 ( ) 2 ( )

(

x x x x

z x x y y

x x y y z

x y z

x x y y x x y yi k m x i k m x

z zz z

m k m i k mH

k m i k m m

m m m m m

k m i k m k m i k mrx e eE m E mE E m E E m

x

1

k +m σ

m m

' '0)

2 ( )

2( )

( ) ( ' )

xik x x y

x x

zx x y y x y

z

k ikt eE mE E m

k mtE mE mk m i k m k ik

E m E m

22

2

'1 Re , cos , sin2

xx x F y y F

kd t k m k k m kE

Hamiltonian

Wavefunctions

Boundary condition gives( 0) ( 0)

Conductance 1 0.9m E

E

Fermi energy

( )a ( )b

xk

yk( )c

xk

yk

Magnetoresistance 2 0m

2 0.9m E

Azimuthal angle of F1

Polar angle of F1

( )a ( )b

Magnetoresistance in pn junction

2 2 2'x yE m k k V

2V E

2 0m 2 0.9m E

Azimuthal angle of F1

Polar angle of F1

( )a

F2F1 F1 F2

( )b

( )c

F2

F1 F1

F2

( )d

Band gap

Fermi level Fermi level

Continuity of wavefunction

Parallel configuration Anti-parallel configuration

nn junction

pn junction

2 2 2 2 2 2 2 2( ) ( ) '' '

, 0, .

z x x y y x y x yUE m k m k m k k m k k V

U L Z UL const

Inclusion of barrier region

We find that transmission coefficient is π-periodic with respect to Z

Due to mismatch effect, some barrier region may be formed near the interface.

( , , )x x y y zk m k m m σ

This indicates spin rotation through the barrier

Magnetoresistance in pn junction

/ 2Z 0Z 2 0.9m E

Opposite tendency due to spin rotation through the barrier region

DiscussionTypical value of induced exchange field due to the magnetic proximity effect would be 5 50 meV (from experiments in graphene and superconductor) ∼

H. Haugen,et al, Phys. Rev. B 77, 115406 (2008).J. Chakhalianet al., Nat. Physics 2, 244 (2006).

E can be tuned by gate electrode or doping below the bulk energy gap ( 100 ∼meV)

Ferromagnet breaks TRS, which would tame the robustness against disoder. However, high quality topological insulator can be fabricated

Y. S. Hor et al., arXiv:0903.4406v210 1Fk l Fkl

Fermi velocityMean free path

Localization does not occur and surface state is stable for exchange field smaller than the bulk energy gap

/F zv m The characteristic length of the wavefuctionThomas-Fermi screening length 21/ ( )e N E

/ / 1zE m for Bi2Se356 10 /Fv m s

H. Zhang et. al, Nat. Phys. 5, 438 (2009).forThus, we have

Conclusion We investigated charge transport in two-dimensional ferromagnet/feromagnet junction on a topological insulator.

The conductance across the interface depends sensitively on the directions of the magnetizations of the two ferromagnets, showing anomalous behaviors compared with the conventional spin-valve.

The conductance depends strongly on the in-plane direction of the magnetization.

The conductance at the parallel configuration can be much smaller than that at the antiparallel configuration.

This stems from the connectivity of wavefunctions between both sides.

( )a ( )b

Overlap integral

†21 2

2

Re '1'

x

x

kT d

k

2 0.9m E

nn junction pn junction

1

2

Incident wavefunction

Transmitted wavefunction