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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
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.
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 e
E 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 ikte
E mE E m
k mt
E mE mk m i k m k ik
E m E m
22
2
'1Re , cos , sin
2x
x x F y y F
kd t k m k k m k
E
Hamiltonian
Wavefunctions
Boundary condition gives( 0) ( 0)
Conductance 1 0.9m E
E
Fermi energy
( )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
Discussion
Typical 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 Fk
lFermi 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.