Download Dirac equation 2.Zeeman field acts like vector potential 1.This

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
B. A. Bernevig and S.-C. Zhang PRL2006
Spin filtered edge states
One to one correspondence between
spin and momentum
Generalization of quantum Spin Hall state to 3D
Fu, Kane & Mele PRL 07
Moore & Balents PRB 07
Roy, cond-mat 06
3D Topological insulator Bi1-xSbx
Observation of Dirac dispersion! H  k  σ
D. Hsieh et al. Nature (2008) & Science (2009)
Motivation
In topological insulator,
electrons obey Dirac equation H  k  σ
1.This corresponds to the infinite mass Rashba model
2.Zeeman field acts like vector potential
k  σ  (k  H)  σ
Therefore, spintronics on topological insulator seems promising!
We study magnetotransport on topological insulator.
Setup
Current
Formalism
Hamiltonian
mz
k x  m x  i ( k y  my ) 

H 
  (k + m)  σ
mz
 k x  m x  i ( k y  my )

m1  (mx , my , mz )  m1(sin cos ,sin sin ,cos  ), m2  (0,0, m)
Wavefunctions
 ( x  0) 
 ( x  0) 
1
2E (E  mz )
e
i ( k x  mx ) x
 k x  mx  i (k y  my ) 
r
 i ( k x  mx ) x  k x  mx  i ( k y  my ) 
e




E

m
E

m
2
E
(
E

m
)

z


z

z
 k '  ik 
e ik 'x x  x y 
2E (E  m )
 E m 
t
Boundary condition  ( 0)   ( 0) gives
t
2(k x  mx )
E  mz
E m
k x  mx  i (k y  my )
 (k ' x  ik y )
E  mz
E m
Conductance
1 2
2
 k 'x 
    d t Re  , k x  mx  kF cos, k y  my  kF sin
2 2
E 
m1  0.9E
E Fermi energy
Magnetoresistance
m2  0
(a )
(b )


(c )
Polar angle of F1
Azimuthal angle of F1
ky
ky
kx
kx
m2  0.9E
Magnetoresistance in pn junction
m2  0
(a )
E   m2  k ' x 2  k y 2  V
V  2E


Polar angle of F1
Azimuthal angle of F1
(b )
m2  0.9E
Continuity of wavefunction
Parallel configuration
Anti-parallel configuration
(b )
(a )
Band gap
nn junction
F1
F2
(c )
F2
(d )
F1
pn junction
F1
F1
Fermi level
Fermi level
F2
F2
Inclusion of barrier region
Due to mismatch effect, some barrier region may be formed near the interface.
E  mz 2  (k x  mx )2  (k y  my )2   k '' x 2  k y 2  U   m 2  k ' x 2  k y 2  V
U  , L  0, Z  UL  const.
We find that transmission coefficient is π-periodic with respect to Z
This indicates spin rotation through the barrier
σ
(kx  mx , ky  my , mz )
Magnetoresistance in pn junction
Z  /2
m2  0.9E
Z 0
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
kF l 10  1
kF Fermi velocity
l Mean free path
Y. S. Hor et al., arXiv:0903.4406v2
Localization does not occur and surface state is stable for exchange field smaller
than the bulk energy gap
The characteristic length of the wavefuction   v F / mz
Thomas-Fermi screening length 1/   e2N(E )
Thus, we have  /   E / mz  1 for v F
6  105 m / s for Bi2Se3
H. Zhang et. al, Nat. Phys. 5, 438 (2009).
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.
Overlap integral
Re  k ' x 
1
†
T   d  1  2

k 'x

2


2
(a )
nn junction
1
Incident wavefunction
2
Transmitted wavefunction
(b )
pn junction
m2  0.9E