Download Spin filtering in mesoscopic systems

Spin filtering in mesoscopic
systems
Shlomi Matityahu, Amnon Aharony,
Ora Entin-Wohlman
Ben-Gurion University of the Negev
Shingo Katsumoto
University of Tokyo
Spintronics (spin electronics)
• Spintronics – Study of spin degrees of
freedom in condensed matter systems
• Typical questions –
• How to polarize and manipulate effectively a spin
system?
• How long is the system able to remember its spin
orientation?
• How can spin be detected?
• Potential applications –
• Giant magnetoresistance (GMR)
read/write head
• Quantum computer??
Disk
Spin filtering
• Generate spin-polarized current out of an
unpolarized source
Unpolarized
beam
Spin
filter
Polarized
beam
• Can we find a system which generates a full
polarization in a tunable direction?
The Aharonov-Bohm (AB) effect
• An electron travelling from point A to point B
in a region with zero magnetic field, but nonzero vector potential A
,
acquires
a
phase
B
 AB 
e
c
 A  dr
A

  2
0
• The phase acquired in a close loop is
 - The magnetic flux through the surface
enclosed by the loop
hc
0 
- Flux quantum
e
Spin-Orbit interaction (SOI)
• Non-relativistic limit of the Dirac equation
H SO 
   p  V 
2
 2mec 
• Rashba SOI - In a 2DEG confined to a plane by
an asymmetric potential along z
H
R
SO
kR

p y x  px y 

me
• This is equivalent to an effective
(momentum-dependent) magnetic field
• The strength of the Rashba term can be
tuned by a gate voltage!
The Aharonov-Casher (AC) effect
• Magnetic moment in an electric field also
acquires a quantum mechanical phase
1 
e

H
p

A
SO 

2me 
c

ASO 
4me c
 E
2
 AC
B
e

ASO  dr

cA
• In contrast to the AB phase, the AC phase is
given by an SU(2) rotation matrix U  eiK 
Quantum networks – Tightbinding approach
• Discrete Schrödinger equation
binding Hamiltonian
   m   m
tight-
  J mnU mn  n
n
 m - 2-component spinor at site m
 m - site energy
J mn - hopping amplitude (a scalar)
U mn  ei eiK  - unitary matrix representing AB
and AC phases
mn
mn
Our spin filter – A simple exercise
in quantum mechanics
Derivation of spin filtering
• In general, one has to solve for the
transmission matrix of the quantum network
and then look for the filtering conditions
n
ikna
 ikna

e


re
r

in

ik  n 1 a
te
t


t t  T in
T  t nˆ ' nˆ  t nˆ ' nˆ
• The main conclusion – we can achieve full
spin filtering in a tunable direction provided
we use both SOI and AB.
Additional conclusions
• The main conclusion – we can achieve full
spin filtering in a tunable direction provided
we use both SOI and AB
• The spin filter can also serve as a spin reader
• Spin filtering is robust against current leakage