Transcript スライド 1
Quantum transport phenomena with
the edge channels in topological superconductors
@Nagoya U. Sept. 5, 2009
Naoto Nagaosa
Department of Applied Physics
The University of Tokyo
and
Cross-Correlated Materials Research Group, RIKEN
Collaborators:
Y. Tanaka, T. Yokoyama, A.V. Balatsky
Phys. Rev. B (Rapid Communications) Vol. 79 060505 (2009)
Phys. Rev. Lett. Vol.102 166801 (2009)
Phys. Rev. Lett. Vol.103 107002 (2009)
Analogy between
chiral superconductor and QHS
Quantum Hall system
Chiral superconductor
Spontaneous
T-symmetry breaking
e2
H n
h
n : Topological integer
Chiral edge channels
??
Chiral p-wave superconductors Sr2RuO4
Maeno (1994),
Sigrist-Rice
Spin-triplet p-wave
Time-reversal
symmetry broken
Topological index for chirality
Volovik
related to the # of edge channels but not to H
Andreev bound state in SRO
Maeno et al. (01)
voltage
V
charge
accumulation
e2
1
compressible
2
h (k F ) ground state
s
H
Furusaki-Matsumoto-Sigrist (2000)
Current
I
Majorana (real) Fermions
f , f
Usual (complex) fermions
( f f )/ 2
2 1
“half” of the usual (complex) fermion
“real” fermion
Chiral Majorana mode at the edge of spinless p+ip SC (A.Furusaki)
k
c.f. Majorna zero energy bound state at vortex
(Read-Green, Kitaev, Ivanov, D.H.Lee etc.)
2D topological insulator (Quantum Spin Hall system)
Time-reversal symmetric system
Spin current instead of charge current
Spin-orbit interaction
helical edge channels
Kane-Mele New topological matter
Quantum Well of HgTe system
Molenkamp-SC.Zhang
3D Topological insulator
3D generalization
of QSH system
Topological insulator
helical edge channels
H ( p)
odd number of 2D
Dirac surface metal
- Robust against
disorder
- New state of matter
From C.L.Kane’s homepage
Proximity effect of SC and topological insulator
Fu-Kane
A
B
A
B
SC
Ferro
Ferro up
SC
0
Ferro
Ferro down
channels
Chiral Majorana
Chiral Fermion
SC
Helical Majorana
Metal
No channel
Non-centrosymmetric Superconductors
CePt3Si
LaAlO3/SrTiO3 interface
Bauer-Sigrist et al.
H 0 ck ( k (k ) )ck
k
(k ) (k ) Time-reversal
(k ) (k )
Space-inversion
Mixture of spin singlet
and triplet pairings
Possible helical
superconductivity
M. Reyren et al 2007
Edge modes of various systems
Majorana
fermion
k k
robust
susceptible
Chiral
Chiral
Helical
Spinless
Majorana Fermion Majorana Fermion
p+ip SC
5/2 FQH
STI+SC
1/3
FQH
Helical
SC
Ferro
wire
Helical
Fermion
Spinful
Fermion
2-Spinful
Fermion
QSHS
Q-wire
Ladder
Purpose of this work
• Charge transport on the surface of topological insulator
via chiral Majorana edge mode(CMM).
• Influence of magnetization on CMM.
• Tunneling conductance in N/FI/S junction
• Josephson current in S/FI/S junctions
• Helical Majorana edge modes in non-centrosymmetric SC
Hamiltonian for the surface state
of Topological insulator
m plays the role of vector potential
N/TI/S
N/TI/S
Chiral Majorana
mode
N/FI/S junction
on top of TI (1)
z
x
y
x
Dispersion of CMM
a
b
Sign change by the direction of mz
c
Chiral Majorana mode (CMM) appears as an Andreev bound state
Change of velocity of Chiral Majorana mode (CMM) by m/mz
Chiral Majorana
mode
N/FI/S junction
on top of TI(2)
z
y
x
a
b
c
Normalized conductance has a peak at zero voltage
x
Chiral Majorana
mode
N/FI/S junction
on top of TI (3)
z
y
x
c
a
b
c
CMM is also influence by my/mz
x
Chiral Majorana
mode
S/FI/S junction
on top of TI (1)
y
x
N: Transparency of the junction
j: Phase difference
CMMs
a
b
c
S/FI/S junction
on top of TI (2)
y
Anomalous current phase relation by mx
a
b
c
Anomalous current phase relation
can be detected by interferometer
Non-centrosymmetric Superconductors
CePt3Si
LaAlO3/SrTiO3 interface
Bauer-Sigrist et al.
H 0 ck ( k (k ) )ck
k
(k ) (k ) Time-reversal
(k ) (k )
Space-inversion
Mixture of spin singlet
and triplet pairings
Possible helical
superconductivity
M. Reyren et al 2007
Rashba superconductor
H k (k k 2 D ) k s ki y k p ki(d (k ) ) y k h.c.
k
Chiral base
k
1
1
i k
(ck e ck ), k
(ck e i k ck )
2
2
H (k | k |)ckck ( s p )eik ckck ( s p )eik ckck h.c.
k
Both + and – bands are p+ip superconductor
ky
k
k
Frigeri et al. 2004
kx
Fu-Kane, 2008
Proximity effect of 3D topological
insulator and s-wave SC
Andreev bound state energy dispersion
Low energy limit
Helical edge modes appear only when
Kramer’s pair of
Majorana edge modes
Angle resolved
Andreev reflection
Normal metal
Helical
superconductor
Doppler shift induces
spin current
Ay (0) H z
Super current
Normal metal
Doppler/Zeeman kF 1
Helical
SC
Magnetic field
-0.4
a : H 0, b : H 0.2H 0 ,
c : H 0.2H 0 , d : H 0.4H 0
0.4
H0
h
e
0.02T
Split electrons into fractions
L
R
R or L
or
positiveor negativeenergy
8 pieces of fractions !!
R xR
R
L
harmonic
oscillator
etc.
Various combination of ' s
can be fixed by el - el interaction
Recombination of pieces
robust
susceptible
Chiral
Chiral
Helical
Spinless
Majorana Fermion Majorana Fermion
p+ip SC
5/2 FQH
STI+SC
1/3
FQH
Helical
SC
Ferro
wire
Helical
Fermion
Spinful
Fermion
2-Spinful
Fermion
QSHS
Q-wire
Ladder
Conclusions
1. Topological insulators and non-centrosymmetric SC
with T-symmetry as new comers
2. Manipulation of the Majorana fermion, Andreev
reflection, and Josephson junction by magnetization
direction transport perpendicular to edge
3. Spintronics functions in superconductors
using helical edge channels
4. All kinds of edge channels
- chiral, helical, Majorana, etc
- electrons are split into 8 pieces
- Recombine some of the pieces to produce a new state