Quantum transport phenomena with the edge channels in topological

superconductors

Naoto Nagaosa Department of Applied Physics

The University of Tokyoand

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)

@Nagoya U. Sept. 5, 2009

Quantum Hall system

Analogy between chiral superconductor and QHS

Chiral superconductor

SpontaneousT-symmetry breaking

nh

eH

2

:n

Chiral edge channels??

Topological integer

Chiral p-wave superconductors Sr2RuO4Maeno (1994), Sigrist-Rice

Spin-triplet p-wave

Time-reversal symmetry broken

Volovik

Topological index for chirality

related to the # of edge channels but not to H

Maeno et al. (01)

Andreev bound state in SRO 　

Current I

V charge accumulation

voltage

2

2

)(

1

F

sH kh

e compressible

ground state

Furusaki-Matsumoto-Sigrist (2000)

“half” of the usual (complex) fermion“real” fermion

Majorana (real) Fermions

ff ,Usual (complex) fermions

2/)( ff 1 2

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 systemSpin current instead of charge currentSpin-orbit interaction

Kane-Mele New topological matter

Molenkamp-SC.Zhang

helical edge channels

Quantum Well of HgTe system

From C.L.Kane’s homepage

3D generalization of QSH systemTopological insulator

helical edge channels

)( pH

odd number of 2D Dirac surface metal- Robust against disorder- New state of matter

3D Topological insulator

A B

Proximity effect of SC and topological insulator

Fu-Kane

A B channels

SC Ferro Chiral Majorana

Ferro up Ferro down Chiral Fermion

SC SC Helical Majorana 0

Ferro Metal No channel

M. Reyren et al 2007

Non-centrosymmetric Superconductors

CePt3Si LaAlO3/SrTiO3 interface

Bauer-Sigrist et al.

kkkk ckcH ))((0

)()( kk

)()( kk

Time-reversal

Space-inversion

Mixture of spin singlet and triplet pairings

Possible helical superconductivity

Edge modes of various systems

Majorana fermion

kk

ChiralMajorana

p+ip SC 5/2 FQHSTI+SC

ChiralFermion

1/3FQH

Helical Majorana

HelicalSC

SpinlessFermion

HelicalFermion

Ferrowire

QSHS

SpinfulFermion

Q-wire Ladder

2-SpinfulFermion

robust susceptible

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

N/TI/S

N/TI/S

m plays the role of vector potential

N/FI/S junction on top of TI (1)

ab

c

Change of velocity of Chiral Majorana mode (CMM) by /mz

xz

Chiral Majorana

mode

x

y

Chiral Majorana mode (CMM) appears as an Andreev bound state

Dispersion of CMM

Sign change by the direction of mz

N/FI/S junction on top of TI(2)

ab

c

Normalized conductance has a peak at zero voltage

xz

Chiral Majorana

mode

x

y

N/FI/S junction on top of TI (3)

ab

c

CMM is also influence by my/mz

xz

Chiral Majorana mode

x

y

c

S/FI/S junction on top of TI (1)

bc

a

Chiral Majorana mode

x

y

CMMs N: Transparency of the junction

: Phase difference

Anomalous current phase relation can be detected by interferometer

bc

a

S/FI/S junction on top of TI (2)

y

Anomalous current phase relation by mx

M. Reyren et al 2007

Non-centrosymmetric Superconductors

CePt3Si LaAlO3/SrTiO3 interface

Bauer-Sigrist et al.

kkkk ckcH ))((0

)()( kk

)()( kk

Time-reversal

Space-inversion

Mixture of spin singlet and triplet pairings

Possible helical superconductivity

..))(()( 2 chkdiikH kykpkykskD

kkk

)(2

1 ),(

2

1

k

ikkk

ikk

ceccec kk Chiral base

..)()(|)|( chcceccecckH kki

pskki

pskkk

kkk

Rashba superconductor

Frigeri et al. 2004

Both + and – bands are p+ip superconductor

xk

yk

k

k Fu-Kane, 2008Proximity effect of 3D topological insulator and s-wave SC

Andreev bound state energy dispersion

Low energy limit Kramer’s pair of

Majorana edge modes

Helical edge modes appear only when

Angle resolved Andreev reflection

Normal metalHelical

superconductor

Normal metalHelical

SC

00

0

4.0 :d ,2.0 :c

,2.0 :b ,0 :a

HHHH

HHH

Doppler shift induces spin current

Magnetic field

Super current

Te

hH 02.00

zy HA )0(

1emanDoppler/Ze Fk

0.4-0.4

Split electrons into fractions

or or LR

energy negativeor positive

8 pieces of fractions !!

R

L

RxR etc.

ninteractio el-elby fixed becan

s' ofn combinatio Various

Recombination of pieces

ChiralMajorana

p+ip SC 5/2 FQHSTI+SC

ChiralFermion

1/3FQH

Helical Majorana

HelicalSC

SpinlessFermion

HelicalFermion

Ferrowire

QSHS

SpinfulFermion

Q-wire Ladder

2-SpinfulFermion

R

L

harmonic oscillator

robust susceptible

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