Robustness of Majorana induced Fractional Josephson Effect

Kam Tuen Law HKUSTJuly 9, 2011 QC11

a)

b)

𝜂𝐿 𝜂𝑅

𝜂𝐿0 𝜂𝑅0

∆0𝑒𝑖𝜙

∆0𝑒𝑖𝜙

∆0

∆0

Single-band superconducting wires

Multi- band superconducting wires

𝛾𝐿𝑛

−𝛾𝐿𝑛

𝛾𝑅𝑛

−𝛾𝑅𝑛

Outline

Majorana Fermions-Properties-In condensed matter systems-Probing Majorana fermions

Fractional Josephson Effect- Single Channel Case- Multi-Channel Case- Chiral Topological Superconductors

Properties of Majorana Fermions

1) Non-locality 3) Satisfy non-Abelian Statistics

Abelian Statistics Non-Abelian Statistics

1|||| ii B 1|||| ii B

1 0

0 1

D. Ivanov PRL (2001)

Properties of Majorana Fermions

𝑆𝑟 2𝑅𝑢𝑂4

Read,Green PRB 2000

𝛾1

𝛾 2

Topological Insulator and Majorana Fermions How to create Majarana Fermions on topological insulator surface?

By proximity effect induced by a superconductor

Vortex hosts Majorana Fermion Fu, Kane PRL 2008

Spin-Orbit Coupled Semiconductors

Alicea PRB 2010

Spin-Orbit Coupled Semiconductors

Majorana Fermion Induced Resonant Andreev Reflection

..2211 chttH DDT

)]0()0()[(1 aitHT

Majorana Fermions at SC/TI interface

Fu, Kane PRL 2009Akhmerov , etc. PRL 2009

Charge Transport Through Neutral Mode

TI

SC

M

)()(0 yxyx iiviivH

*

sH

MMHm

h

eG

2

2121 ii

Resonant Andreev Reflection

TML HHHH

dxxxivH xfL )()(

)]0()0()[(1 aitHT

dxxxivH xmM )()(

KTL, Lee, Ng PRL 2009

2e

e

h

NM Lead Sc

Andreev Reflection

Even number of flux Quantum 0G

Odd number of flux Quantumh

eG

22

Resonant Andreev Reflection

Superconductorelectron hole

Cross Andreevreflection

Even number of vortices

IedtItIIP *2)0(ˆ)(ˆ)0(ˆ

IeP 2

IeP

Local Andreev reflection

Cross Andreevreflection

Fractional Josephson Effect 1

0 2 3

2 2 1 .0

0 .5

0 .0

0 .5

1 .0E

𝜙

𝜙

E

0 2 3

2 2 1

0 .5

0

0 .5

1

𝜙=0

𝜙

Kitaev 2000Kwon etc. 2004

Fractional Josephson Effect 2

𝛾1 𝛾 2

0 2 3

2 2 1

0 .5

0

0 .5

1E

𝜙

¿1>¿

¿0>¿

Robust because for any Vwhich preserve fermion parity

1. Why changes sign?2. Can the crossing survives in the multi-channel case?

Fractional Josephson Effect 3

Potter, Lee PRL 2010

a)

b)

𝜂𝐿 𝜂𝑅

𝜂𝐿0 𝜂𝑅0

∆0𝑒𝑖𝜙

∆0𝑒𝑖𝜙

∆0

∆0

Single-band superconducting wires

Multi- band superconducting wires

𝛾𝐿𝑛

−𝛾𝐿𝑛

𝛾𝑅𝑛

−𝛾𝑅𝑛

KTL,LEE Arxiv:1103.5013

Fractional Josephson Effect 4

Change by 2

0 2 3

2 2 1

0 .5

0

0 .5

1

Fractional Josephson Effect 5

Fractional Josephson Effect 6

To be published KLT, LEE

Conclusion

1.Fractional Josephson effect is robust in the multi-channel case

2.It survives also at finite temperatures

3.It can be used to probe the fermion parity of the Josephson current

4. FJE is immune to disorder in the chiral superconductor case

5.Current phase relation in the chiral superconductor case

shows distinctive features

Thank you