Anomalous Josephson current through interacting

quantum dots and Majorana nanowires

���� �����

Institut für Theoretische Physik, Düsseldorf

Overview

• Josephson effect in S-TS-S junctions

AZ & Egger (PRB 2012)

• Anomalous Josephson current and Majorana bound states

in spin-orbit coupled quantum dots

Brunetti, AZ, Kundu, Egger (PRB 2013)

AZ, Egger, Jonckheere, Martin (PRL 2009)

Vg

∆ e-iφ/2

∆ eiφ/2

Spin-orbit coupled QD

• QD formed by a confinement potential V within a 2DEG

• semiconductor (InAs, InSb) nanowires as few-level dots

• both SO and Zeeman are (i) spatially uniform, (ii) localized within QD region

• in-plane Zeeman field B (affects only electron spins but not orbital motion)

• gate-tunable Rashba (and/or Dresselhaus) SO coupling α

General model

• multilevel QD with charging energy Ec

• tunnel contacted by two s-wave BCS leads

orbital quantum number n = 1,…, Mspin projection σ=↑,↓

• spin-conserving & spin-indpenedent tunneling

Vg

∆ e-iφ/2

∆ eiφ/2

M x M matrix in orbital space

AZ, Egger, Jonckheere, Martin (PRL 2009)

Anomalous supercurrent in the limit of weak SO, Zeeman

and tunnel asymmetry (off-diagonal Γ):

Necessary conditions for anomalous Josephson effect:

• finite SO

• suitably oriented Zeeman field (B || x)

• QD is a chiral conductor; at least two orbitals are required

Exactly solvable case Ec=0

Equilibrium Josephson current-phase relation:

Basic explanation: Cooper pair transfer through a 2-level QD

+ reverse process

Summing up all relevant processes:

Necessary condition

also holds in the presence

of Coulomb interactions

Full numerical results for 2-level QD

• Current jumps are associated with changes in occupation of relevant Andreev levels

• Inset: Numerical vs analytical results (accurate even for moderate αB)

B/∆ = {0.3, 0.7, 0.9}

B/∆ = 0.5

αL=0.4

Ec ≠ 0: Cotunneling regime (PT in Γ)

In the rotated basis:

depends only on 1-particle couplings(SO & Zeeman are treated exactly)

encodes Coulomb interactions

Anomalous supercurrent:

Interactions can be crucial in breaking balance between TR processes

Necessary condition for Ia ≠ 0:

consistent with

• sizable anomalous supercurrent for arbitrarily weak Zeeman or/and SO

• onset behavior triggered by Coulomb interactions (if ground state is of odd parity)

Incipient TRS breaking for low T<B

QD with two orbital levels(minimal model for SO effects)

Basic explanation of the onset behavior for Ec ≠ 0

µ

• Energy gap to states with different N grows with Ec:

N = 1 state is robust even for very small Zeeman & SOTR partner

Two Kramer‘s partner states

contribute with opposite sign to Ia

Ia

N

• No complete cancelation of TR contributions:

one dominates (while the other is strongly suppressed

by the Coulomb gap), and hence Ia remains finite

Majorana bound states in a double QD

• proximity-induced superconductivity

• SO coupling α

• TRS breaking magnetic field B

Superconducting atomic limit:• ∆ is largest energy scale

• exact result for free energy (beyond PT cotunneling regime)

• effective dot Hamiltonian

Two-site Kitaev‘s chain (effectively spinless fermions)

∆ >> ε+B >> α, Γ, µ, Ec

Effective pairing amplitude (~Γ) = Andreev level energy in 1-channel QPC

… and large-B limit:

“sweet spot” (quadratically

but not topologically protected)

BdG quasiparticles:

Majorana regime

Josephson CPR (for Ec=0):

Conditions for MBS formation:

spatially separated MBSs Re η , Im η- -

CPR through a double dot in the atomic limit

α = 0.4 ΓT = 0.99

MBSs are formed at the boundaries between

ground states with opposite fermion parity

φ/π

I/Γ

Brunetti, AZ, Kundu, Egger (PRB 2013)

0

Josephson junction with a Majorana wire

• topologically nontrivial superconducting (TS) wire

with two Majorana fermion end states

• tunnel coupled to s-wave BCS electrodes

• Coulomb interaction via charging energy (floating geometry)

• bulk TS quasiparticles must be accessible to have a Josephson effect

No supercurrent through isolated Majorana state

Sufficient condition for supercurrent blockade:

S-TS junction

(collinear time-reversal states)Josephson current is blocked unless

noncollinear TR states are available in TS wire

To lowest nontrivial order in tunnel couplings:

Noninteracting S-TS-S junction (Ec=0)

• Cooper-pair transport simultaneously involves MBS & TS continuum states

• JC is additionally suppressed compared to the ususal cotunneling limit

• π-periodic CPR with negative critical current

Interacting S-TS-S junction

Nonlocal auxiliary complex fermionCooper-pair number

Cooper pairs are transported through TS by creation or annihilation of charge 2e

or

(plus conjugate processes)

2nd order PT in Γ:

Anomalous CPR (2π-periodic), no destructive interference of order Γ 2

… may provide experimentally

detectable signatures for MBS

• resonance enhancement

at half-integer δ

(charge degeneracy points

for even- & odd-parity GSs)

• pronounced for large Ec

Anomalous supercurrent in cotunneling regime

even-parity GS odd-parity GS

• parity sensitivity of critical current

(„π-shift“ vs. δ)

AZ & Egger (PRB 2012)

Summary

� Josephson effect in S-TS-S junctions

• mediated by Majoranas in conspiracy with bulk TS quasiparticles,

otherwise „supercurrent blockade“

• charging effects cause anomalous (cosine) CPR

� Combined effects of SO, Zeeman and Coulomb interactions on

Josephson CPR for a multilevel QD

� Majorana bound states (spatially separated but topologically

unprotected) in a double dot Josephson junction

• conditions for anomalous supercurrent

• incipient time-reversal symmetry breaking