Splitting Kramers degeneracy with superconducting phase difference - - PowerPoint PPT Presentation

Splitting Kramers degeneracy with superconducting phase difference

Bernard van Heck, Shuo Mi (Leiden), Anton Akhmerov (Delft)

arXiv:1408.1563

ESI, Vienna, 11 September 2014

Plan

Using phase difference in a Josephson junction as a means of breaking time reversal symmetry.

◮ What does ‘breaking time reversal’ mean? ◮ Why it won’t work. ◮ How to make it work (and why 3 is much better than 2)?

Time reversal breaking in a mesoscopic JJ

Several manifestations:

◮ Splitting of Kramer’s degeneracy (Chtchelkatchev&Nazarov, B´ eri&Bardarson&Beenakker) ◮ Closing of the induced gap ◮ Protected zero energy level crossings (switches in the ground

state fermion parity) P = Pf(iH)

◮ Spectral peak in the DOS (Ivanov, Altland&Bagrets)

ρ(E) = ρ0

• 1 + sin(2πE/δ)

2πE/δ

Setup and formalism

Scattering matrices of electrons and holes: Sh(−E) = S∗

e (E)

Andreev reflection matrix: rA = ieiφi Bound state condition: Se(E)rASh(E)r∗

Aψ = e−2i arccos(E/∆)ψ

Setup and formalism

Scattering matrices of electrons and holes: Sh(−E) = S∗

e (E)

Andreev reflection matrix: rA = ieiφi Bound state condition: S(E)rAS∗(−E)r∗

Aψ = e−2i arccos(E/∆)ψ

Short junction limit

S(E) ≈ S(0) Lowest density of Andreev states, strongest effect phase difference

• n a single state.

Due to unitarity and time reversal symmetry of S the energies are given by En = ±∆

• 1 − Tn sin2(φ/2) (Beenakker)

Why it won’t work

◮ Splitting of Kramers degeneracy ◮ Closing of the gap

1 E/∆ DOS

◮ Protected zero energy level crossings (switches in the ground

state fermion parity)

◮ Spectral peak in the DOS

Why it won’t work

◮ Splitting of Kramers degeneracy :-(

δE ∼ E 2/ET < ∆2/ET

◮ Closing of the gap

1 E/∆ DOS

◮ Protected zero energy level crossings (switches in the ground

state fermion parity)

◮ Spectral peak in the DOS

Why it won’t work

◮ Splitting of Kramers degeneracy :-(

δE ∼ E 2/ET < ∆2/ET

◮ Closing of the gap :-(

|E| ≥ ∆ cos(φ/2)

1 E/∆ DOS

◮ Protected zero energy level crossings (switches in the ground

state fermion parity)

◮ Spectral peak in the DOS

Why it won’t work

◮ Splitting of Kramers degeneracy :-(

δE ∼ E 2/ET < ∆2/ET

◮ Closing of the gap :-(

|E| ≥ ∆ cos(φ/2)

1 E/∆ DOS

◮ Protected zero energy level crossings (switches in the ground

state fermion parity) :-(

◮ Spectral peak in the DOS

Why it won’t work

◮ Splitting of Kramers degeneracy :-(

δE ∼ E 2/ET < ∆2/ET

◮ Closing of the gap :-(

|E| ≥ ∆ cos(φ/2)

1 E/∆ DOS

◮ Protected zero energy level crossings (switches in the ground

state fermion parity) :-(

◮ Spectral peak in the DOS :-(

A big improvement

All the special properties of the spectrum originate from the small number of leads!

Kramers degeneracy splitting

Take a Rashba quantum dot with E ∼ ESO, R lSO, and λ R

15 15 15 15

Kramers degeneracy splitting

Take a Rashba quantum dot with E ∼ ESO, R lSO, and λ R Calculate the splitting between the lowest two Andreev levels

15 15 15 15

Kramers degeneracy splitting

Take a Rashba quantum dot with E ∼ ESO, R lSO, and λ R Calculate the splitting between the lowest two Andreev levels

π 2π φ1 π 2π φ2 0.2 δE/∆

Kramers degeneracy is strongly broken.

Protected level crossings

Once again, try a random quantum dot:

π 2π φ1 π 2π φ2 1 Emin

Protected level crossings

Once again, try a random quantum dot:

π 2π φ1 π 2π φ2 1 Emin

Level crossings are allowed.

Protected level crossings

Are level crossings allowed for any (φ1, φ2)?

π 2π φ1 π 2π φ2 0.2 Podd

Protected level crossings

Are level crossings allowed for any (φ1, φ2)?

π 2π φ1 π 2π φ2 0.2 Podd

No: the gap may only close when all the clockwise phase differences are smaller (or larger) than π.

(Note that this result holds for any junction)

Proof

• 1. The expression for Andreev spectrum:

SrAS∗r∗

Aψ = ω2ψ,

E = ∆ Im ω

Proof

• 1. The expression for Andreev spectrum:

SrAS∗r∗

Aψ = ω2ψ,

E = ∆ Im ω

• 2. The simplified expression for Andreev spectrum:

(SrA − rAST)ψ = 2eiα |E|

∆ ψ

Proof

• 1. The expression for Andreev spectrum:

SrAS∗r∗

Aψ = ω2ψ,

E = ∆ Im ω

• 2. The simplified expression for Andreev spectrum:

(SrA − rAST)ψ = 2eiα |E|

∆ ψ

• 3. S = −ST due to time reversal.

Proof

• 1. The expression for Andreev spectrum:

SrAS∗r∗

Aψ = ω2ψ,

E = ∆ Im ω

• 2. The simplified expression for Andreev spectrum:

(SrA − rAST)ψ = 2eiα |E|

∆ ψ

• 3. S = −ST due to time reversal.
• 4. This means Sψ ≡ ψ′,

S(rAψ) = 2|E|eiα

∆

ψ − (rAψ′)

Proof

• 1. The expression for Andreev spectrum:

SrAS∗r∗

Aψ = ω2ψ,

E = ∆ Im ω

• 2. The simplified expression for Andreev spectrum:

(SrA − rAST)ψ = 2eiα |E|

∆ ψ

• 3. S = −ST due to time reversal.
• 4. This means Sψ ≡ ψ′,

S(rAψ) = 2|E|eiα

∆

ψ − (rAψ′)

• 5. The necessary and sufficient condition for existence of a

unitary S: ∃ψ, ψ′ : ψ| rA |ψ + ψ′| rA |ψ′ = 2|E|

∆

eiχ ψ′|ψ .

Proof II

• 1. We get:

|E| ≥ 1

2 ∆ |ψ| rA |ψ + ψ′| rA |ψ′| .

Proof II

• 1. We get:

|E| ≥ 1

2 ∆ |ψ| rA |ψ + ψ′| rA |ψ′| .

• 2. Graphical solution:
• 3. The lower bound on the gap:

E ≥ ∆ min

i,j cos φi−φj 2

Gap closing and the spectral peak

Both phenomena are visible In the ensemble (averaging over random antisymmetric S)

0.0 0.2 0.4 0.6 0.8

ǫ/∆

10 20

ρ(ǫ) × ∆

RMT

Gap closing and the spectral peak

Both phenomena are visible In the ensemble (averaging over random antisymmetric S)

0.0 0.2 0.4 0.6 0.8

ǫ/∆

10 20

ρ(ǫ) × ∆

RMT

In a single realization (averaging over chemical potential)

π 2π φ2 ≡ 2π − φ1 ∆ ǫ

Rashba dot, lso/R = 0.2, l/R = 0.4

ρ(ǫ) [a.u.]

Conclusions

◮ Superconducting phase difference can strongly break time

reversal symmetry in a Josephson junction.

◮ This requires more than two superconducting leads. ◮ Spin degeneracy is split by a large fraction of ∆. ◮ The induced superconducting gap only closes in a a finite

subregion of the phase space.

Conclusions

Thank you all. The end.