Quantum charge fluctuation in a superconducting grain

Manuel Houzet

SPSMS, CEA Grenoble

In collaboration with

L. Glazman (University of Minnesota)

D. Pesin (University of Washington)

A. Andreev (University of Washington)

Ref: Phys. Rev. B 72, 104507 (2005)

Isolated superconducting grains

• In "large" grains, conventional Bardeen-Cooper-Schrieffer theory applies:

The gap in the grain obeys the self consistency equation:

Thermal fluctuations (Ginzburg-Levanyuk criterion):

Same criterion:

Mean level spacing:

Bulk gap

at

Anderson, 1959

gapped spectrum

normal spectrum

gap in the grain

at

Parity effect in isolated superconducting grains

• The number of electrons in the grain is fixed → parity effect

Parity effect subsists till ionisation temperature:

Averin and Nazarov, 1992 Tuominen et al., 1992

Free energy difference at low temperature:

N

Coulomb blockade in almost isolated grains

S NCharge transfered in the grain:

Energy:

Coulomb blockade requires

• low temperature

• large barrier

Lafarge et al., 1993Junction Al/Al2O3/CuExperiment

Finite temperature:

vanishes at

The thermal width remains small

Quantum charge fluctuations at finite coupling

Even side

SS NN

2

Odd side

SS NN

e

Competing states near degeneracy point

SS NN SS NN

e

"vaccum corrections" to ground state energy are different:

We calculate them in perturbation theory with Hamiltonian:

This gives a correction to the step position (odd plateaus are narrower)

e he

h

SS NN

e h

2

Effective Hamiltonian for low energy processes near (even side)

Tunnel coupling Quasiparticule scattering

Electron-hole pair creation in the lead

Schrieffer-Wolf transformation:

Even state(0 electron = 0 q.p.)

odd state (1 electron = 1 q.p.)

Shape of the step (1)

Simplification :

For a large junction, only the states with 0 ou 1 electron/hole pair are important in all orders.

The difficulty :

creates n electron/hole pairs

diverges at

Perturbation theory diverges in any finite order

Shape of the step (2)

Fermi sea in lead

Analogy with Fano problem:

Continuum of states with excitation energies:

Discrete state with energy U < 0 without coupling

1/20 3/2

2e

1

e

Scenario for even/odd transition

quantum mechanics for a single particule in 3d space + potential well

• The bound state forms only if the well is deep enough:

• Its energy dependence is close to

Quantum width of the step:

Corrections are small for large junctions

N

ionisation temperature of the bound state

Finite temperature

Ste

p po

sitio

n

width

• Step position hardly changes at T<Tq

• Width behaves nonmonotonically with T

Excited Fermi sea in lead

Continuum of statesDiscrete state

not sufficient to test Matveev’s prediction:

Conclusion

quantum phase transition in presence of electron-electron interactions

• N-I-N multichannel Kondo problem (idem for S-I-N at Δ>Ec)Matveev, 1991

• S-I-S Josephson coupling → avoided level crossingBouchiat, 1997

• N-I-S abrupt transitionMatveev and Glazman, 1998

• S-I-N at Δ<Ec = new class: charge is continuous, differential capacitance is not

Physical picture of even/odd transition:

bound state formed by an electron/hole pair across the tunnel barrier.

Experimental accuracy?

Lehnert et al, 2003

N N