Quantum charge fluctuation in a superconducting grain

1. Quantum charge fluctuation in a superconducting grain Manuel Houzet SPSMS, CEA Grenoble In collaboration with L. Glazman (University of Minnesota) D. Pesin (University ofWashington) A. Andreev (University of Washington) Ref: Phys. Rev. B 72, 104507 (2005)

2. Isolated superconducting grains Anderson, 1959 • In "large" grains, conventional Bardeen-Cooper-Schrieffer theory applies: Mean level spacing: normal spectrum Bulk gap gap in the grain gapped spectrum The gap in the grain obeys the self consistency equation: at Thermal fluctuations (Ginzburg-Levanyuk criterion): Same criterion: at

3. Parity effect in isolated superconducting grains • The number of electrons in the grain is fixed → parity effect Free energy difference at low temperature: Parity effect subsists till ionisation temperature: Averin and Nazarov, 1992 Tuominen et al., 1992

4. Coulomb blockade requires • low temperature • large barrier Coulomb blockade in almost isolated grains N S N Charge transfered in the grain: Energy:

5. Finite temperature: The thermal width remains small vanishes at Experiment Junction Al/Al2O3/Cu Lafarge et al., 1993

6. "vaccum corrections" to ground state energy are different: e e e h h S N S N We calculate them in perturbation theory with Hamiltonian: This gives a correction to the step position (odd plateaus are narrower) Quantum charge fluctuations at finite coupling Competing states near degeneracy point Even side Odd side e 2D N S N S

7. Even state(0 electron = 0 q.p.) odd state (1 electron = 1 q.p.) Schrieffer-Wolf transformation: Quasiparticule scattering Electron-hole pair creation in the lead Tunnel coupling Shape of the step (1) Effective Hamiltonian for low energy processes near (even side) e h 2D S N

8. diverges at Perturbation theory diverges in any finite order 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 Analogy with Fano problem: Fermi sea in lead Discrete state with energy U < 0 without coupling Continuum of states with excitation energies: Shape of the step (2)

9. 2e e Quantum width of the step: 0 3/2 1/2 1 Scenario for even/odd transition Corrections are small for large junctions 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

10. Finite temperature Excited Fermi sea in lead Discrete state Continuum of states Step position width ionisation temperature of the bound state • Step position hardly changes atT<Tq • Width behaves nonmonotonically withT

11. N N 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 crossing • Bouchiat, 1997 • N-I-S abrupt transition • Matveev 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 pairacross the tunnel barrier. Experimental accuracy? not sufficient to test Matveev’s prediction: Lehnert et al, 2003