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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
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
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
Coulomb blockade in almost isolated grains
N
S
N
Charge transfered in the grain:
Energy:
Coulomb blockade requires
• low temperature
• large barrier
Experiment
Junction Al/Al2O3/Cu
Finite temperature:
Lafarge et al., 1993
The thermal width remains small
vanishes at
Quantum charge fluctuations at finite coupling
Competing states near degeneracy point
Even side
Odd side
e
2D
S
N
N
S
"vaccum corrections" to ground state energy are different:
e
h
e
e
h
S
N
S
We calculate them in perturbation theory with Hamiltonian:
This gives a correction to the step position (odd plateaus are narrower)
N
Shape of the step (1)
Effective Hamiltonian for low energy processes near
e
(even side)
h
2D
S
Even state(0 electron = 0 q.p.)
odd state (1 electron = 1 q.p.)
N
Schrieffer-Wolf transformation:
Tunnel coupling
Quasiparticule
scattering
Electron-hole pair creation in
the lead
Shape of the step (2)
The difficulty :
creates n
electron/hole pairs
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.
Analogy with Fano problem:
Fermi sea in lead
Discrete state with energy
U < 0 without coupling
Continuum of states with
excitation energies:
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
2e
e
Quantum width of the step:
0
1/2
1
3/2
N
Scenario for even/odd transition
Corrections are small for large junctions
Finite temperature
Excited Fermi sea in lead
Continuum of states
ionisation temperature of the bound state
• Step position hardly changes at T<Tq
• Width behaves nonmonotonically with T
width
Step position
Discrete state
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 pair across the tunnel barrier.
Experimental accuracy?
N
N
Lehnert et al, 2003
not sufficient to test Matveev’s
prediction: