Quantum charge fluctuation in a superconducting grain Manuel Houzet SPSMS, CEA Grenoble In...
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Transcript of Quantum charge fluctuation in a superconducting grain Manuel Houzet SPSMS, CEA Grenoble In...
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