Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

154 C.W.J. Beenakker
P(x)
0.4
0.3
0.2
0.1
0
0 1 2
E/E T
2
1
0
–6 –4 –2 0 2 4 6
x
Fig. 12. Main plot: Gap distribution for the Andreev kicked rotator with parameters
M =2π/δ = 8192, kicking strength K = 45, and M/N = τdwell =10(⋄), 20 (•), 40
(+), and 50 (×). There is no magnetic field. The solid line is the RMT prediction
(62). Inset: Average density of states for the same system. The solid line is the RMT
prediction (49). (Deviations from perturbation theory are not visible on the scale of
the inset.) Adapted from [27]
6.7 Coulomb Blockade
Coulomb interactions between electron and hole quasiparticles break the
charge-conjugation invariance (37) of the Hamiltonian. Since Andreev reflection
changes the charge on the billiard by 2e, this scattering process becomes
energetically unfavorable if the charging energy EC exceeds the superconducting
condensation energy (Josephson energy) EJ. ForEC > ∼ EJ one obtains the
Coulomb blockade of the proximity effect studied by Ostrovsky, Skvortsov,
and Feigelman [55].
The charging energy EC = e 2 /2C is determined by the capacitance C of
the billiard. The Josephson energy is determined by the change in free energy
of the billiard resulting from the coupling to the superconductor,
� ∞
EJ = − [ρ(E) − 2/δ] EdE . (71)
0
The discrete spectrum E < Egap contributes an amount of order E2 gap/δ
to EJ. In the continuous spectrum E > Egap the density of states ρ(E),
calculated by RMT, decays ∝ 1/E2 to its asymptotic value 2/δ. This leads
to a logarithmic divergence of the Josephson energy [37, 56], with a cutoff set
by min(∆, �/τerg):
EJ = E2 gap
δ
� �
min(∆, �/τerg)
ln
. (72)
Egap