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

136 C.W.J. Beenakker
F
E x 8d/hv
gap
0.1
0.05
normal metal
superconductor
0
0 2 4 6 8 10 12 14 16 18 20
Fig. 3. Excitation gap Egap of a disordered NS junction, as a function of the ratio of
the thickness d of the normal metal layer and the mean free path l. The curve in the
bottom panel is calculated from the disorder-averaged Green function (for ξ0 ≪ d, l).
The top panel illustrates the geometry. The normal metal layer has a specularly
reflecting upper surface and an ideally transmitting lower surface. Adapted from
[20]
�
0.43 �vF /l , if d/l ≪ 1 ,
Egap =
0.78 �D/d2 (9)
, if d/l ≫ 1 ,
with D = vF l/3 the diffusion constant in the normal metal.
The minigap in a ballistic quantum dot (Andreev billiard) differs from
that in a disordered NS junction in two qualitative ways:
1. The opening of an excitation gap depends on the shape of the boundary,
rather than on the degree of disorder [22]. A chaotic billiard has a gap
at the Thouless energy ET � �/τdwell, like a disordered NS junction. An
integrable billiard has a linearly vanishing density of states, like a ballistic
NS junction.
2. In a chaotic billiard a new time scale appears, the Ehrenfest time τE, which
competes with τdwell in setting the scale for the excitation gap [23]. While
τdwell is a classical �-independent time scale, τE ∝|ln �| has a quantum
mechanical origin.
Because one can not perform a disorder average in Andreev billiards, the
Green function formulation is less useful than in disordered NS junctions.
Instead, we will make extensive use of the scattering matrix formulation, explained
in the next section.
d/l
d