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

Andreev Billiards 147
〈ρ(E)〉 = 2E
δ (E2 − E 2 T ) −1/2 ,E>Egap = ET . (51)
In this limit the density of states of the Andreev billiard has the same form
as in the BCS theory for a bulk superconductor [44], with a reduced value of
the gap (“minigap”). The inverse square-root singularity at the gap is cut off
for any finite Γ , cf. Fig. 6.
6.3 Effect of Impurity Scattering
Impurity scattering in a chaotic Andreev billiard reduces the magnitude of the
excitation gap by increasing the mean time τdwell between Andreev reflections.
This effect was calculated by Vavilov and Larkin [40] using the method of
impurity-averaged Green functions [21]. The minigap in a disordered quantum
dot is qualitatively similar to that in a disordered NS junction, cf. Sect. 3.
The main parameter is the ratio of the mean free path l and the width of the
contact W . (We assume that there is no barrier in the point contact, otherwise
the tunnel probability Γ would enter as well.)
For l ≫ W the mean dwell time saturates at the ballistic value
τdwell = 2π�
Nδ
= πA
vF W
, if l ≫ W. (52)
In the opposite limite l ≪ W the mean dwell time is determined by the twodimensional
diffusion equation. Up to a geometry-dependent coefficient c of
order unity, one has
τdwell = c A
vF l ln(A/W 2 ) , if l ≪ W. (53)
The density of states in the two limits is shown in Fig. 7. There is little
difference, once the energy is scaled by τdwell. Forl ≫ W the excitation gap
is given by the RMT result Egap =0.300 �/τdwell, cf.(49). For l ≪ W Vavilov
and Larkin find Egap =0.331 �/τdwell.
6.4 Magnetic Field Dependence
A magnetic field B, perpendicular to the billiard, breaks time-reversal symmetry,
thereby suppressing the excitation gap. A perturbative treatment remains
possible as long as Egap(B) remains large compared to δ [45].
The appropriate RMT ensemble for the isolated billiard is described by
the Pandey-Mehta distribution [9, 46]
�
P (H) ∝ exp
×
M�
i,j=1
− π2 (1 + b 2 )
4Mδ 2
� (Re Hij) 2 + b −2 (Im Hij) 2� �
. (54)