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

Andreev Billiards 145
The average over the distribution (38) can be done diagrammatically
[41, 42]. To leading order in 1/M and for E ≫ δ only simple (planar) diagrams
need to be considered. Resummation of these diagrams leads to the
selfconsistency equation [22, 37]
G =[E + W−(Mδ/π)σzGσz] −1 � �
1 0
, σz =
. (40)
0 −1
This is a matrix-generalization of Pastur’s equation in the RMT of normal
systems [43].
The matrices in (40) havefourM × M blocks. By taking the trace of each
block one obtains an equation for a 4 × 4 matrix,
G = 1
M�
� �−1 πE/Mδ − G11 ˜wm + G12
, (41)
M ˜wm + G21 πE/Mδ − G22
m=1
�
2 π wm/M δ if m =1, 2,...N ,
˜wm =
(42)
0 if m = N +1,...M .
Since G22 = G11 and G21 = G12 there are two unknown functions to determine.
For M ≫ N these satisfy
G 2 12 =1+G 2 11 , (43a)
2πE
δ G12
N�
= G11 (−G12 +1− 2/Γn) −1 , (43b)
n=1
where we have used the relation (31) between the parameters wn and the transmission
probabilities Γn. Equation (43) has multiple solutions. The physical
solution satisfies limE→∞〈ρ(E)〉 =2/δ, when substituted into
〈ρ(E)〉 = −(2/δ)ImG11(E) . (44)
In Fig. 6 we plot the density of states in the mode-independent case Γn ≡
Γ , for several vaues of Γ . It vanishes as a square root near the excitation
gap. The value of Egap can be determined directly by solving (43) jointly with
dE/dG11 = 0. The result is
k6 − k4 (1 − k) 6 x6 − 3k4 − 20k2 +16
(1 − k) 4 x 4 + 3k2 +8
(1 − k) 2 x2 =1,
x = Egap/ET ,k=1− 2/Γ , ET = NΓδ/4π . (45)
For later use we parametrize the square-root dependence near the gap as
〈ρ(E)〉 → 1
�
E − Egap
π ∆3 , E → Egap . (46)
gap