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

112 M. Pustilnik and L.I. Glazman
for the average value of the elastic co-tunneling contribution to the conductance.
This result is easily generalized to gate voltages tuned away from the
middle of the Coulomb blockade valley. The corresponding expression reads
Gel ∼ GLGR
e 2 /h
�
δE
EC
1
N0 − N ∗ 0
+
1
N ∗ 0 − N0 +1
�
. (46)
and is valid when N0 is not too close to the degeneracy points N0 = N ∗ 0 and
N0 = N ∗ 0 +1(N ∗ 0 is a half-integer number):
min
�
|N0 − N ∗ 0 | , |N0 − N ∗ �
0 − 1|
≫ δE/EC
Comparison of (45) with (40) shows that the elastic co-tunneling mechanism
dominates the electron transport already at temperatures
T � Tel = � ECδE , (47)
which may exceed significantly the level spacing. However, as we will see
shortly below, mesoscopic fluctuations of Gel are strong [41], of the order of
its average value. Thus, although Gel is always positive, see (46), the samplespecific
value of Gel for a given gate voltage may vanish.
The key to understanding the statistical properties of the elastic cotunneling
contribution to the conductance is provided by the observation that
there are many (∼EC/δE ≫ 1) terms making significant contribution to the
amplitude (42). All these terms are random and statisticaly independent of
each other. The central limit theorem then suggests that the distribution of
Ael is Gaussian [20], and, therefore, is completely characterised by the first
two statistical moments,
A el = A ∗ el , A el A el = A∗ el A∗ el = δ β,1 A el A ∗ el (48)
with A∗ elAel given by (43). This can be proven by explicit consideration of
higher moments. For example,
|A4 el | =2�AelA∗ el
�
�2 + δ |A4 el | . (49)
�2 + � � A el A el
The non-Gaussian correction here, δ |A 4 el |∼ |A2 el | (δE/EC), is by a factor of
δE/EC ≪ 1 smaller than the main (Gaussian) contribution.
It follows from (44), (48), and (49) that the fluctuation of the conductance
δGel = Gel − Gel satisfies
δG 2 el
2 � �2
= Gel . (50)
β
Note that breaking of time reversal symmetry reduces the fluctuations by a
factor of 2, similar to conductance fluctuations in bulk systems, whereas the
average conductance (46) is not affected by the magnetic field.