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

Low-Temperature Conduction of a Quantum Dot 111
The smallest energy of the electron-hole pair is of the order of δE. At
temperatures below this threshold the inelastic co-tunneling contribution to
the conductance becomes exponentially small. It turns out, however, that even
at much higher temperatures this mechanism becomes less effective than the
elastic co-tunneling.
4.2 Elastic Co-Tunneling
In the process of elastic co-tunneling, transfer of charge between the leads
is not accompanied by the creation of an electron-hole pair in the dot. In
other words, occupation numbers of single-particle energy levels in the dot in
the initial and final states of the co-tunneling process are exactly the same,
see Fig. 3(b). Close to the middle of the Coulomb blockade valley (at almost
integer N0) the average number of electrons on the dot, N ≈ N0, is also
an integer. Both an addition and a removal of a single electron cost EC in
electrostatic energy, see (15). The amplitude of the elastic co-tunneling process
in which an electron is transfered from lead L to lead R can then be written
as
Ael = �
t
n
∗ sign(ɛn)
LntRn (42)
EC + |ɛn|
The two types of contributions to the amplitude Ael are associated with virtual
creation of either an electron if the level n is empty (ɛn > 0), or of a hole if
the level is occupied (ɛn < 0); the relative sign difference between the two
types of contributions originates in the fermionic commutation relations.
As discussed in Sect. 2, the tunneling amplitudes tαn entering (42) are
Gaussian random variables with zero mean and variances given by (21). It is
then easy to see that the second moment of the amplitude (42) is given by
|A2 ΓLΓR
el | =
(πν) 2
�
(EC + |ɛn|) −2 .
n
Since for EC ≫ δE the number of terms making significant contribution to
the sum over n here is large, and since the sum is converging, one can replace
the summation by an integral which yields
|A2 ΓLΓR
el |≈
(πν) 2
1
ECδE
Substitution of this expression into
and making use of (22) gives [39]
Gel = 4πe2 ν 2
�
Gel ∼ GLGR
e 2 /h
�
� A 2 el
. (43)
�
� (44)
δE
. (45)
EC