Fig. 2.15 shows the energy diagram of the
inelastic and
elastic co-tunneling.

Electrons are allowed to tunnel via an intermediate virtual state where first order tunneling would be suppressed. Suppose an electron cannot tunnel directly from jail to ocean. Also tunneling from jail to top is impossible because of missing energy. Nevertheless, an electron will escape to ocean via an intermediate virtual state, where two simultaneous tunnel events have an overall negative change in free energy. One could picture this process in the following simplified way. An electron starting at jail overcomes the energy difference to top, violating the energy conservation for a very short time allowed by Heisenberg's uncertainty principle. If a different electron from top tunnels in the same very short time to ocean, then overall an electron escaped from jail to ocean. This process is called inelastic co-tunneling, because it produces an electron-hole excitation in the island which is eventually dissipated through carrier-carrier interactions. A second process, the elastic co-tunneling corresponds to the same electron tunneling into and out of a virtual state. An electron tunnels through either one junction, travels through the island, and finally tunnels out of the island through the other junction. The phase of the electron is preserved which makes elastic co-tunneling a coherent process. Elastic co-tunneling strongly depends on the internal structure of the island. Usually inelastic co-tunneling is dominant in comparison to elastic co-tunneling except at very small bias voltages and temperatures or very low energy state densities in the quantum dot [10] [11]. Inelastic co-tunneling in small normal-metal tunnel junctions, as opposed to superconducting tunnel junctions, was first experimentally observed by L. Geerligs et al. [40], in a silicon quantum dot by H. Matsuoka and S. Kimura [84], and elastic co-tunneling by A. Hanna et al. [50]. Co-tunneling is a major source of errors in SET devices. Especially in SET logic devices that rely on the presence or absence of a single or few number of electrons, co-tunneling is an important issue to consider.

Using Fermi's golden rule
(Appendix E) [30],
the second order co-tunneling rate may be written
as [11]

where
is the difference in free energy for tunneling through
the *x*th barrier. The two terms,
and
,
represent
the fact, that the process
could start either with a tunneling in junction one or two. Summing over
all possible initial and final states, the total
tunnel rate for the general
case of *N*th-order co-tunneling is given by [10] [12]
[43]

where the
are the intermediate energy levels electrons tunnel
to and from, and
denotes all permutations of the
numbers
.
An *N*th-order co-tunneling event starts at energy level *F*_{0} and passes through
levels
,
as shown in Fig. 2.16.
denotes the difference to the
initial level,
.
The quantum mechanical amplitudes
of all co-tunneling sequences with the same initial and final states are added
coherently to give the total rate (2.44). For different sequences
with the same initial and final level, the intermediate levels *F*_{x} need
not be necessarily the same. It is not possible
to solve (2.44) in the general case. Second order co-tunneling at
zero temperature is one of the special cases where an analytic solution
exists (see Appendix F).

An approximation introduced by H. Jensen and J. Martinis [56]
for (2.44) is to assume that the free energy difference
is, on the average, equally divided over the electron-hole
excitation energies. Thus,
,
and
(2.46) becomes

The co-tunneling rate may then be analytically calculated as

The result of the perturbative calculation (2.44)
diverges if
there are states with energies *F*_{k} in the window
.
A consequence of this is the divergence of (2.49) for
.
L. Fonseca et al. [32] suggested to
move all
levels *F*_{k} that fall in this window a distance *k*_{B}*T* over the nearest
border in order to avoid all divergences. Another possibility to regularize
the singularities is to take the finite lifetime of
island states into
account [69] or to partially re-sum the infinite perturbation
expansion [72]. A theory for
inelastic co-tunneling through a quantum
dot in the regime of strong tunneling, where one or both contacts are close
to perfect transmission was developed in [36].