4 Coulomb blockade

4.5 Cotunneling
4.5 Cotunneling 87
Until now we have considered only the sequential tunneling regime, which is
described by the first-order perturbation in tunneling. At low temperatures
the sequential tunneling is suppressed by the Coulomb blockade (excepting
the degeneracy points), and the conductance of the system become exponentially
small. In this case the second-order processes start to be important.
These processes include the events, when two electrons with different energy
participate in tunneling simultaneously (so-called inelastic cotunneling), or
one electron tunnels coherently twice (elastic cotunneling), which is equivalent,
however, to the simultaneous tunneling of two electrons with the same
energy. Note, that both terms ”inelastic” and ”elastic” have no relation to the
presence or absence of the additional inelastic interactions inside the system.
The standard second-order perturbation theory yields the transition rate
from some initial state |i〉 to some final state |f〉
Γ (2) 2π
i→f =
¯h
f
ˆHT
λ λ ˆHT
2
i
Eλ − Ei
δ(Ei − Ef ). (4.62)
λ
Theintermediatestates|λ〉 are virtual states with Eλ >Ei. Inthispointwe
already assume that the system is in the Coulomb blockade regime with fixed
number of electrons n, and the temperature is low T ≪ ∆E + n , because in the
opposite case thermally excited quasiparticles in the leads can overcome the
energy barrier, and the sequential tunneling through high-energy states will
determine the current.
Fig. 4.13. Cotunneling processes, elastic (a,c) and inelastic (b,d).
(from Aleiner et al.).