4 Coulomb blockade

4.3 Single-electron transistor 75
At zero bias voltage the system is equivalent to the single-electron box
with two contacts instead of one, and the minimum energy, as well as the
average number of excess electrons, are defined in the same way. At finite bias
voltage V = VL − VR the current flows through the system. To calculate this
current we use the sequential tunneling approach and master equation, valid
in the limit of weak system-to-lead coupling (see sec. 3.4). To describe the
transport we should calculate the nonequilibrium probability p(n) of different
states n, which is not described now by the Gibbs distribution as for the
single-electron box. First of all, let us determine the transition rates in the
sequential tunneling regime.
4.3.1 Tunneling transition rates
The transition rate is determined by the golden-rule expression. For example,
the transition from the state |n〉 to the state |n +1〉 due to the coupling to
the left lead, is determined by the full probability of tunneling of one electron
from any state |k〉 in the left lead to any single-particle state |α〉:
n+1 n
ΓL = 2π
¯h
2π
¯h
n +1| ˆ
2
HTL|n δ(Ei − Ef) =
|Vkα| 2 fk (1 − fα) δ(Eα + ∆E + n − Ek), (4.31)
kα
the sum here is over all electronic states |k〉 in the left lead and all singleparticle
states |α〉 in the system in the sense of the constant-interaction model
(4.17), ˆHTL is the part of the tunneling Hamiltonian describing coupling to
the left lead. All single-particle states are assumed to be thermalized and incoherent.
The Fermi distribution functions fk and (1 − fα) describe probability
CL
L System
R
Gate
VG
CR
VL R V
CG
Fig. 4.4. A single-electron transistor.