Biomedical Engineering Reference
In-Depth Information
7.3
Blocking States
The dynamics of SETs is essentially described by sequential tunnelling events at the
source and drain lead which connect the many-body eigenstates of the system. It is
natural to define, in this picture, a blocking state as a state which the system can
enter but from which it cannot escape. When the system occupies a blocking state
the particle number cannot change in time and the current vanishes. If degenerate
states participate in transport, they can lead to interference since, like the two arms
of an electronic interferometer, they are populated simultaneously (see Fig.
7.1
). In
particular, depending on the external parameters they can form linear superpositions
which behave as blocking states. If a blocking state is the linear combination of
degenerate states, we call it
interference blocking state
.
We present in this section the general criteria for the identification of a blocking
state and more specifically of an interference blocking state. First of all we will
proceed to a classification of the tunnelling processes needed for a many-body
description of the electron transport through a nanojunction.
7.3.1
Classification of the Tunnelling Processes
For the description of the tunnelling dynamics contained in the superoperator
L
tun
(see Eqs. (
7.5
)and(
7.6
)), it is convenient to classify all possible tunnelling
events according to four categories: (i) Creation (Annihilation) tunnelling events
that increase (decrease) by one the number of electrons in the system, (ii) Source
(Drain) tunnelling that involves the lead with the higher (lower) chemical potential,
(iii)
) tunnelling that involves an electron with spin up (down) with respect of
the corresponding lead quantization axis, (iv) Gain (Loss) tunnelling that increases
(decreases) the energy in the system.
Using categories (i)-(iii) we can efficiently organize the matrix elements of the
system component of
H
tun
in the matrices:
↑
(
↓
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
γ
S
↑
γ
S
↓
γ
D
↑
γ
D
↓
γ
S
↑
γ
S
↓
γ
D
↑
γ
D
↓
T
N
,
EE
=
T
N
,
EE
=
(7.13)
where
S
,
D
means source and drain, respectively, and
γ
χσ
=
∑
i
(
t
i
σ
)
∗
N
+
1
,{
,
τ
},
E
|
d
i
σ
|
N
,{,
τ
},
E
(7.14)
is a matrix in itself, defined for every creation transition from a state with particle
number
N
and energy
E
to one with
N
1 particles and energy
E
. We indicate
correspondingly in the following transitions involving
+
γ
S
σ
γ
D
σ
and
as source-
creation and drain-creation transitions. The compact notation
{,
τ
}
indicates all
