Biomedical Engineering Reference
In-Depth Information
measure the energy starting from the equilibrium chemical potential.
1
The potential
V
g
of the gate electrode defines the particle number of the global minimum and, by
sweeping
V
g
at zero bias, one can change the number of electrons on the system
one by one. This situation, the Coulomb blockade, remains unchanged until the bias
is high enough to open a gain transition that unblocks the global minimum. Then,
the current can flow. Depending on the gate this first unblocking transition can be
of the kind source-creation or drain-annihilation. Correspondingly, the current is
associated with
N
↔
N
+
1or
N
↔
N
−
1 oscillations, where
N
is the particle number
of the global minimum.
7.3.3
Blocking Conditions
At
finite
bias
the
condition
which
defines
a
blocking
state
becomes
more
elaborate:
1. The blocking state must be achievable from the global minimum with a finite
number of allowed transitions.
2. All matrix elements corresponding to energetically allowed transitions outgo-
ing from the blocking state should vanish: in particular all matrix elements
corresponding to
E
f
eV
2
only the ones
corresponding to the drain-annihilation and source-creation transitions.
eV
b
2
−
E
block
< −
and for
|
E
f
−
E
block
| <
The first condition ensures the blocking state to be populated in the stationary
regime. The second is a modification of the generic definition of blocking state
restricted to energetically allowed transitions and it can be written in terms of the
tunnelling matrices
T
N
,
EE
and
T
N
,
EE
. For each many-body energy level
|
NE
,the
space spanned by the blocking states reads then:
B
N
,
E
=
B
(
1
)
N
,
E
∩B
(
2
)
N
,
E
∩C
N
,
E
(7.17)
with
P
NE
ker
B
(
1
)
T
N
,
EE
,
N
,
E
=
(
T
D
)
∩
E
NE
ker
T
N
,
EE
,
P
(
T
S
)
ker
T
N
,
EE
∩
B
(
2
)
ker
T
N
,
EE
N
,
E
=
.
(7.18)
E
1
If the equilibrium chemical potential is not set to zero, the many-body energy spectrum should be
substituted with the spectrum of the many-body free energy (
H
sys
−
μ
0
N
)where
μ
0
is the chemical
potential of the leads at zero bias. The rest of the argumentation remains unchanged.
