Biomedical Engineering Reference
In-Depth Information
Fig. 7.2 Energetically
available transitions from an
N particle level. The
patterned rectangles indicate
the energy range of
energetically available source
(S) and drain (D) transitions
both to states with N
+
1and
N
1 particles. The arrows
show examples of both
allowed and forbidden
transitions. From [ 22 ]
These are states with zero transition elements to all other relevant states. Within the
subspace with N particles and energy E the decoupled states span the vector space:
ker T N , EE
ker T N , EE
D N , E =
(7.16)
E
where E is the energy of a relevant state with N
1 particles, respectively.
The function ker M returns the null space of the linear application associated with
the matrix M .
The decoupled space
+
1or N
D N , E as presented in Eq. ( 7.16 ) is constructed as follows.
Let us consider a generic many-body state
| ψ NE
with N particles and energy E and
. The vector T N , EE v has thus
let v be the vector of its components in the basis
|
N
τ
E
E )
4
×
mul
(
N
+
1
,
components and consists of all possible transition amplitudes
1 particles and energy E . Consequently
ker T N , EE contains the vectors v associated with states with N particles and energy
E which are decoupled from all possible states with N
from
| ψ
to all possible states with N
+
NE
1 particles and energy E .
Analogously holds for the significance of ker T N , EE . The intersections in ( 7.16 )and
the condition on E ensure that
+
D N , E contains only states decoupled at the same time
from all other states relevant for transport in the stationary regime. We emphasize
that, due to the condition on the energy E , the decoupled space
D N , E is a dynamical
concept that depends on the applied gate and bias across the I-SET. The coupled
space
D N , E in the Hilbert space with N
particles and energy E . The blocking states belong to it.
As a first simple application of the ideas presented so far, let us consider the SET
at zero bias. According to Table 7.1 the system can only undergo loss tunnelling
events and the global energy minimum is the only blocking state, in accordance
with the observation that the system is in equilibrium with the leads and that we
C N , E is the orthogonal complement of
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