Biomedical Engineering Reference
In-Depth Information
Tabl e 7. 1
Energy conditions for tunnelling transitions
between the many-body eigenstates of the system. The
quantity
E
i
is the difference between the
energies of the final and initial many-body states of
the system involved in the transition. The bias energy
eV
b
is assumed to be positive. From [
22
]
Tunnelling process
Δ
E
=
E
f
−
Energy condition
Δ
E
≤
+
eV
b
/
Source-Creation
2
Δ
E
≤−
eV
b
/
Source-Annihilation
2
Drain-Creation
Δ
E
≤−
eV
b
/
2
Drain-Annihilation
Δ
E
≤
+
eV
b
/
2
possible combinations of the quantum numbers
and
τ
. It follows that the
γ
χσ
E
)
×
size of
gives
the degeneracy of the many-body energy level with
N
particles and energy
E
.
Analogously
is mul
(
N
+
1
,
mul
(
N
,
E
)
where the function mul
(
N
,
E
)
t
i
σ
γ
χσ
=
∑
i
,{
,
τ
},
E
|
N
−
1
d
i
σ
|
N
,{,
τ
},
E
(7.15)
accounts for the annihilation transitions.
The fourth category concerns energy and it is intimately related to the first and
the second. Not all transitions are in fact allowed: due to the energy conservation
and the Pauli exclusion principle holding in the fermionic leads, the energy gain
(loss) of the system associated with a gain (loss) transition is governed by the bias
voltage. These energy conditions, for the case of equal potential drop at the source
and drain lead (
c
=
1
/
2), are summarized in Table
7.1
and illustrated in Fig.
7.2
.
E
i
is the difference between the energy of the final
and initial state of the system and the condition
The quantity
Δ
E
:
=
E
f
−
is in reality smoothed due to the
thermal broadening of the Fermi distributions. For simplicity we set the zero of the
energy at the chemical potential of the unbiased device. In Table
7.1
one reads, for
example, that in a source-creation tunnelling event the system can gain at maximum
eV
b
2
or that in a source-annihilation and drain-creation transition the system loses at
least an energy of
eV
2
.
From Table
7.1
one also deduces that, from whatever initial state, it is always
possible to reach the lowest energy state (the global minimum) via a series of
energetically allowed transitions. Vice versa, not all states can be reached starting
from the global minimum. Thus, the only
relevant states
for the transport in the
stationary regime are the states that can be reached from the global minimum via a
finite number of energetically allowed transitions.
≤
7.3.2
Subspace of Decoupled States
In the process of detecting the blocking states we observe first that some states do
not participate in the transport and can be excluded a priori from any consideration.
