Biomedical Engineering Reference
In-Depth Information
Theoretical models show that the typical asymmetric line profile is composed of a
Lorentzian centered at the bonding energy, and a Fano line shape at the energy of the
anti-bonding state [
18
]. In recent experiments a QD molecule has been embedded
between the arms of an AB interferometer, that has a number of tunable parameters,
allowing the transmission to exhibit a large variety of behaviors as a function of
them [
6
]. It has also been experimentally demonstrated that, within the Coulomb
blockade regime and keeping the number of electrons in the dots constant, there is
a phase-coherent contribution that can dominate the measured conductance when
tunneling times are shorter than dephasing times [
9
,
10
,
14
].
8.2.3
Non-Equilibrium Approach to Electronic Transport
One of the approaches for the study of the electron transport is through the use of
Green functions. The Green function of a quantum system
G
t
;
x
,
t
)
propagates
a solution of the Schrodinger equation from a position
x
at the instant
t
to a position
x
at time
t
through
(
x
,
G
x
,
t
)=
x
,
t
;
x
ψ
(
(
,
t
)
ψ
(
x
,
t
)
dx
.
(8.2)
x
,
t
;
x
G
(
,
t
)
is the resolvent of the Hamiltonian
x
,
t
;
x
x
−
t
−
[
E
−
H
(
x
)]
G
(
,
t
)=
δ
(
x
)
δ
(
t
)
.
(8.3)
For time-independent Hamiltonians,
G
depends on
t
and
t
only through their
t
−
difference
t
, in which case it is convenient to Fourier transform to the
energy space. Then,
G
τ
=
x
,
x
|
(
x
,
E
)
can be written as the matrix element
G
(
E
)
|
x
of the operator
G
(
E
)
, formally defined as
G
r
,
a
]
−
1
(
E
)=[(
E
±
i
0
)
−
H
,
(8.4)
where
i
0 has to be understood as an infinitesimal imaginary part added to
E
,
giving the retarded or advanced Green function, respectively. Inserting the identity
resolution,
±
∑
k
|
ψ
k
ψ
k
|
=
1, in terms of the eigenstates
ψ
k
of
H
,wegetthe
spectral
representation
of the Green function
|
ψ
k
ψ
k
|
E
G
r
,
a
)=
k
(
E
i
0
.
(8.5)
−
ε
k
±
, the Green
function of the new Hamiltonian (suppressing, for abbreviation, the indexes labeling
the advanced or retarded function)
If an interaction term is added to the Hamiltonian,
H
→
H
+
V
