Biomedical Engineering Reference
In-Depth Information
−V
]
−
1
)
−
1
V
]
−
1
)
−
1
V
)
−
1
G
G
=[(
E
±
i
0
)
−
H
=[
1
−
(
E
±
i
0
−
H
(
E
±
i
0
−
H
=(
1
−
G
,
(8.6)
is related to the old one by the Dyson equation
G
=
G
+
GV
G.
(8.7)
Usually
will be the interaction that couples the QD array to the terminals.
Therefore,
G
and
V
will represent the Green functions of the isolated and connected
device, respectively.
Let us consider in the following the electron transport through a quantum dot
array under steady-state conditions only. The application of a bias voltage between
the leads gives rise to a non-equilibrium situation, with the electrons of both
leads being at different chemical potentials
G
μ
R
. A great variety of effects
can occur as the electrons passes throughout the system. They are subjected to
interactions with the substrate, losing coherence; with phonons, producing heating
and dissipation; and with each other, changing the occupation within the accessible
states, even precluding their occupation (Coulomb blockade effect). We shall focus
here on the regime of coherent transport where the relevant physics of interference-
based devices works.
The current through the device can be calculated with the Landauer equation
μ
L
and
dE T
2
e
h
=
(
)[
(
)
−
(
)]
,
I
E
f
L
E
f
R
E
(8.8)
where
f
L
and
f
R
are the Fermi distributions at the L and R leads [
1
]. At low
temperatures, the transmission function represents the dimensionless conductance
(in units of the quantum
e
2
/
2
h
) and is calculated as
L
G
r
R
G
a
T
(
E
)=
4Tr
(
(
E
)
(
E
))
,
(8.9)
where
G
a
and
G
r
are the matrix representation of the advanced and retarded Green
functions of the system isolated from the terminals, and
R
are proportional
to the spectral densities of the leads. The Green functions of the system connected
to the leads can be determined by using Dyson equation
L
and
a
r
†
i
0
+
)
a
L
a
R
]
−
1
G
(
E
)=[
G
(
E
)]
=[(
E
−
1
−
H
−
†
−
†
,
(8.10)
where
H
is the matrix representation of the Hamiltonian of the central region
connected to the terminals, and
†
L
and
†
R
are the self-energies, with
=
L
,
R
Im
R
.
In theoretical investigations, typically, two different approaches can be followed.
In first-principles calculations, the confinement potential is modeled as accurately
as possible, and the various effects are included into the Hamiltonian from funda-
mental interactions, like the electron-electron Coulomb potential. In contrast, the
†
L
,
