Biomedical Engineering Reference
In-Depth Information
where H sys represents the central system and also contains the energy shift operated
by a capacitively coupled gate electrode at the potential V g . The Hamiltonian H sys
is invariant with respect to a set of point symmetry operations that defines the
symmetry group of the device. This fact ensures the existence of degenerate states.
To fix the ideas, the system Hamiltonian describing the triple dot and the benzene
molecule considered later in this chapter (see Figs. 7.3 and 7.9 )isofthePariser-
Parr-Pople form [ 44 - 46 ]:
n i
n i
d i σ
1
2
1
2
d i σ
d i +
= ξ i σ
i σ
i
d i σ +
d i + +
+
H sys
b
d i σ
U
i n i + n i 1 n i + 1 + n i + 1 1 ,
+
V
(7.3)
where d i σ
creates an electron of spin
σ
in the p z orbital of site i or in the ground
state of the quantum dot i and i
=
1
,...,
6
(
3
)
runs over the six carbon atoms (three
d i σ
quantum dots) of the system. Moreover, n i σ =
d i σ
counts the number of electrons
of spin
on site i . The effect of the gate is included as a renormalization of the on-
site energy
σ
eV g with V g being the gate voltage. The parameters U and V
describe the Coulomb interaction between electrons, respectively, on the same and
on neighboring sites.
We leave a detailed analysis of the many-body spectrum of ( 7.3 ) to the Sects. 7.4
and 7.5 . Here we just mention that, for these planar structures belonging to the D n
group, the (non-accidental) orbital degeneracy is at maximum twofold and can be
resolved using the eigenvalues
ξ = ξ 0
of the projection of the angular momentum along
the principal axis of rotation. A generic eigenstate is then represented by the ket
|
is the spin, and E the
energy of the state. The size of the Fock space can make the exact diagonalization
of H sys a numerical challenge in its own. We will not treat here this problem and
concentrate instead on the transport characteristics. H leads describes two reservoirs
of non-interacting electrons with a difference eV b between their electrochemical
potentials. Finally, H tun accounts for the weak tunnelling coupling between the
system and the leads, characteristic of SETs:
N
σ
E
where N is the number of electrons on the system,
σ
t ik c
H tun = χ ki σ
d i σ +
h
.
c
.,
(7.4)
χ k σ
where c
χ
creates an electron with spin
σ
and momentum k in lead
χ =
L
,
R and
k
σ
t ik
is the bare tunnelling amplitude of a k electron in the lead
χ
to the site i .We
t ik |
|
assume it for simplicity independent of the spin
σ
. Naturally,
is highest for
the atom (quantum dot) closest to the lead
, due to the exponential decay on the
atomic scale of the tunnelling probability with the distance between the system and
the lead. Moreover, in the case of atomically localized coupling where the tunnelling
from the lead is most probable only to a small part of the system it is also reasonable
to assume a very weak k -dependence of the tunnelling amplitude. We will simply
neglect it in Sect. 7.3 when discussing the general criteria for the identification of
blocking states.
χ
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