Biomedical Engineering Reference
In-Depth Information
X
k
2
L
ð
R
Þ
2
m
G
L
ð
R
Þ
ðoÞ¼
2
p
j
V
km
j
dðo e
k
Þ
½
2
:
17
where H
L
, H
R
are the Hamiltonians for the left (top) and right (bottom)
leads, H
QD
is the Hamiltonian for the quantum dot and H
T
the dot-lead
interaction. Within the quantum dot,
ε
m
σ
represents the orbital energy levels
and
is the level broadening due to coupling to the leads.
This model exhibits the required even-odd filling. In the case of an odd
total number of electrons, the last excess electron occupies the next higher
orbital level, which is the lowest unoccupied level, yielding a net spin of 1/2
for the quantum dot as desired. In this situation of an excess odd electron
with spin 1/2, the Anderson Hamiltonian gives rise to different scenarios
depending on the ratio of the energy of last occupied orbital measured, from
the chemical potential of the leads, |
G
ε
o
|, to the level broadening due to
coupling to the leads,
. When this ratio lies between 0 and 0.5,
charge fluctuates readily on and off of the quantum dot and the system is in
the mixed valence regime. For a ratio in excess of 0.5, charge is essentially
quantized on the dot and Kondo physics becomes relevant. Under this
condition, the Anderson Hamiltonian can be mapped into the Kondo
Hamiltonian, familiar in the context of a magnetic impurity imbedded in a
host metal:
, i.e. |
ε
o
|/
G
G
X
i
;
j
;
s
t
ij
c
is
c
js
þ
J
X
i
s
ci
H
¼
s
fi
T
K
!
exp
ð
1
=r
J
Þ
½
2
:
18
The corresponding Kondo energy scale in the quantum dot case is given by
E
K
=k
B
T
K
:
exp
p½e
0
j
jð
U
þ e
0
Þ
1
=
2
T
K
¼ð
U
GÞ
½
2
:
19
G
U
Going from the individual quantum dot to the coupled DQD case involves
an additional dot-dot tunnel-coupling term:
t
X
s
ð
d
L
s
d
R
s
þ
h
:
c
:Þ
H
dot-dot
¼
½
2
:
20
Here we are taking the simplest case of full symmetry between the two (left/
right or top/bottom) dots, in their electronic energy level structure and level
splitting
, for
respective couplings to the left (top) and right (bottom) leads. Only the
coupling term between the excess, last odd electron on each dot needs be
included as the relevant coupling term. The inter-dot tunnel coupling, t,
gives rise to an effective antiferromagnetic coupling, J=4t
2
/U, between the
two excess spins. According to theoretical analysis, the following scenarios
arise as t/
Δ
E, on site Coulomb repulsion U and level broadening
G
and J/T
K
are varied.
Assuming U to be the largest energy scale as is usually the case in
G
