04_hopping


Marco and Villagonzalo

10

Hopping in a Molecular Wire

R. Marco*, R. Banzon, and C. Villagonzalo
Structure and Dynamics Group,

National Institute of Physics, University of the Philippines,
Diliman, Quezon City 1101

E-mail: rmarcojr@nip.upd.edu.ph

Science Diliman (July-December 2004 ) 16:2, 10–11

*Corresponding author

We utilize the Hubbard model approach in analyzing
the effect of the hopping parameter in the transport
properties of a two-atom molecular device (Kostyrko
& Bulka, 2003). The system consists of a molecule
with two identical atoms connecting two metal
electrodes. In this model, two important and realistic
energy parameters are considered which affects the
transport properties. The first is the kinetic energy
required to “hop” from one atomic site to the next. We
call this parameter the hopping term denoted by t, in
arbitrary units of energy. The next is an on-site
Coulombic repulsion, U, between two electrons on the
same atomic site.

The molecular wire is represented by the Hubbard
Hamiltonian

          . (1)

We then used the nonequilibrium Green’s function
formalism (Datta, 2000) to obtain the transport
equations for the system. These transport equations are
solved self-consistently. We utilized a root finding
method to implement such calculations. Our results for
the current at t = 1.0 agree with that in the literature
(Kostyrko & Bulka, 2003).

The behavior of the current-voltage characteristics was
shown to depend on the position of the molecular self-
consistent energy levels with respect to the Fermi energy
levels. For U > 0, the molecular levels are pinned
between the Fermi levels of the source and drain
electrode for values of t ≤ 0.5 . The threshold voltage
is fixed for these values of t ( Fig. 1) and continuous  to

increase for values to t < 0.5. For U = 0, the molecular
levels continue to decrease as t decreases. The threshold

Fig. 1. Current-voltage characteristics for U = 4 for different
values of t.

1.5

1

0.5

0
1 2 3 4 5 6

V (t/e)

J
 (

e
ΓΓΓΓ Γ

/2
h

)

0

Fig. 2. Current-voltage characteristics for U = 0 for different
values of t.

2

1

0
1 2 3 4 5 6

V (t/e)

J
 (

e
ΓΓΓΓ Γ

/2
h

)

0

( ) †, , , , ,
,

'mol i i j i j i i
i j i

H E t c c U n nσ σ
σ

δ ↑ ↓= + +∑ ∑ ∑



Hopping in a Molecular Wire

11

voltage reaches its minimum at t = 0.5 (Fig. 2)
corresponding to the point where the antibonding
energy levels at V = 0 are located between the Fermi
level of the source and drain. When both molecular
levels are below the Fermi level of the drain, transport
does not occur.

REFERENCES

Kostyrko, T. & B. Bulka, 2003. Hubbard model approach for
the transport properties of short molecular  chains. Phys.
Rev. B. 67: 205331.

Datta, S., 2000. Nanoscale device modeling: The Green’s
function method. Superlattices and Microstructures. 28: 4.