Quantum Interference Effects on the Electronic Transmission Through Quantum Dot Molecules - Quantum Dot Molecules - page 264

Therefore, it suffices to know the pole structure of the Green functions of
the disconnected device to predict its transport properties when connected to the
terminals. At the energies of the poles of G lr there exists peaks of finite (or even
perfect) transmission, while at the energies at which G ll or G rr do have poles (but
G lr do not), the electron transmission is forbidden.

8.2.7 Transmission Pathways

Spatially discretized models with hopping terms, allowing transitions from a site to
its neighbor, like the present one, suggest the appealing interpretation of an electron
successively jumping along a path. This interpretation allows one to recover the
image of electrons moving along trajectories through the device. A similar approach
has been used in an analytical treatment of quantum interference in a benzene
ring [ 48 ].
Partitioning of a Hilbert space is usually employed for isolating the effects on
the part of interest from the rest of the system [ 49 ]. Here, we apply a partitioning
technique to define spatial transmission pathways. The 4

× 4 Hamiltonian can be

partitioned in terms of 2 × 2 matrices as follows:

H P

H Q

U

= ,

H (8.21)

U †

where H P

PHP is the part of the Hamiltonian projected on the subspace of orbitals
centered on the sites of connection 1 and n ,where P

=

, whilst H Q is = | 1 1 | + | n n |

P . Matrices U and U †
contain matrix elements connecting states belonging to P and Q . The Green function
can be obtained by block matrix decomposition

the projection of H on the complementary subspace Q = 1 −

E

H Q − 1

H P

− − U

) − 1

g =( E − H = . (8.22)

U †

− E −

We are interested here in the Green function projected on the subspace of the
connection sites, i.e., its P -block. Hence, g P can be obtained from the inverse of

H Q ,

the Schur complement of the Q -block, E −

g P H P H Q ) − 1 U † ] − 1 H P Ug Q U † ) − 1 =[ E − − U ( E − =( E − − , (8.23)

from which an effective Hamiltonian can be defined as

g P ) − 1 H P Ug Q U † H P H eff = E − ( = + = + Σ , (8.24)