Antiresonance Condition

Antiresonance Condition

The above section outlined a method for numerically evaluating the transmission probability for a MW system. We now focus on an interesting physical feature of the transmission probability, namely the occurrence of antiresonances. These are characterized by the perfect reflection of electrons incident on the molecule at a certain energy E. The basis of our analysis in this section will be the Lippmann-Schwinger (LS) equation applied to the scattering problem in a highly idealized model. It will lead to an analytic formula for T(E).

Before we proceed further we wish to highlight a problem that arises in most quantum chemistry applications. In studying quantum systems, such as MW, which are composed out of atomic building blocks, it is customary to solve the problem by expressing the electron's wavefunction in terms of the atomic orbitals on the various atoms. A problem that arises is that these orbitals are not usually orthogonal to each other. Including this nonorthogonality complicates the solution and so it is often neglected. In the analysis that follows we have utilized a transformation which removes the nonorthogonality and allows for a straightforward solution[19]. This transformation leads to an energy dependent Hamiltonian which will play an important role in determining the presence of antiresonances.

The LS equation we consider has the form

$\begin{displaymath}\vert\Psi ' \rangle = \vert\Phi ' \rangle + G'(E) W^E \vert\Psi ' \rangle . \end{displaymath}$

(6)

Here $\vert\Psi'\rangle$ is the scattered electron wavefunction of the transformed Hamiltonian H^E. W^E couples the molecule to the adjacent lead sites. $\vert\Phi' \rangle$ is the initial electron state which is a propagating Bloch wave that is confined to the left lead. G'(E)=(E - H₀^E)^-1 is the Green's function of the decoupled system.

We consider the LS equation in the tight binding approximation and solve for $\vert\Psi'\rangle$ . This gives us the value for $\Psi_1$ which is the value of the wavefunction on the first atomic site on the right lead. The transmission probability, T is simply $\vert\Psi_1\vert^2$ . The result is

$\begin{displaymath}\Psi_{1} = \frac{P \Phi'_{-1}}{[(1-Q)(1-R)-PS]} \end{displaymath}$

(7)

where

$\begin{eqnarray*}P &=& G'_{1,1} \sum_j W^E_{1,j} G'_j W^E_{j,-1} \\ Q &=& G'_{1... ...1,j})^2 G'_j \\ S &=& G'_{1,1} \sum_j W^E_{-1,j} G'_j W^E_{j,1} \end{eqnarray*}$

The sum over j is over only the MO's. In the above, W^E_1,j = H_1,j - E S_1,j is the energy-dependent hopping element of H^E between the first lead site and the j^th MO in terms of the hopping element of the original Hamiltonian H and the overlap S in the non-orthogonal basis. The Green's function on the molecule is expanded in terms of its molecular eigenstates and this gives $G'_j = 1/(E-\epsilon_j)$ for the j^th MO with energy $\epsilon_j$ . G'_1,1 is the diagonal matrix element of the Green's function G'(E) at the end site of the isolated lead.

Antiresonances of the MW occur where the transmission T is equal to zero. These occur at Fermi energies E that are the roots of Eq. (7), namely

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